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 |
|---|---|---|---|---|
3,491,028 | <p>Problem:<br>
Suppose that <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> are independent uniformly distributed on the interval <span class="math-container">$[1,3]$</span>. What is the probability that
<span class="math-container">$x_... | V.J. | 736,680 | <p>As @Leander point out, you missed in the limits that the information of <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> must lie on <span class="math-container">$[1,3]$</span>. Thus your correct integral should be</p>
<p><span class=... |
823,055 | <p>This may be a naive question. I am reading the definition of differetiablity of a function $f:\mathbb{R^n}\rightarrow \mathbb{R^m}$ in the book Calculus Manifolds. I already know that all norms on $\mathbb{R}^n$ induce the same metric topology. If we change the norms in the definition (for example we can use the man... | Ted Shifrin | 71,348 | <p>Unless you mean a Möbius band that's modeled on an open interval $(-1,1)$ rather than the usual closed interval $[-1,1]$, this will be (in the smooth world) a manifold with corners, not a manifold with boundary; in the topological world, no problem. </p>
<p>In general, $\partial(M\times N) = \partial M\times N \cup... |
3,068,197 | <p>I'm working through this problem and I haven't been able to make any progress. The textbook provides the answer of <span class="math-container">$ {9 \choose 4}$</span> but I'm not sure as to how they got this result. </p>
| Yuval Filmus | 1,277 | <p>Here is a more sophisticated way to solve this, using DFAs. We can construct a state machine accepting the language of all strings without two consecutive zeroes as follows:</p>
<ul>
<li>There are two states, <span class="math-container">$q_0$</span> and <span class="math-container">$q_1$</span>.</li>
<li>The initi... |
965,851 | <p>I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to simplify it?</p>
<p>$$\sin^2\beta ⋅ d^4 + c^2\left(\cos^2\beta⋅\cos^2\alpha-\frac{\cos^2\beta}{2}-\frac12\right)d^2 + ... | egreg | 62,967 | <p>The equation can be written $x\log2=2\log|x|$. Let's consider the function
$$
f(x)=x\log2-2\log|x|
$$
defined for $x\ne0$.
We have easily
$$
\lim_{x\to-\infty}f(x)=-\infty,
\qquad
\lim_{x\to\infty}f(x)=\infty
$$
and
$$
\lim_{x\to0}f(x)=\infty.
$$
Moreover
$$
f'(x)=\log2-\frac{2}{x}=\frac{x\log2-2}{x}
$$
Set $\alpha=... |
1,522,929 | <p>For every fixed $t\ge 0$ I need to prove that the sequence $\big\{n\big(t^{\frac{1}{n}}-1\big) \big\}_{n\in \Bbb N}$ is non-increasing, i.e.
$$n\big(t^{\frac{1}{n}}-1\big)\ge (n+1)\big(t^{\frac{1}{n+1}}-1\big)\;\ \forall n\in \Bbb N$$
I'm trying by induction over $n$, but got stuck in the proof for $n+1$:
<br/>
For ... | TravorLZH | 748,964 | <p>This answer is dedicated to give an alternative proof of</p>
<p><span class="math-container">$$
\lim_{R\to+\infty}\int_\Gamma e^{iz^2}\mathrm dz=0
$$</span></p>
<p>where <span class="math-container">$\Gamma$</span> denotes a circular arc connecting <span class="math-container">$R$</span> and <span class="math-contai... |
2,691,266 | <p>The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. </p>
<p>I am trying to show $\mathcal{O}/\mathfrak{p}^n$ is principal ring. Let $\mathfrak{p}^i/\mathfrak{p}^n$ be an ideal and choose $\pi\in\mathfrak{p}\setminus\mathfrak{p}^2$. Then how ... | Notone | 408,724 | <p>You have
$(\pi)^i \not\subset \mathfrak{p}^{i+1}$, but $(\pi)^i\subset \mathfrak{p}^i$. </p>
<p>Thus $(\pi)^i=J\cdot \mathfrak{p}^i$ for an ideal $J$ with $\mathfrak{p}\nmid J$ . </p>
<p>Now we have $(\pi)^i+\mathfrak{p}^{n}=\mathfrak{p}^{i}(J+\mathfrak{p}^{n-i})$. But $J$ and $\mathfrak{p}^{n-i}$ have no common... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | Tom Collinge | 98,230 | <p>A vector space is a generalisation of the 3D space we live in and the behaviour of vector addition and scalar multiplication is modelled on that.</p>
<p>Another consideration is that mathematical structures are generally <em>closed</em> - groups, rings, fields, vector spaces are all closed. This means that the oper... |
825,703 | <p>I have been working with vector spaces for a while and I now take for granted what the vector space does. I feel like I dont really understand why multiplication and addition must be defined on a vector space. For example, it feels like adding two vectors and having their sum contained within the space is just a nam... | Lee Mosher | 26,501 | <p>To see why this property is important, consider an example where the property fails. Let $X$ be the union of the $x$ and $y$ axes in the Cartesian plane, so
$$X = \{(x,0) \, | \, x \in \mathbb{R}\} \cup \{(0,y) \, | \, y \in \mathbb{R}\}
$$
Is $X$ a vector space? If not, what axiom fails? And (perhaps of more impor... |
3,027,286 | <p>I am a little confused as to proving that <span class="math-container">$(C^*)^{-1} = (C^{-1})^*$</span> where <span class="math-container">$C$</span> is an invertible matrix which is complex. </p>
<p>Initially, I thought that it would have something to do with the identity matrix where <span class="math-container">... | Acccumulation | 476,070 | <p>Another counterexample can be derived from the <a href="https://en.wikipedia.org/wiki/St._Petersburg_paradox" rel="nofollow noreferrer">St. Petersburg paradox</a>. Suppose that <span class="math-container">$p_j=\frac1j$</span> if <span class="math-container">$j$</span> is a power of <span class="math-container">$2$<... |
2,245,631 | <blockquote>
<p>$x+x\sqrt{(2x+2)}=3$</p>
</blockquote>
<p>I must solve this, but I always get to a point where I don't know what to do. The answer is 1.</p>
<p>Here is what I did: </p>
<p>$$\begin{align}
3&=x(1+\sqrt{2(x+1)}) \\
\frac{3}{x}&=1+\sqrt{2(x+1)} \\
\frac{3}{x}-1&=\sqrt{2(x+1)} \\
\frac{(3-x... | fleablood | 280,126 | <p>I wouldn't have gotten there is such a convoluted way (and I wouldn't have divided by $x$ without checking that $x \ne 0$ first) but that would really make a difference.</p>
<p>And I'd watch out for extraneous roots.</p>
<p>$x +x\sqrt{2x+2} = 3$</p>
<p>$x\sqrt{2x+2} = 3-x$ $2x+2 > 0$ so $x > -1$</p>
<p>$x... |
4,271,909 | <p>I'm struggling with improper integrals (Calc I). I've calculated the following:</p>
<p>If <span class="math-container">$a = 0$</span>:
<span class="math-container">$$\int_{0}^{\infty}\cos(x)dx =\lim\limits_{R\to\infty} \int_{0}^{R}\cos(x)dx =\lim\limits_{R\to\infty} \sin(R) $$</span>
Which diverges?</p>
<p>If <span ... | Giorgos Giapitzakis | 907,711 | <p>For <span class="math-container">$a > 0$</span> the integral evaluates to <span class="math-container">$\frac{a}{a^2+1}$</span> (you forgot a square in the denominator of the first fraction). Now set <span class="math-container">$g(a) = \frac{a}{a^2+1}$</span>, <span class="math-container">$a \in (0, \infty)$</sp... |
4,271,909 | <p>I'm struggling with improper integrals (Calc I). I've calculated the following:</p>
<p>If <span class="math-container">$a = 0$</span>:
<span class="math-container">$$\int_{0}^{\infty}\cos(x)dx =\lim\limits_{R\to\infty} \int_{0}^{R}\cos(x)dx =\lim\limits_{R\to\infty} \sin(R) $$</span>
Which diverges?</p>
<p>If <span ... | Robert Lee | 695,196 | <p>You can actually continue your method to get the correct solution! You want to evaluate the limit
<span class="math-container">$$
\lim_{R\to\infty} \left[\frac{a}{a^{\color{red}{2}}+1} + \left(\frac{\sin(R)-a\cos(R)}{a^{2}+1}\right)e^{-aR}\right]
$$</span></p>
<p>By the <a href="https://mathworld.wolfram.com/Harmoni... |
4,271,909 | <p>I'm struggling with improper integrals (Calc I). I've calculated the following:</p>
<p>If <span class="math-container">$a = 0$</span>:
<span class="math-container">$$\int_{0}^{\infty}\cos(x)dx =\lim\limits_{R\to\infty} \int_{0}^{R}\cos(x)dx =\lim\limits_{R\to\infty} \sin(R) $$</span>
Which diverges?</p>
<p>If <span ... | Claude Leibovici | 82,404 | <p><span class="math-container">$$I=\int_{0}^{\infty}e^{-ax}\cos(x)dx=\Re \int_{0}^{\infty}e^{-ax}e^{ix}dx=\Re \int_{0}^{\infty}e^{-(a-i)x}dx=\Re\left(\frac 1{a-i}\right)$$</span>
<span class="math-container">$$I=\Re\left(\frac{a}{a^2+1}+\frac{i}{a^2+1}\right)=\frac{a}{a^2+1}$$</span></p>
|
1,685,895 | <blockquote>
<blockquote>
<p>Question: Find a value of $n$ such that the coefficients of $x^7$ and $x^8$ are in the expansion of $\displaystyle \left(2+\frac{x}{3}\right)^{n}$ are equal.</p>
</blockquote>
</blockquote>
<hr>
<p>My attempt:</p>
<p>$\displaystyle \binom{n}{7}=\binom{n}{8} $</p>
<p>$$ n(n-1)(n-... | Community | -1 | <p>$\displaystyle \left(2+\frac{x}{3}\right)^{n}=\sum_{i=0} ^n {n \choose k}({\frac{x}{3}})^i 2^{n-i}$<br>
Now for $i =7\implies{n \choose 7}({\frac{x}{3}})^7 2^{n-7}$ </p>
<p>and for $i =8\implies{n \choose 8}({\frac{x}{3}})^8 2^{n-8}$ </p>
<p>now to get the coefficients of $x^8$ & $x^7$ equal. $\implies$ </... |
2,554,153 | <p>I have some problem with writing character table of a group. For instance, a group $S_4$. When we write character table, we write irreducible representations of group. So, how can I quickly find them? Then how to Fill the table? Can someone explain me upon this example?</p>
| Somos | 438,089 | <p>You want to find the character table of a small finite group. The quickest way is to look them up in a book or <a href="http://brauer.maths.qmul.ac.uk/Atlas/v3/" rel="nofollow noreferrer">website</a>, or use a program like <a href="https://www.gap-system.org/" rel="nofollow noreferrer">GAP</a>. Failing that, you can... |
2,264,614 | <p>Is there a way to evaluate, </p>
<p>$$
\large \cos x \cdot \cos \frac{x}{2} \cdot \cos \frac{x}{4} ... \cdot \cos \frac{x}{2^{n-1}} \tag*{(1)}
$$</p>
<p>I asked this to one of my teachers and what he told is something like this, </p>
<p>Multiply and divide the last term of $(1)$ with $\boxed{\sin \frac{x}{2^{n-... | Jack D'Aurizio | 44,121 | <p>By the sine duplication formula $\sin(2z)=2\sin(z)\cos(z)$ we have $\cos(z)=\frac{1}{2}\cdot\frac{\sin(2z)}{\sin(z)}$.<br>
In particular
$$ \prod_{k=0}^{n-1}\cos\left(\frac{x}{2^k}\right)=\frac{1}{2^n}\prod_{k=0}^{n-1}\frac{\sin\frac{x}{2^{k-1}}}{\sin\frac{x}{2^k}}=\frac{\sin(2x)}{2^n\sin\frac{x}{2^{n-1}}} \tag{1} $... |
108,060 | <p>Suppose:
$$\sum_{n=2}^{\infty} \left( \frac{1}{n(\ln(n))^{k}} \right) =\frac{1}{ 2(\ln(2))^{k} } +\frac{1}{ 3(\ln(3))^{k} }+...,
$$
by which $k$ does it converge?</p>
<p>When I use comparison test I get inconclusive result:</p>
<p>$\lim_{n\rightarrow\infty} \frac{u_{n+1}}{u_{n}}=\frac{n\ln(n)^{k}}{(n+1)\ln(n+1)^{... | Selim Ghazouani | 24,713 | <p>You can use integral/series comparison.</p>
|
327,860 | <p>Let <span class="math-container">$A$</span> be a symmetric <span class="math-container">$d\times d$</span> matrix with integer entries such that the quadratic form <span class="math-container">$Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$</span>, is non-negative definite. For which <span class="math-container">$d$</... | WKC | 29,241 | <p>I should add that there have been many results on this problem since the 1990s. The existence of the number <span class="math-container">$g(n)$</span> was established by Maria Icaza in her Ohio State 1992 thesis. You should check out her papers. A bit later Myung Kwan Kim and Byeong Kweon Oh published a few paper... |
2,778,422 | <p>The generalized $\lambda-\text{eigenspace}$ is defined by:
$V^f_{(\lambda)}=\bigl\lbrace v\in V\mid\exists j\,\text{ such that }\,(f-\lambda)^jv=0 \bigr\rbrace$.
Suppose that $V$ is a vector space over the field $k$ and $f,g\in \operatorname{End}_k(V)$ satisfy $f\circ g=g\circ f$. Show that $g(V^f_{(\lambda)})\subse... | mechanodroid | 144,766 | <p>For $x \in V^f_{(\lambda)}$ take $i \in \mathbb{N}$ such that $(f-\lambda I)^ix = 0$. We have</p>
<p>$$(f-\lambda I)^i(g(x)) = g((f-\lambda I)^ix) = g(0) = 0$$</p>
<p>because $g$ commutes with $f$. Therefore $g(x) \in V^f_{(\lambda)}$. </p>
|
1,423,252 | <p>The proposition is:</p>
<blockquote>
<p>If $\lim S_n = L$ and for every $n$, $S_n$ is in the interval $[a,b]$, then $L$ is also in $[a,b]$.</p>
</blockquote>
<p>I have proved this effectively, but now the question is to provide a counterexample to the stronger assumption, for the interval $(a,b)$. </p>
<p>Basi... | Dontknowanything | 240,480 | <p>What about $1/(n+1)$ in $(0,1)$?</p>
|
3,096,115 | <p>By using a sieve created by Prime Number Tables set up by the formula PN+(PNx6) for numbers generated by 6n+or-1, takes 182 calculations to identify 170 composite numbers. Using the Sieve of Eratosthenes would take around 1600 calculations. The Prime Number Tables identify all the composite numbers on the the list o... | D Left Adjoint to U | 26,327 | <p>What you are looking for is a computational function that returns the list of primes less than 1000, and works as efficiently as possible, on just that problem.</p>
<p>There are <a href="https://www.wolframalpha.com/input/?i=pi(1000)" rel="nofollow noreferrer">168 primes</a> between 1 and 1000, so any method will t... |
2,041,839 | <p>$$\int_{1}^{x}\frac{dt}{\sqrt{t^3-1}}$$ does this have a closed form involving jacobi elliptic functions of parameter $k$?</p>
<p><strong>N.B</strong> I tried with the change of variables $t=1+k\frac{1-u}{1+u}$. But this leads no where. <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html" rel="nofoll... | B. Goddard | 362,009 | <p>We don't have good intuition about what time-squared means, so maybe you should use length, since length squared is area. If you had a square which was $x$-by-$x$ feet and then increased both sides by $dx$, then the area of your new square is $x^2 +2x \; dx+ (dx)^2$. The change in the area is two skinny $x$-by-$dx... |
617,163 | <p>I need to find a proper definition of a quantile.
It says:
a p-th quantile $x_p$ is a number, that satisfies the following conditions:
$$
0<p<1
$$
and
$$
P(X \le x_{p}) \ge p
$$
and
$$
P(X \ge x_{p}) \ge 1-p
$$
is this definition right?</p>
| Eric Stucky | 31,888 | <p>The total amount of money he brought in was the selling price of $250$ chairs. But $50$ of those chairs were pure gain, so when <em>he</em> bought the chairs, he only spent the equivalent of the selling price of $200$ chairs.</p>
<p>The profit percent is the amount of gain divided by the amount he spent. This is di... |
1,038,198 | <p>How do you prove that
$8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?</p>
| Petite Etincelle | 100,564 | <p>Multiply both sides by $\sin x$, then</p>
<p>$$2\cos x \sin x = \sin 2x$$</p>
<p>$$2\cos 2x \sin 2x = \sin 4x$$</p>
<p>$$2\sin 4x \cos 3x = \sin 7x + \sin x$$</p>
|
2,275,604 | <p>We can break up the circle into an infinite amount of rings with perimeter $2\pi r$. For a given circle $r$, the outside ring has perimeter of $2\pi r$ and the smallest one has of course perimeter $0$. We can add up all the area of the infinite rings using the arithmetic series concept; we can get the average of the... | Paramanand Singh | 72,031 | <p>You started off with the right idea. Area of any plane region is found by splitting into smaller regions and adding their areas. The assumption here is that the smaller regions can be chosen with familiar shapes like squares, rectangles or even triangles whose areas can be calculated using well known formulas. Natur... |
316,965 | <p>How do I interpret following types of matrices as special types of transformations?
I mean what are the transformative properties of following types of matrices, from $\mathbb{R}^n $ to $ \mathbb{R}^n$, or $\mathbb{C^n}$ to $\mathbb{C^n}$?</p>
<p><strong>Normal and Anti Hermitian Matrices</strong>?</p>
<p><strong>... | Branimir Ćaćić | 49,610 | <p>Let $V$ be a finite-dimensional complex inner product space. </p>
<ol>
<li><p>By the finite-dimensional spectral theorem, <a href="http://en.wikipedia.org/wiki/Normal_matrix" rel="noreferrer">one has that</a> $T \in L(V) := L(V,V)$ is normal if and only if there exists an orthonormal basis for $V$ consisting of eig... |
1,936,260 | <p>We have a binary sequence of 1s and 0s, and the length is 10.
I wonder how many binary sequence of length 10 with four 1's can be created such that the 1's do not appear consecutively?</p>
| carmichael561 | 314,708 | <p>Since $\log_{10}(xy)=\log_{10}(x)+\log_{10}(y)$, it's enough to prove that $\lfloor a+b\rfloor \geq \lfloor a\rfloor +\lfloor b\rfloor$ for all real numbers $a,b$.</p>
|
4,037,697 | <p><span class="math-container">$a_n $</span> is a sequence defined this way:
<a href="https://i.stack.imgur.com/nHdiD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nHdiD.png" alt="enter image description here" /></a></p>
<p>and we define: <a href="https://i.stack.imgur.com/b01hX.png" rel="nofollow... | Beyond Infinity | 683,570 | <p>Let <span class="math-container">$\mathbf {x, y} \in \mathbb{R}^n$</span> be unit vectors.<br />
Then, the angle <span class="math-container">$\theta_1$</span> between them is given by <span class="math-container">$ \cos \theta_1 = \mathbf{x^\top y}$</span>.</p>
<p>Also, angle between <span class="math-container">$\... |
2,525,573 | <p>Find the domain and range of $y=\sqrt {x-2}$</p>
<p>My Attempt:
$$y=\sqrt {x-2}$$
For $y$ to be defined,
$$(x-2)\geq 0$$
$$x\geq 2$$
So $dom(f)=[2,\infty)$.</p>
| randomgirl | 209,647 | <p>Write $5e^{2x}$ as $e^{ \ln(5)}e^{2x}=e^{\ln(5)+2x}$</p>
|
2,525,573 | <p>Find the domain and range of $y=\sqrt {x-2}$</p>
<p>My Attempt:
$$y=\sqrt {x-2}$$
For $y$ to be defined,
$$(x-2)\geq 0$$
$$x\geq 2$$
So $dom(f)=[2,\infty)$.</p>
| Sarvesh Ravichandran Iyer | 316,409 | <blockquote>
<p>$\ln ab = \ln a + \ln b$, $\ln b^a = a \ln b$.</p>
</blockquote>
<p>You seem to have confused $\ln 5 \times e^{2x}$ with $\ln (5e)^{2x}$. The first would simplify according to the first rule, as $\ln 5 + 2x$, while the second would simplify as $2x \ln (5e)$. You should have obtained the first express... |
121,403 | <blockquote>
<p>A manifold $M$ of dimension n is a topological space with the following properties:<br>
a) $M$ is Hausdorff<br>
b)$M$ is locally Euclidean of dimension n<br>
c) $M$ has a countable basis of open sets. </p>
</blockquote>
<p>Why is the first property necessary? I do not have much experience with ... | Samuel Reid | 19,723 | <p>Let us first review our topological notions to ensure there isn't any confusion there.</p>
<p>A topological space is a set $X$ together with a collection $U$ of subsets of $X$, called a topology on $X$, so that the following hold,</p>
<ul>
<li>The empty set $\varnothing$ and the set $X$ are both open.</li>
<li>$U$... |
1,742,768 | <p>How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?</p>
| Clement C. | 75,808 | <p>You have, from Taylor expansion of $e^u$ around $0$ to second order,
$$\begin{align}
\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2} &=
e^{\frac{1}{n}\ln a} - \frac{1}{2}\left(e^{\frac{1}{n}\ln c}+e^{\frac{1}{n}\ln b}\right)\\ &=
1+\frac{1}{n}\ln a + \frac{1}{2n^2}\ln ^2 a \\&\qquad- \frac{1}{2}\left(2+... |
2,516,023 | <blockquote>
<p>Why does taking logarithms on both sides of $0<r<s$ reverse the inequality for logarithms with base $a$, $0<a<1$?</p>
</blockquote>
<p>I would like some intuition on why this works. I tried graphing $\log_{0.5}(x)$ on Desmos, for example, and if the graph were true this would be evident f... | Ross Millikan | 1,827 | <p>Because logs with bases less than $1$ are decreasing, not increasing functions. As an example, we have $8 \lt 16$, but $\log_{0.5} 8=-3 \gt \log_{0.5}16=-4$. It is like multiplying by a negative number, which is a decreasing function and reverses the inequality.</p>
|
148,313 | <p>Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost.</p>
<p>His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( [hhv]^*-[h^*h^*v^*]^*)$ where h is (sinh(x),cosh(x),0,0). He derived this odd transform by substituting the hyperbolic sine and cosi... | Jeremy Stein | 31,954 | <p>Not mathematically rigorous, but perhaps more practical:</p>
<p>Let's assume we can narrow down the numbers to one of ten values. Perhaps you're willing to round your salaries to the nearest 5K and you think they're between 50K and 95K, or you were both willing to announce that you had less than 10 speeding ticket... |
1,979,226 | <p>Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.
Y1, Y2, Y3 form a partition of S.</p>
<p>P(X | Y1) = .8, P(X | Y2) = .1, P(X | Y3) = .9, P(Y1) = .1, P(Y2) = .4. </p>
<p>Find P(Y1 | X).</p>
<p>P(Y1 | X) =</p>
<p>For this one I thought that all... | mercio | 17,445 | <p>Since every $u_n$ is positive, we have $u_n \ge \sqrt {u_0}$ for $n \ge 1$, and then $u_n \ge \sqrt {(n-1)\sqrt {u_0}}$ for $n \ge 2$, which shows that $(u_n)$ diverges.</p>
|
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | PDE | 300,754 | <p>My attempt to phrase Brian Drake's answer even more simply:</p>
<p>The original problem can be phrased geometrically as, "What is the smallest angle <span class="math-container">$\theta$</span> by which we can rotate the standard parabola <span class="math-container">$P$</span> around the origin in order to mak... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | Narasimham | 95,860 | <p>A rigid parabolic arc <em>remains a parabola</em> <strong>for any arbitrary displacement or rotation</strong> imparted to the parabolic arc.</p>
<p>That is why we can describe any parabola in the plane by an intrinsic or natural equation</p>
<p><span class="math-container">$$ \kappa = \dfrac{\cos^3\phi}{2f} $$</span... |
4,492,566 | <blockquote>
<p>To which degree must I rotate a parabola for it to be no longer the graph of a function?</p>
</blockquote>
<p>I have no problem with narrowing the question down by only concerning the standard parabola: <span class="math-container">$$f(x)=x^2.$$</span></p>
<p>I am looking for a specific angle measure. O... | Blue | 409 | <p>A little more generally ... Consider a <em>non-degenerate</em> (and, for the sake of this discussion, non-circular) conic with eccentricity <span class="math-container">$e>0$</span>, focus <span class="math-container">$(f_x,f_y)$</span>, focus-to-directrix length <span class="math-container">$d>0$</span>, and ... |
2,880,566 | <p>In an optimization problem I finally get to the point where I have to solve</p>
<p>$$x +\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x)) =0$$</p>
<p>which obviously leads to</p>
<p>$$x=-\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x))$$</p>
<p>Nevertheless, this couldn't in any case help knowing the optimal size of the angl... | mfl | 148,513 | <p>We have that $\cos 2x=\cos^2x-\sin^2x=2\cos^2x-1,\sin 2x=2\sin x\cos x.$ Now</p>
<p>$$\tan(x)\cos(2x)+\tan(x)= \tan x(\cos 2x+1)=2\tan x\cos^2x=2\sin x\cos x.$$</p>
<p>Thus</p>
<p>$$\tan(x)\cos(2x)+\tan(x)-2\sin(2x)=-2\sin x\cos x.$$</p>
<p>So</p>
<p>$$\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x))=-2\sin x.$$</p>
... |
48,864 | <p>I can't resist asking this companion question to the <a href="https://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking"> one of Gowers</a>. There, Tim Dokchitser suggested the idea of Grothendieck topologies as a fundamentally new insight. But Gowers' original motivation is to ... | Michael Renardy | 12,120 | <p>Grothendieck seems to be still alive. So should not the question be: Is Grothendieck a computer? (Ask him, good luck!) Or perhaps: How did he morph from a computer to ... whatever it may be?</p>
|
349,309 | <p>I seem to be short on examples for $I$-adic completions of rings.</p>
<p>I know that a ring is $I$-adically complete if the canonical homomorphism into the inverse limit is an isomorphism. My thinking and searching on the internet has been surprisingly fruitless, though, for examples where the map is either surject... | Andreas Caranti | 58,401 | <p>For injective but not surjective, take the ring $R = \Bbb{Z}$ and the ideal $I = \langle p \rangle$, for a prime $p$, to obtain the $p$-adic integers $\Bbb{Z}_{p}$. </p>
<p>For surjective but not injective, I think you might take the ring $R = \Bbb{Z}_{p} \oplus B$, where $B$ is any Boolean ring, and $I = \langle (... |
3,903,774 | <p><span class="math-container">$30$</span> red balls and <span class="math-container">$20$</span> black balls are being distributed to <span class="math-container">$5$</span> kids, so that each kid gets at least one red ball. In how many ways can we distribute balls?</p>
<p>Circle the correct answers:</p>
<p>a) <span ... | PierreCarre | 639,238 | <p>The subscript is relative to the norm being used. The "2" stands for the euclidean norm. The superscript is a power.
<span class="math-container">$$
\|X\|_2^2 = \left(\left(\sum_{i=1}^n X_i^2 \right)^{1/2}\right)^2 = \sum_{i=1}^n X_i^2
$$</span></p>
<p>Other examples of norms could be</p>
<p><span class="m... |
735,015 | <p>I don't have to write a proof I just have to show that $(^{n}_{k})(^k_m)=(^n_m)(^{n-m}_{k-m})$
But I am struggling to expand this.
$$\frac{n!}{k!(n-k)!}*\frac{k!}{m!(k-m)!}$$</p>
<p>Once I get it to this point I am having no luck multiplying it out to make it look like the other side should look</p>
| Thanos Darkadakis | 105,049 | <p>Just use the expansion of $(^{n}_{k})$ and simplify the fractions..</p>
<p>$(^{n}_{k})(^k_m)=(^n_m)(^{n-m}_{k-m})$</p>
<p>$\frac{n!}{k!(n-k)!}\cdot \frac{k!}{m!(k-m)!}=\frac{(n-m)!}{(k-m)!(n-k)!}$</p>
<p>$\frac{n!}{m!}=(n-m)!$, which is true.</p>
|
735,015 | <p>I don't have to write a proof I just have to show that $(^{n}_{k})(^k_m)=(^n_m)(^{n-m}_{k-m})$
But I am struggling to expand this.
$$\frac{n!}{k!(n-k)!}*\frac{k!}{m!(k-m)!}$$</p>
<p>Once I get it to this point I am having no luck multiplying it out to make it look like the other side should look</p>
| Nigel Overmars | 96,700 | <p>LHS: You're forming a committee of $k$ people that can be chosen out of $n$ people, $n \choose k$ combinations. Then, from those $k$ people you choose a board of $m$ people, $k \choose m$ combinations. So in total there are ${n \choose k} {k \choose m}$ combinations.</p>
<p>RHS: Reversed order of the LHS, first yo... |
1,211,978 | <p>I cannot find the roots of the characteristic equation to get a solution. I only know the basic way to solve these equations. I factored out an $r^2$.</p>
<p>$2r^5-7r^4+12r^3-8r^2 = 0$</p>
<p>$r^2(2r^3-7r^2+12r-8) = 0$</p>
| parsiad | 64,601 | <p>Assuming $y^{\left(n\right)}\equiv\frac{\partial^{n}y}{\partial t^{n}}$.
Let $w=y^{\prime\prime}$ so that the equation becomes
$$
2w^{\prime\prime\prime}-7w^{\prime\prime}+12w^{\prime}-8w=0.
$$
The characteristic polynomial, as you pointed out, is
$$
2r^{3}-7r^{2}+12r-8
$$
with imaginary and real roots (solvable in ... |
2,532,280 | <p>If a N×N (N≥3) Hermitian matrix <strong>A</strong> meets the following conditions: </p>
<ol>
<li><strong>A</strong> is positive semi-definite (not positive definite, i.e. <strong>A</strong> has at least M zero eigenvalue, where M is a given paremeter with 1≤M≤N-1).</li>
<li>The sum of each off diagonal results in 0... | Michael Rozenberg | 190,319 | <p>For $$ax^2+bx+c=0,$$ where $a\neq0$ we need
$b=0$ and $\frac{c}{a}<0$.</p>
<p>Thus, $k=-\frac{3}{5}$ and check that $2k^2-1<0$.</p>
|
1,764,106 | <p>In my book I have the following definition for subgroups of a group $G$ generated by $A$, a subset of G:</p>
<p>$$\langle A\rangle=\{x_1^{\epsilon_1}x_2^{\epsilon_2}...x_n^{\epsilon_n}\mid x_i\in A,~\epsilon_i\in\mathbb Z,~ x_i \neq x_{i+1} ,~ n=1,2,3...\}$$</p>
<p>I have no trouble understanding this. But then we... | Christian Gaetz | 75,296 | <p>The commutative property doesn't follow from that definition per se. However since $G$ is abelian and the elements of $A$ multiply according to the multiplication rules for $G$, we get commutativity for free.</p>
|
4,482,707 | <p><a href="https://math.stackexchange.com/questions/942263/really-advanced-techniques-of-integration-definite-or-indefinite/1885401#1885401">Here</a>, I saw the following formula:</p>
<p><span class="math-container">$$\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$$</span></p... | j4nd3r53n | 446,918 | <p>As Theo Bendt says in his comment, a relation between sets <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> is simply a subset of <span class="math-container">$X \times Y$</span>, that is, if you choose <span class="math-container">$R \subseteq X \times Y$</span>, then you can defi... |
2,414,011 | <p>In my recent works in PDEs, I'm interested in finding a family of cut-off functions satisfying following properties:</p>
<p>For each $\varepsilon >0$, find a function ${\psi _\varepsilon } \in {C^\infty }\left( \mathbb{R} \right)$ which is a non-decreasing function on $\mathbb{R}$ such that:</p>
<ol>
<li>${\psi... | Rigel | 11,776 | <p>You can start with the piecewise affine Lipschitz function
$$
h_\varepsilon(x) :=
\begin{cases}
0, & \text{if}\ |x| \leq 5\varepsilon/4,\\
1, & \text{if}\ |x| \geq 7\varepsilon/4
\end{cases}
$$
(and affine for the remaining values of $x$).
For this function you have $|x h_\varepsilon'(x)| \leq 4$.</p>
<p>Le... |
69,948 | <p>Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are bounded by an integer K, if useful) :</p>
<p>n m > n' m' => p(n,m) > p(n',m') </p>
<p>If yes what does it look lik... | Gerhard Paseman | 3,206 | <p>To pull this observation out of the comments, suppose we had a pairing function which was monotonic increasing with respect to products of 2 or more integers each larger than 1, and which had nice inverses, say one of them was F(p) and had a nice formula for it which was quickly computable and returned an integer gr... |
1,611,730 | <p>I am a linguist, not a mathematician, so I apologize if there's something wrong with my terminology and/or notation.</p>
<p>I have two structures that I want to merge (partially or completely). To generate a list of all possible combinations, I compute the Cartesian product of the two sets of objects, which gives m... | JasonRobinson | 333,310 | <p>If you want to fully enumerate all of those combinations (and save it to csv format along with the sets that generated that Cartesian product) and you need to account for variable size sets (set A has 3 distinct members, set B has 5 distinct members, C has 2 distinct members, etc) then I have written an algorithm th... |
209,116 | <p>Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?</p>
| Alexander Gruber | 12,952 | <p>$GL(V)$ is the group of <em>linear</em> transformations over a vector space $V$. You can also, as you have, write it $GL_n(K)$ if $V$ is an $n$-dimensional vector space over a field $K$, and thus isomorphic to $K^n$.</p>
<p>So, the "linear" part refers to the linearity property of the transformations: given vector... |
209,116 | <p>Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?</p>
| Tobias Kildetoft | 2,538 | <p>The term linear here refers to the fact that it is a group consisting of linear transformations of some vector space.
In some sense, all groups are "linear" like this, but usually if one refers to something as a linear group, then a specific realization as a group of linear transformations is usually (at least impli... |
324,385 | <p>I'm going through Wallace Clarke Boyden's <a href="http://books.google.com/books?id=OhMAAAAAYAAJ&pg=PA71#v=onepage&q&f=false" rel="noreferrer">A First Book in Algebra</a>, and there's a section on finding the square root of a perfect square polynomial, eg. <span class="math-container">$4x^2-12xy+9y^2=(2x... | Will Jagy | 10,400 | <p>It's just saying start with the highest degree and work down. This may not be the fastest but will work or tell you that the thing is not really a square. So, begin with $x^3,$ since the square must be $x^6$ and we get one free choice, $\pm x^3.$ Next, $(x^3 + A x^2)^2 = x^6 + 2 A x^5 + \mbox{more}.$ So $2A = -2, A ... |
4,565,728 | <p>Given a collection of topological spaces <span class="math-container">$X_i$</span> indexed by the elements <span class="math-container">$i$</span> of a set <span class="math-container">$I$</span>, we consider the set product <span class="math-container">$P = \prod_{i \in I} X_i$</span> with projections <span class="... | Paul Frost | 349,785 | <p>Here is a fairly obvious alternative characterization of the product topology:</p>
<blockquote>
<p>The product topology is the coarsest topology on <span class="math-container">$P$</span> such that all products of the form <span class="math-container">$\prod_{i \in I} A_i$</span> with closed <span class="math-contai... |
1,040,136 | <p>Just a quick question:</p>
<p>Is the size of the set of real numbers from 1 to 2 greater, or equal in size to the number of real numbers between 1 and 10?</p>
<p>I'm a Physicist so I'm not totally clued up on Mathematical jargon pertaining to set theory...</p>
| Adriano | 76,987 | <p>They're equal. Indeed, it's not too hard to show that the function $f\colon [1, 2] \to [1, 10]$ defined by:
$$
f(x) = 9x - 8
$$
is a bijection (which shows that the two sets have equal cardinality).</p>
|
1,555,548 | <p>There are $8$ people and they want to sit in a bus which has $2$ single front seats and $4$ sets of $3$ seats with $1$ person that is always the designated driver. How many ways are there for the people to sit in the bus?</p>
<p>I solved it by using:</p>
<p>$6!*(\binom{9}{3}) - 4((6*5*4*3)*2(\binom{4}{2})+(6*5*4*3... | fleablood | 280,126 | <p>If it's the same person driving then the answer is $14!/6!$ or 14*13*... 7. There are 14 choices for the first person, 13 for the second, and so on. </p>
<p>The arrangements of the seats and the driver are utterly irrelevant.</p>
<p>====</p>
<p>Why isn't it just $8*14!/7!$? One of the eight people has to drive... |
2,774,923 | <blockquote>
<p>$ABC$ is a triangle where $AE$ and $EB$ are angle bisectors, $|EC| = 5$, $|DE| = 3$, $|AB| = 9$. Find the perimeter of the triangle $ABC$.
<a href="https://i.stack.imgur.com/4nsiM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4nsiM.jpg" alt="enter image description here"></a></p... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/NtslB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NtslB.png" alt="enter image description here"></a></p>
<p>$|DE|=3=u$,
$|EC|=5=v$,
$|AB|=9=c$.</p>
<p>The question is not that innocent, as it looks at first glance.</p>
<p>Yes, following
the angle bisector... |
1,548,771 | <p>I have come up with the following constrained minimization problem:
\begin{eqnarray}
\min\ \sum_{i=1}^\infty x_i^2\\
\sum_{i=1}^\infty a_ix_i=1
\end{eqnarray}
If it were a finite-dimensional case it would be easily solved via Lagrange multipliers; in this case I ask your help since I don't know where to begin.</p>
| Kay K. | 292,333 | <p>Assuming the Cauchy's inequality still holds for infinite series,
$$\left(\sum_{i=1}^{\infty}x_i^2\right)\left(\sum_{i=1}^{\infty}a_i^2\right)\geq \sum_{i=1}^{\infty}x_ia_i=1$$
$$\sum_{i=1}^{\infty}x_i^2\geq \frac{1}{\sum_{i=1}^{\infty}a_i^2}$$</p>
|
3,193,696 | <p>Could someone explain what are (at least the four first) moments ? (normalized moment to be more precise) Let <span class="math-container">$X$</span> a r.v. </p>
<ul>
<li><p>So the first moment is the expectation. This will correspond to <span class="math-container">$\mathbb E[X]$</span> and is going to be the "bar... | BigBendRegion | 472,987 | <p>Actually, the geometric interpretation of kurtosis is quite simple: Following J.G.'s notation, let <span class="math-container">$V = Z^4$</span>, and let <span class="math-container">$p_V(v)$</span> denote the pdf of <span class="math-container">$V$</span>. Then the kurtosis of <span class="math-container">$X$</span... |
52,841 | <p>In classical Mechanics, momentum and position can be paired together to form a symplectic manifold. If you have the simple harmonic oscillator with energy $H = (k/2)x^2 + (m/2)\dot{x}^2$. In this case, the orbits are ellipses. How is the vector field determined by the (symplectic) gradient, then? </p>
<p>Also, ... | Yakov Shlapentokh-Rothman | 4,345 | <p>For one interpretation of the area inside a curve in phase space, see Arnold's <em>Mathematical Methods of Classical Mechanics</em> page 20. In case you do not have a copy of the book, he defines a function $S: (E_0 - \epsilon, E_0 + \epsilon) \to \mathbb{R}$ which gives the area inside the curve associated to an en... |
4,172,964 | <p>EDIT: Agreed, this isn't a well formed question. But responses below have at least given me a different way to think about it.</p>
<p>EDIT2: Thanks the answers I discovered the Google S2 library (<a href="http://s2geometry.io/devguide/s2cell_hierarchy" rel="nofollow noreferrer">http://s2geometry.io/devguide/s2cell_h... | Harsh joshi | 940,974 | <p>This problem can be solved by simple geometry.</p>
<p>Let <span class="math-container">$z^4=w$</span></p>
<p>Using <span class="math-container">$|z^4|=|z|^4$</span>, as <span class="math-container">$|z|=1$</span> then <span class="math-container">$|w|=1$</span>.</p>
<p>Now <span class="math-container">$w$</span> sat... |
85,841 | <p><a href="http://reference.wolfram.com/language/ref/Binomial.html" rel="nofollow"><code>Binomial[n, k]</code></a> is converted to a polynomial only for <code>k</code> less than 6.</p>
<pre><code>Table[Binomial[n, k], {k, 1, 8}]
(* {n,
1/2 (-1 + n) n,
1/6 (-2 + n) (-1 + n) n,
1/24 (-3 + n) (-2 + n) (-1 + n) n,
1/120 ... | Vaclav Kotesovec | 29,655 | <p>Solution of my problem is</p>
<pre><code>FunctionExpand[Table[Binomial[n, k], {k, 1, 8}]]
</code></pre>
|
8,699 | <p>I love your site.... but the your question does not meet our quality standards thing is really annoying... I have wasted lots of time trying to figure out what this message means.....maybe someone could explain it to me.....whats wrong with this question:</p>
<p>Find numbers a and b such that: </p>
<p>$ lim =((sq... | zyx | 14,120 | <p>As this is turning into a kind of case study of how the meta operates, I did a tiny bit of research that merits separate posting.</p>
<p>The source of this question shows that the OP did use TeX/MathJax dollar signs for the formulas, and tried to render the $x \to 0$ below the limit by placing it on the next line. ... |
1,879,129 | <p>If $0 < y < 1$ and $-1 < x<1$, then prove that $$\left|\frac{x(1-y)}{1+yx}\right| < 1$$</p>
| lab bhattacharjee | 33,337 | <p>This will hold true iff $$x^2(1-y)^2<(1+xy)^2$$</p>
<p>$$\iff0>x^2-2x^2y-2xy-1=(x+1)(x-1-2xy)$$</p>
<p>$$\iff0>x-1-2xy\iff1>x(1-2y)$$</p>
<p>WLOG let $y=\sin^2A,x=\cos B$</p>
<p>$x(1-2y)=\cos B\cos2A$ which is $<1$</p>
<p>OR</p>
<p>as $0<y<1\iff0>-2y>-2\iff1>1-2y>-1$ and we hav... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | fleablood | 280,126 | <p>If <span class="math-container">$A \subset B$</span> then <span class="math-container">$A' \subset B'$</span>.</p>
<p>Pf: If <span class="math-container">$a \in A'$</span> then <span class="math-container">$a$</span> is a limit point of <span class="math-container">$A$</span>. So every neighborhood of <span class... |
2,410,517 | <p>I feel like I'm missing something very simple here, but I'm confused at how Rudin proved Theorem 2.27 c:</p>
<p>If <span class="math-container">$X$</span> is a metric space and <span class="math-container">$E\subset X$</span>, then <span class="math-container">$\overline{E}\subset F$</span> for every closed set <spa... | Community | -1 | <p>$F\supset F'$ because $F $ is closed. $F'\supset E'$ because $F\supset E $, by assumption. Therefore $F\supset E' $.</p>
|
1,611,390 | <p>How to show that the following function is an injective function?</p>
<p>$ \varphi : \mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N} \\
\varphi(\langle n, k\rangle) = \frac{1}{2}(n+k+1)(n+k)+n$</p>
<p>I'm starting with $ \frac{1}{2}(a+b+1)(a+b)+a = \frac{1}{2}(c+d+1)(c+d)+c$, but how am I supposed to show from ... | Sam Birns | 291,254 | <p>It might be easier to start by letting $(n, k)$ be such that $\varphi( \langle n, k\rangle ) = 0$. Then, $(n + k + 1)(n+k) + 2n = 0$, and since $n, k \in \mathbb{N}$, $(n + k + 1)(n + k), 2n \ge 0$, so $2n = n = 0$, so $k = 0$, and since $(n, k) = (0, 0)$, $\varphi$ is injective. </p>
|
1,611,390 | <p>How to show that the following function is an injective function?</p>
<p>$ \varphi : \mathbb{N}\times \mathbb{N} \rightarrow \mathbb{N} \\
\varphi(\langle n, k\rangle) = \frac{1}{2}(n+k+1)(n+k)+n$</p>
<p>I'm starting with $ \frac{1}{2}(a+b+1)(a+b)+a = \frac{1}{2}(c+d+1)(c+d)+c$, but how am I supposed to show from ... | BrianO | 277,043 | <p>Note that
$$\begin{align}
\varphi(0, n+k) &= \frac 1 2 (n+k)(n+k+1) \\
&\le \varphi(n,k) \\
&= \frac 1 2 (n+k)(n+k+1) + n \\
&= n + \sum_{i \le (n+k)} i \\
&< \sum_{i \le (n+k+1)} i \\
&= \varphi(0, n+k+1).
\end{align}$$</p>
<p>Suppose $\varphi(a,b) = \varphi(c,d)$. If $a+b < c+d$, th... |
894,159 | <p>I was assigned the following problem: find the value of $$\sum_{k=1}^{n} k \binom {n} {k}$$ by using the derivative of $(1+x)^n$, but I'm basically clueless. Can anyone give me a hint?</p>
| Caddyshack | 168,872 | <p>Notice that $\displaystyle S = \sum_{k=0}^{n} k \binom {n} {k} = \sum_{k=0}^{n}(n-k)\binom {n} {n-k} = n\sum_{k=0}^{n}\binom {n} {n-k}-S$</p>
<p>so, as $\displaystyle \binom {n} {n-k}=\binom {n} {k}$ we have $\displaystyle 2S = n\sum_{k=0}^{n}\binom {n} {k} = n2^n$ and so $\displaystyle S = n2^{n-1}.$</p>
|
512,590 | <p>According to the definition my professor gave us its okay for a matrix in echelon form to have a zero row, but a system of equations in echelon form cannot have an equation with no leading variable.</p>
<p>Why is this? Aren't they supposed to represent the same thing?</p>
| egreg | 62,967 | <p>Your hypothesis means that
$$
z^2 - 2z\cos\alpha + 1 = 0
$$
Solve the quadratic equation:
$$
z=\cos\alpha\pm\sqrt{\cos^2\alpha-1}
$$
so
$$
z=\cos\alpha+i\sin\alpha
\quad\text{or}\quad
z=\cos\alpha-i\sin\alpha=\cos(-\alpha)+i\sin(-\alpha).
$$
Apply de Moivre's formula.</p>
|
325,765 | <p>Is there any method which allows us to describe all continuous functions (maps to $\mathbb{R}$) on the quotient space?</p>
<p>For examle, how could I classify all continuous functions on $\mathbb{R}/[x\sim2x]$?</p>
| Community | -1 | <p>Let $Y=X/\sim$ be a quotient space equipped with the quotient topology. $f:Y\to \mathbb{R}$ is continuous if and only if $f \circ \pi$ is continuous, where $\pi:X\to Y$ is the natural quotient map. The reason is $U\subseteq Y$ is open in the quotient topology if and only if $\pi^{-1}(U)$ is open in $X$.</p>
|
1,826,964 | <blockquote>
<p>A fair die is tossed n times (for large n). Assume tosses are independent. What is the probability that the sum of the face showing is $6n-3$?</p>
</blockquote>
<p>Is there a way to do this without random variables explicitly? This is in a basic probability theory reviewer, and random variables was n... | Emre | 321,157 | <p>Your way is correct, but your calculations have some flaws. For (2), you choose $1$ place for $4$ and $1$ place for $5$. So, the binomial coefficients should be ${n\choose 1},{n-1\choose 1}$. So, $(2)={n\choose 1}(1/6)^1{n-1\choose 1}(1/6)^1(1/6)^{n-2}$</p>
<p>For three you need to have $(1/6)^{n-1}$, instead of $(... |
2,909,480 | <p>Please notice the following before reading: the following text is translated from Swedish and it may contain wrong wording. Also note that I am a first year student at an university - in the sense that my knowledge in mathematics is limited.</p>
<p>Translated text:</p>
<p><strong>Example 4.4</strong> Show that it ... | Dr. Sonnhard Graubner | 175,066 | <p>Since we have assumed the $$n^3-n$$ is divisible by $3$ we can write $$n^3-n=3b$$ with $b$ is an integer number.
The proof becomes very easy if we write $$n^3-n=(n-1)n(n+1)$$</p>
|
409,220 | <p>$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$
$$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$
$$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$
Points, in which partial derivatives ar equal to 0 are: (3,2), (x,0), (0,y), x,y are any real numbers. Now I find second derivatives
$$\Delta_1=\frac{\delta f}{\delta x^2}=36xy^2... | PierreCarre | 639,238 | <p>You just need to use the definition of maximum/minimum, together with the knowledge of the sign of <span class="math-container">$f$</span>.</p>
<p>For instance, if you take a point of the form <span class="math-container">$(x,0), x<0$</span>, you see that <span class="math-container">$f(x,0)=0$</span> and <span c... |
1,416,972 | <p>Recently I started reading about <em>graph embeddings</em>, but I am unable to grasp its definition from <a href="https://en.wikipedia.org/wiki/Graph_embedding" rel="nofollow">Wikipedia</a>. Can anyone explain this term with an example.</p>
| Mike Pierce | 167,197 | <p>A <em>graph</em> is just an abstract idea, whereas a <em>graph embedding</em> is an actual physical instance of a graph that has to be drawn (or embedded) onto some surface.</p>
<hr />
<p>In order to understand what a <em>graph embedding</em> is, it helps to first define what a <em>graph</em> is.</p>
<blockquote>
<p... |
1,416,972 | <p>Recently I started reading about <em>graph embeddings</em>, but I am unable to grasp its definition from <a href="https://en.wikipedia.org/wiki/Graph_embedding" rel="nofollow">Wikipedia</a>. Can anyone explain this term with an example.</p>
| elmodeer | 523,196 | <p>I think the best example to understand the nuanced difference between Topological vs Embedded Graphs is the example Steven Skiena gave in his <a href="https://www.google.com/search?q=The%20Algorithm%20Design%20Manual&rlz=1C5CHFA_enDE999DE999&sxsrf=ALiCzsbGZ7iG-u3J7kVPWVSjEg_SKrvJFQ%3A1654439483011&ei=O76... |
3,006,511 | <p>Let <span class="math-container">$c = \{ x = \{x_k\}_{k=1}^{\infty} \in l^\infty \vert \exists \lim_{k \to \infty} x_k \in \mathbb{C} \}$</span>. </p>
<p>Let <span class="math-container">$x_n \in c$</span>, with <span class="math-container">$x_n \to x = \{x_k\}$</span> with the sup norm. </p>
<p>I want to prove th... | DanielWainfleet | 254,665 | <p>With the notation of the Answer given by JustDroppedIn.</p>
<p>Since you want to show that <span class="math-container">$c$</span> is closed in <span class="math-container">$l^{\infty},$</span> you can prove that <span class="math-container">$l^{\infty}\setminus c$</span> is open, as follows: </p>
<p>Let <span cla... |
677,241 | <p>Let $A$ be a list of $n$ numbers in range $[1,100]$ (numbers can repeat).
I'm looking for the number of permutations of $A$ which start with a non-decreasing part, where this part ends with the first instance of the highest number, call this "index $i$" (1 based)from the left. After $i$, the remaining permutation is... | Marc Romaní | 179,483 | <p>Let's say you have $t_i$ times integer $i$, with $i$ from $0$ to $9$. The number of permutations of the tuple $(a_1, \ldots, a_{t_1 + \cdots + t_9})$ with $0 \leq a_i \leq 9$, such that $a_i \leq a_j$ if $i < j$, is $$t_1!\cdots t_9!$$</p>
|
1,274,816 | <p>It seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: <a href="https://math.stackexchange.com/questions/1192338/number-theory-taxicab-number">Number Theory Taxicab Number</a>).</p>
<p>We know that</p>
... | Jibran | 515,854 | <p>As for the question regarding whether or not there are infinitely many numbers that can be expressed as the sum of two cubes in two different ways or not, there's a very quick and simple way to prove this. Since you have
<span class="math-container">$$ 1729 = 10^3 + 9^3 = 12^3 + 1^3 $$</span>
Multiply both sides b... |
2,498,628 | <p>This was a question in our exam and I did not know which change of variables or trick to apply</p>
<p><strong>How to show by inspection ( change of variables or whatever trick ) that</strong></p>
<p><span class="math-container">$$ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx \tag{I} $$</span></p>
<p>Co... | Giuseppe Negro | 8,157 | <blockquote>
<p><strong>This is an extended comment not a proper answer.</strong></p>
</blockquote>
<p>The question can be rewritten as follows: to show that
$$\tag{1}
\Re \int_{-\infty}^\infty e^{-i|\xi|^2}\, d\xi + \Im \int_{-\infty}^\infty e^{-i|\xi|^2}\, d\xi=0,$$
where the integral is in the principal-value s... |
3,792,954 | <p>For vector space <span class="math-container">$V$</span> and <span class="math-container">$v \in V$</span>, there is a natural identification <span class="math-container">$T_vV \cong V$</span> where <span class="math-container">$T_vV$</span> is the tangent space of <span class="math-container">$V$</span> at <span cl... | LSpice | 87,579 | <p>Note that <span class="math-container">$1 + x y$</span> is positive for all <span class="math-container">$x, y \in (-1, 1)$</span>; so, since our function is odd in <span class="math-container">$(x, y)$</span> and hence takes values in <span class="math-container">$(-1, 1)$</span> if it takes values in <span class="... |
959,201 | <p>I am confused about the following.</p>
<p>Could you explain me why if $A=\varnothing$,then $\cap A$ is the set of all sets?</p>
<p>Definition of $\cap A$:</p>
<p>For $A \neq \varnothing$:</p>
<p>$$x \in \cap A \leftrightarrow (\forall b \in A )x \in b$$</p>
<p><strong>EDIT</strong>:</p>
<p>I want to prove that... | Mauro ALLEGRANZA | 108,274 | <p>See Herbert Enderton, <a href="http://rads.stackoverflow.com/amzn/click/0122384407" rel="noreferrer">Elements of Set Theory</a> (1977), page 24 :</p>
<blockquote>
<p>Suppose we want to take the intersection of infinitely many sets $b_0, b_1, \ldots$. Then where $A = \{ b_0, b_1,\ldots \}$ the desired intersection... |
784,258 | <p>I understand that this is an induction question. </p>
<p>I start with the base case (n=1):</p>
<p>$$1 < 2 \tag{That works!}$$</p>
<p>Induction step: Assume the statement works for all $n = k$, Prove for all $n = k+1$</p>
<p>Assume $1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}}+ ... +\frac{1}{\sqrt{k+1}}\le 2\s... | Srinivasa Ramanujan | 147,778 | <p>Compare the area below the red curve ($y=1/\sqrt{x}$) and the blue curve from $x=0$ to $x=\sqrt{n}$.</p>
<p><img src="https://i.stack.imgur.com/elri5.png" alt="enter image description here"></p>
|
87,948 | <p>Let $\mu_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue meas... | The Bridge | 2,642 | <p>Hi, </p>
<p>This is not the general answer you are looking for but it might be sufficient for your needs, here are my two cents. </p>
<p>If you are given a Semimartingale then there exists conditions that ensures that there exists a Markov process such that this markov process has the same marginal distributions t... |
2,353,272 | <p>Suppose that we are given the function $f(x)$ in the following product form:
$$f(x) = \prod_{k = -K}^K (1-a^k x)\,,$$
Where $a$ is some real number. </p>
<p>I would like to find the expansion coefficients $c_n$, such that:
$$f(x) = \sum_{n = 0}^{2K+1} c_n x^n\,.$$</p>
<p>A closed form solution for $c_n$, or at le... | Vassilis Markos | 460,287 | <p>Well, we have that - we need to suppose that $a\neq0$:
$$f(x)=\prod_{k=-K}^K(1-a^kx)=(1-a^{-K}x)(1-a^{-K+1}x)\dots(1-a^Kx)$$
We can expand this product to a sum of powers of $x$, by considering in how many ways can we get $x^n$, where $n=0,1,2,\dots,2K+1$.</p>
<p>While expanding this product, we take each time exac... |
2,353,272 | <p>Suppose that we are given the function $f(x)$ in the following product form:
$$f(x) = \prod_{k = -K}^K (1-a^k x)\,,$$
Where $a$ is some real number. </p>
<p>I would like to find the expansion coefficients $c_n$, such that:
$$f(x) = \sum_{n = 0}^{2K+1} c_n x^n\,.$$</p>
<p>A closed form solution for $c_n$, or at le... | epi163sqrt | 132,007 | <p>We derive a representation of $c_j$ in
\begin{align*}
f_K(x)=\prod_{k=-K}^K(1-a^kx)=\sum_{j=0}^{2K+1}\color{blue}{c_j}x^j\qquad\qquad \qquad K\geq 0
\end{align*}
based upon the <em><a href="https://en.wikipedia.org/wiki/Q-Pochhammer_symbol" rel="nofollow noreferrer">q-Pochhammer symbol</a></em> $(x;a)_K$.</p>
<bloc... |
4,013,559 | <p>In the video <a href="https://youtu.be/eI4an8aSsgw?t=16354" rel="nofollow noreferrer">https://youtu.be/eI4an8aSsgw?t=16354</a> the professor says that the following equation
<span class="math-container">$$\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2+y^2}=2a$$</span>
simplifies to
<span class="math-container">$$
(a^2-c^2)x^2... | Z Ahmed | 671,540 | <p>Let
<span class="math-container">$$\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2+y^2}=2a=\sqrt{P}+\sqrt{Q}~~~(1)$$</span>
<span class="math-container">$$\implies P-Q=4cx \implies \sqrt{P}-\sqrt{Q}=2cx/a~~~(2)$$</span>
Adding (1) and (2) get
<span class="math-container">$$\sqrt{P}=xc/a+a \implies (x+c)^2+y^2=(cx/a+a)^2 \impli... |
681,737 | <p>What is the simplest way we can find which one of $\cos(\cos(1))$ and $\cos(\cos(\cos(1)))$ [in radians] is greater without using a calculator [pen and paper approach]? I thought of using some inequality relating $\cos(x)$ and $x$, but do not know anything helpful.
We can use basic calculus. Please help. </p>
| Yiorgos S. Smyrlis | 57,021 | <p>This statement is correct is "bounded" is replaced by "compact".</p>
<p>Also, it is correct if $\mathbb R$ is replaced by "a compact metric space".</p>
|
3,852,952 | <p>Given a projective space <span class="math-container">$\mathbb{P}^n(\mathbb{C})$</span>, I can consider the Grasmannian of lines <span class="math-container">$G(2,n+1)$</span>, which has a structure of projective variety inside <span class="math-container">$\mathbb{P}^N$</span>, where <span class="math-container">$... | Gustavo Labegalini | 356,961 | <p>End up here with the same question, but I could not understand the answers very well. I'll let a proposal of an answer out here, and see what's up.</p>
<p><strong>Plucker relations for <span class="math-container">$k=2$</span>:</strong> An element <span class="math-container">$\omega \in \bigwedge^2 V$</span> is dec... |
926,804 | <p>Is there a word for the quality of a number to be either positive or negative? Consider this question:</p>
<p><em>What's the ... (sign/positivity/negativity, but a word that could describe either) of number <strong>x</strong>?</em></p>
<p>Also, is there an all-encompassing word for the sign put in front of a numbe... | k170 | 161,538 | <p>The word that you are looking for is the very word that you are using and that word is <strong>sign</strong>.</p>
<p>If you are looking for something more "academic", then you can use its Latin variant, <strong>signum</strong>.</p>
<p>Context can also effect what word you might choose. For instance, in phy... |
926,804 | <p>Is there a word for the quality of a number to be either positive or negative? Consider this question:</p>
<p><em>What's the ... (sign/positivity/negativity, but a word that could describe either) of number <strong>x</strong>?</em></p>
<p>Also, is there an all-encompassing word for the sign put in front of a numbe... | Neville | 493,521 | <p>The search is for a PHILOSOPHICAL term to denote positive / negative. I agree that "sign" lacks something - hence, consider "polarity" and "charge".</p>
|
4,346,338 | <p>In Loring Tu Introduction to Manifolds, pg. 328:</p>
<blockquote>
<p>The projection <span class="math-container">$\pi:X \times Y \rightarrow X$</span>, <span class="math-container">$\pi(x,y)=x$</span>, is
continuous.</p>
<p>Proof: Let <span class="math-container">$U$</span> be open in <span class="math-container">$X... | Intelligenti pauca | 255,730 | <p>Relabelling <span class="math-container">$z=x-x_0$</span> and <span class="math-container">$y=f(z)$</span> your first equation reads
<span class="math-container">$$
y={a\over2}\big(z-\sqrt{z^2+b^2}\big).
$$</span>
Rearranging and squaring yields
<span class="math-container">$$
y^2-ayz={a^2b^2\over4},
$$</span>
that ... |
4,346,338 | <p>In Loring Tu Introduction to Manifolds, pg. 328:</p>
<blockquote>
<p>The projection <span class="math-container">$\pi:X \times Y \rightarrow X$</span>, <span class="math-container">$\pi(x,y)=x$</span>, is
continuous.</p>
<p>Proof: Let <span class="math-container">$U$</span> be open in <span class="math-container">$X... | user2603428 | 1,010,688 | <p>Thanks to @Intelligenti pauca for pointing to my error and how to get to the answer that I find satisfactory.</p>
<p>First I redefine f(x) such that it the independent and dependent variables are unitless. Analytic geometry requires that all dimensions have the same unit for anything to be sensible. What is the leng... |
643,560 | <p>Let $\{x_n\}_{n=1}^{\infty}$ and $\{y_n\}_{n=1}^{\infty}$ be sequences of real numbers. Does the following hold:</p>
<p>$$
\limsup x_n +\liminf y_n \le \limsup\,(x_n+y_n).
$$ </p>
<p>This is what I have tried but I am not quite sure if it is correct.
$\text{Fix } K>1. \text{ Let }L=\inf_{1\le i \le k}y_i$. Now... | taritari | 122,374 | <p>$$\limsup (x_n+y_n) = \lim_{n\to\infty} \left(\sup_{k>n}(x_k+y_k)\right) \leq \lim_{n\to\infty}\left(\sup_{k>n}x_k+\sup_{k>n}y_k\right)=\\=\limsup(x_n)+\limsup (y_n)$$</p>
<p>The inequality is because a property of supreme: $\sup (a+b) \leq \sup (a) + \sup (b)$.</p>
|
268,152 | <p>I often see proofs, that claim to be <em>by induction</em>, but where the variable we induct on <em>doesn't</em> take value is $\mathbb{N}$ but only in some set $\{1,\ldots,m\}$.</p>
<p>Imagine for example that we have to prove an equality that encompasses $n$ variables on each side, where $n$ can <em>only</em> ran... | Hagen von Eitzen | 39,174 | <p>If the statement in question really does not "work" if $n>m$, then necessarily the induction step $n\to n+1$ at least <em>somewhere</em> uses that $n<m$. You may view this as actually proving by induction
$$\tag1\forall n\in \mathbb N\colon (n>m\lor \phi(m,n))$$
That is, you first show
$$\tag2\phi(m,1)$$
(... |
1,530,848 | <p>Let $F(\mathbb{R})$ be the set of all functions $f : \mathbb{R} → \mathbb{R}$. Define pointwise addition and multiplication as follows. For any $f$ and $g$ in $F(\mathbb{R})$ let:</p>
<p>(i) $(f + g)(s) = f(x) + g(x)$ for all $x \in \mathbb{R}$</p>
<p>(ii) $(f · g)(s) = f(x) · g(x)$ for all $x \in \mathbb{R}$</p>
... | beroal | 7,011 | <p>“This is not really an answer, but it was getting too long to be a comment.”</p>
<blockquote>
<p>I see in each new chapter that I study in calculus or linear algebra,
there is an inherent geometrical intuition that can be easily
visualized</p>
</blockquote>
<p>…And you expect it to generalize to the whole ma... |
2,214,137 | <p>How many positive integer solutions does the equation $a+b+c=100$ have if we require $a<b<c$?</p>
<p>I know how to solve the problem if it was just $a+b+c=100$ but the fact it has the restriction $a<b<c$ is throwing me off.</p>
<p>How would I solve this?</p>
| Arthur | 15,500 | <p>Hint: For each possible value of $a$, count the number of possible values of $b$ and $c$. It's a quite regular pattern, so you don't have to brute force much to find it.</p>
|
2,214,137 | <p>How many positive integer solutions does the equation $a+b+c=100$ have if we require $a<b<c$?</p>
<p>I know how to solve the problem if it was just $a+b+c=100$ but the fact it has the restriction $a<b<c$ is throwing me off.</p>
<p>How would I solve this?</p>
| Christian Blatter | 1,303 | <p>By stars and bars there are ${99\choose2}$ nonnegative solutions to the equation $x_1+x_2+x_3=97$. Among these there are $3\cdot49$ having two equal $x_i\in[0\ ..\ 48]$, so that there are $${1\over6}\left({99\choose2}-3\cdot49\right)=784$$
solutions satisfying $x_1<x_2<x_3$. For each of these the numbers $a:=1... |
186,146 | <p>On a finite dimensional vector space, the answer is yes (because surjective linear map must be an isomorphism). Does this extend to infinite dimensional vector space? In other words, for any linear surjection $T:V\rightarrow V$, AC guarantees the existence of right inverse $R:V\rightarrow V$. Must $R$ be linear?</p>... | Robert Israel | 8,508 | <p>Let $T$ be any map of $V$ to $W$ that is onto but not one-to-one. A right inverse for $T$ is any $R: W \to V$ such that $T(R(x)) = x$ for every $x \in W$. In particular, for any $x$ such that there are $y_1 \ne y_2$ with $T(y_1) = T(y_2) = x$, you are free to make $R(x) = y_1$ or $R(x) = y_2$. If $x = u + v$ (and... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.