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 |
|---|---|---|---|---|
1,796,056 | <p>I'm reading <a href="https://proofwiki.org/wiki/Non-Trivial_Discrete_Space_is_not_Connected" rel="nofollow noreferrer">this proof</a> that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton <span class="math-container">${x... | Hew Wolff | 34,355 | <p>You're a little muddled about what open and closed mean. In a metric space, open and closed sets are defined using the concept of a ball. This proof does not deal with metric spaces, but with topological spaces, which are more general, so there is no such thing as a ball here. Here, to know whether the two halves... |
3,738,646 | <p>I think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence:</p>
<p>The sequence <span class="math-container">$\{q_n\}$</span> of rationals is a Cauchy sequence if, for every <span class="math-container">$\epsilon>0$</span>, where <span class="math-container">$\epsilon$</spa... | Community | -1 | <p><span class="math-container">$x$</span> is increasing<span class="math-container">$\implies\sqrt x$</span> is increasing <span class="math-container">$\implies2+\sqrt x$</span> is increasing<span class="math-container">$\implies\sqrt{2+\sqrt x}$</span> is increasing.</p>
<p>Now if the sequence converges, it converge... |
1,044,096 | <p>Let $\mathscr { S } $ the class of all closed intervals $[a,b]$, where $a$ and $b$ are rational , i.e. $a,b\in \mathbb{Q}$ and $a<b$ . Show that $\mathscr{S}\cup \{\{p\}:p\in\mathbb{ Q }\}$ is a basis of the topology $\tau$ of the real line $\mathbb{R}$ generated by $\mathscr {S}$ .</p>
<p>Having problems... | Jonathan Gleason | 10,109 | <p>Let us define $\mathscr{T}:=\mathscr{S}\cup \left\{ \{ p\} :p\in \mathbb{Q}\right\}$. You first should check that each element of $\mathscr{T}$ is actually open in this topology. Of course, every element in $\mathscr{S}$ is, and $\{ p\} =[a,p]\cap [p,b]$ for $a<p<b$, and so $\{ p\}$ is also open in this topo... |
98,451 | <p>I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in 1977. The paper is at <a href="http://link.aps.org/doi/10.1103/PhysRevB.16.4466" rel="nofollow noreferrer">this page</a>... | Robert Israel | 13,650 | <p>The paper is not using residue theory, he is evaluating the contour integral numerically. I don't see a justification given for deforming the path of integration from a vertical line to a circle, but assuming sufficiently nice properties at infinity that should be OK.</p>
|
999,227 | <p>I have convinced myself that this true, however I'm at a loss of where I should start with this proof. Looking at a similar proof with 3 instead of 9, I saw the use of the basis representation theorem, but I'm not 100% comfortable with that, so it was hard for me to follow. Is there a way to do this with mod?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>let $z=a_0+10a_1+100a_2\equiv a_0+a_1+a_2 \mod 9$</p>
|
1,884,791 | <p>I'm trying to prove the convergence of </p>
<p>$$\int^{\infty}_{0} \frac{1}{e^x+1}dx$$</p>
<p>Any suggestion or hint for how to deal with it?</p>
<p>Thanks in advance!</p>
| TBTD | 175,165 | <p>Rewrite the integral as
\begin{equation}
\lim_{n \to \infty} \int_{0}^n \frac{1}{1+e^x} dx = \lim_{n \to \infty} \int_{0}^n \left(1- \frac{e^x}{1+e^x}\right) dx
\end{equation}</p>
<p>Now, evaluate the integral above to obtain
\begin{equation}
n - \ln(1+e^n) + \ln2 = \ln \left( \frac{e^n}{1+e^n}\right) + \ln2
\end{... |
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | Paramanand Singh | 72,031 | <p>This one is easy and an immediate consequence of the fundamental inequality satisfied by $\log$ function: $$\log x\leq x-1,x>0\tag{1}$$ For the current question we have $x\geq 0$ and hence $$0\leq \sqrt{N+1}\log\left(1+\frac{x}{N+1}\right) \leq \frac{x} {\sqrt{N+1}}$$ and the result follows via Squeeze Theorem. <... |
2,998,342 | <p>Find <span class="math-container">$y$</span> such that </p>
<p><span class="math-container">$$\lfloor y \rfloor + \lfloor 3y \rfloor = 5$$</span></p>
<p>First, I use the properties that <span class="math-container">$$n \leq y <n+1$$</span>
And suppose </p>
<p><span class="math-container">$$\lfloor y\rfloor = 5... | Hagen von Eitzen | 39,174 | <p>As <span class="math-container">$p-1\mid p^q-1=2^n$</span>, we see that <span class="math-container">$p$</span> must be one more than a power of <span class="math-container">$2$</span>, so <span class="math-container">$p=2^k+1$</span> (with <span class="math-container">$k\ge1$</span> because <span class="math-contai... |
3,237,995 | <p>Let <span class="math-container">$\Bbb L$</span> be a positive integer and define <span class="math-container">$\Bbb Z_L$</span> and <span class="math-container">$\Bbb Z^*_L$</span> as <span class="math-container">$$\Bbb Z_L=\{ 0,1,...,L-1\},\qquad\Bbb Z^*_L=\{i\in \Bbb Z_L\mid \gcd(i,L)=1\}$$</span> where <span cla... | Ender Wiggins | 424,790 | <p>First of all, for the future, it is always a good thing to add some attempt of yours to a question, to show that you already put some effort. </p>
<p>Secondly, try to be a bit more precise. I assume that you are looking at <span class="math-container">$Z_{L}^*$</span> as endowed with the multiplication, so you are ... |
1,527,183 | <p>This is likely a duplicate, but can't find it on MSE.</p>
<p>Let's say I have a normally distributed population with $\mu=2.75$ and $\sigma=0.25$. If $x$ is a value in the population of interest, using the normal probability distribution function I find that</p>
<p>$$P(x=3)\approx 0.96788,$$</p>
<p>and using the... | Guillaume A. | 289,508 | <p>It is unlikely but could it be the probability of being within 3 standard deviations of the mean ?
If you computed statistics you generated using few samples, outliers may have more weight than they should ie 1-96.8% instead of 1-99.7%.</p>
<p>How did you find 96.788% ?</p>
|
614,045 | <p>I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone explain what that actually is, and how one uses the specified methods to solve this sort of problem?</p>
| amWhy | 9,003 | <p>It would help you to factor $$f(x) = -x^2 + 4x - 3 = -(x - 3)(x - 1)$$</p>
<p>Then $$f(x) = 0 \iff x - 3 = 0 \;\text{ or } \; x - 1 = 0$$</p>
<p>As suggested, did you graph the function? You can then visually see where $f(x)$ intersects the $x$-axis: those are the "zeros" of the function.</p>
|
477,834 | <p>What is the transformation such that a general cubic polynomial to be made a cube,</p>
<p>$$ax^3+bx^2+cx+d = y^3\tag{1}$$</p>
<p>can be transformed to Weierstrass form,</p>
<p>$$x^3+Ax+B = t^2\tag{2}$$</p>
<p>(The special case $b = c = 0$ is easier.) Given an initial rational point, I know a method how to find s... | duje | 82,393 | <p>General algorithm for transforming a cubic into Weierstrass form, with several worked examples, can be found in Section 1.4 of Ian Connell's <a href="http://pendientedemigracion.ucm.es/BUCM/mat/doc8354.pdf" rel="nofollow">Elliptic Curve Handbook</a> </p>
|
4,516,512 | <p>The finite subgroups of <span class="math-container">$\mathrm{GL}_2(\mathbb{C})$</span> are pretty much classified in several expository papers in the web as well as posts on this site. However, I would like to know that how does one show that the Heisenberg group <span class="math-container">$H_3$</span> cannot be ... | Qiaochu Yuan | 232 | <p>Here is yet another argument. You can compute that the abelianization of <span class="math-container">$H_3$</span> has order <span class="math-container">$9$</span>, so <span class="math-container">$H_3$</span> has nine <span class="math-container">$1$</span>-dimensional irreducible representations. If <span class="... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | Martín-Blas Pérez Pinilla | 98,199 | <p>Let be $\omega\in\Bbb C$, $\omega^3 = 1$, $\omega\ne 1$. $\omega^n$ gives:
$$\omega^0 = 1,$$
$$\omega^1 = \omega,$$
$$\omega^2 = \omega^{-1},$$
$$\omega^3 = 1,$$
$$\cdots$$
"But I want $1,2,3,\dots$!" Solution: compose with a <a href="https://en.wikipedia.org/wiki/Polynomial_interpolation" rel="noreferrer">polynomia... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | yo' | 43,247 | <p>Well, there are many answers to this actually, for instance
$$ f(n) = n + 1 - 3\lfloor n/3\rfloor $$
(assuming $\mathbb Z^+$ starts with $0$).</p>
<p>How to come to this? The basic idea is that $s(x)=x-\lfloor x\rfloor$ is the sawtooth function with range $[0,1)$. Then $ns(x/n)$ is the same function scaled $n$ time... |
2,222,898 | <p>A friend of mine recently asked me this question.</p>
<p>For a positive integer $n$, a function $f(n)$ is defined as:</p>
<p>$f(n)=$ sum of digits in $n$.</p>
<p>given $f(n)=5$ find the maximum value of $f(n^5)$.</p>
<p>I tried solving this problem by putting random values, but my friend gave me a hint that the ... | Robert Israel | 8,508 | <p>$f(xy) \le f(x) f(y)$, with equality only if there are no carries in the long multiplication of $x$ and $y$. So $f(n^5) \le f(n)^5 = 5^5$, with equality only if there are no carries. Well, you can't quite get away with no carries because ${5 \choose 2} = {5 \choose 3} = 10$, but you can try to reduce them.
For exa... |
208,419 | <blockquote>
<p>Every group of order $231$ is the direct product of a group of order $11$ and a group of order $21$.</p>
</blockquote>
<p>By Sylow's theorem, we know there are one Sylow-7 subgroup, one Sylow-11 subgroup (these two are normal, for sure), and some Sylow-2 subgroups.</p>
<p>What does the "direct produ... | Martin Argerami | 22,857 | <p>It means that there exist two normal subgroups $K_1$ and $K_2$ such that $K_1\cap K_2=\{e\}$ and every element in the group can be written as $k_1k_2$, with $k_1\in K_1$, $k_2\in K_2$. </p>
<p>This is equivalent to saying that the group is isomorphic to $K_1\times K_2$ with the entrywise operation. </p>
<p>(if you... |
83,507 | <p>Is there any shorter / more handy / more flexible way to get directory tree as a nested <code>Rules/Associations</code> list?</p>
<p>I tried to incorporate <code>FileNames</code> + <code>GatherBy</code> and <code>GroupBy</code>, but nothing was as short and handy as old fashioned:</p>
<pre><code>ClearAll @ filesIn... | Mr.Wizard | 121 | <p>Update: much simpler now. If this works right I think I would consider it superior.</p>
<p>Update 2: I attempted to add specific handling of files and directories. Please tell me if this appears to work and produce a format that is acceptable to you.</p>
<pre><code>fn[x : {___, {_, __}, ___}] := Normal @ GroupBy... |
135,717 | <p>I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. </p>
<p>The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equi... | Rasmus | 1,291 | <p>Daniel Dugger and Brooke Shipley give an example in their paper</p>
<p><strong>A curious example of triangulated-equivalent
model categories which are not Quillen equivalent</strong></p>
<p>available <a href="http://homepages.math.uic.edu/~bshipley/dugger.shipley.curious.example.pdf" rel="noreferrer">here</a>.</p>... |
1,835,519 | <p>If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed.<br>
Does this mean that $V$ is always dense in $V$$^o$? </p>
<p>Thanks!</p>
| GoodChessPlayer | 259,427 | <p>You could edit the image to have a horizontal line run through the vase, and then use any photo editor to identify points along the curve of the vase. Then, you could plot some of the pixel coordinates relative to the horizontal line on a new graph, and then, using polynomial regression, identify a piecewise functio... |
1,782,032 | <blockquote>
<p>Find the value of <span class="math-container">$$\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\cdots}}}}$$</span></p>
</blockquote>
<p>I know how to solve when all surds are of the same order, but what if they are different?</p>
<p>Technically, (as some users wanted to know <strong><em>exactly</em></str... | 5xum | 112,884 | <p>Any function defined on a point $x_0$ is continuous on the "interval" $\{x_0\}$.</p>
<p>This is because, by definition, a function is continuous on $\{x_0\}$ if:</p>
<blockquote>
<p>For every $\epsilon >0$, there exists $\delta >0$ sucht that for every $x\in \{x_0\}$ such that $|x-x_0|<\delta$, we have ... |
2,525,055 | <p>How can the differential of $\mathbf r\cdot\mathbf r$ be written as $2\mathbf r\cdot d\mathbf r$? That is,</p>
<p>$$d(\mathbf r\cdot\mathbf r)=2\mathbf r\cdot d\mathbf r$$</p>
<p>How? Here, $\mathbf r$ is a vector.</p>
| Acccumulation | 476,070 | <p>AlkaKadri and Trevor Gunn's answers are given in terms of coordinates, and David K's is given in terms of a particular basis. For a more abstract proof, we can consider the limit as h goes to zero of $\frac{\mathbf r \cdot \mathbf r - (\mathbf r+h \mathbf u) \cdot (\mathbf r+h \mathbf u)}{h}$ where h is a scalar and... |
477,559 | <p>I understand there is propositional logic, first-order logic, second-order logic higher-order logic, and type theory, where the latter logics are extensions of the former logics. Can someone explain the differences between these types of logics for someone who knows some basic linear algebra, set theory, calculus, a... | digilante | 620,978 | <p>Here is my attempt to answer this question in a clear and concise way (without going into formal or philosophical discussions), but may be flawed:</p>
<p>Propositional logic (PL):
Can only be used to talk about truth values/booleans (B) (true, false)</p>
<p>First-order logic (FOL):
Can be used talk about objects/... |
4,145,243 | <p><a href="https://i.stack.imgur.com/1XO2v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1XO2v.png" alt="enter image description here" /></a></p>
<p>The way I approached the problem is first by solving a system of equations:
<span class="math-container">$$
r_x \frac{dx}{dt} + r_y\frac{dy}{dt} = \f... | Sofía Marlasca Aparicio | 482,670 | <p>There are as many non-isomorphic modules as conjugacy classes, so in your notation <span class="math-container">$r \geq |G|$</span> since the list is complete. By a dimension argument, you can see that each is <span class="math-container">$1$</span>-dimensional and can appear only once.</p>
|
844,528 | <p>I cannot figure out an aesthetic way to do this.</p>
<p>Can someone give a beautiful solution to this ugly question?</p>
<hr>
<p>This is what I have tried yet. </p>
<p>I used the fact that $$x = \dfrac{1}{1-\left(1-\dfrac{1}{x}\right)}$$</p>
<p>Hence $$x = \sum_{n=0}^{\infty} \left(1-\dfrac{1}{x}\right)^n$$</p>... | Tunk-Fey | 123,277 | <p><strong>Hint :</strong></p>
<p><a href="http://en.wikipedia.org/wiki/Taylor_series#List_of_Maclaurin_series_of_some_common_functions">Maclaurin series for Binomial series</a> is
$$
(1+y)^n=\sum_{k=0}^\infty\binom{n}{k} y^n,
$$
for $|y|<1$ and all $n$. Also use identities binomial coefficients:
$$
\binom{n}{0}=1... |
4,179,661 | <p><strong>Question:</strong> Prove that <span class="math-container">$p(x^q-1) + q(x^{-p}-1) \geq 0$</span> for <span class="math-container">$x > 0$</span> and <span class="math-container">$p,q \in \mathbb{Z}^+$</span>.</p>
<p><strong>My attempts:</strong> I initially thought to make the expression <span class="mat... | Swan Klein | 940,564 | <p>Q9y5 gives a quicker explanation, but the answer can still be seen starting from something similar to your approach. If <span class="math-container">$q\equiv1\mod(p-1)$</span>, then <span class="math-container">$q=1+n(p-1)$</span> for some <span class="math-container">$n\in\mathbb{N}$</span>, which leads to <span cl... |
4,179,661 | <p><strong>Question:</strong> Prove that <span class="math-container">$p(x^q-1) + q(x^{-p}-1) \geq 0$</span> for <span class="math-container">$x > 0$</span> and <span class="math-container">$p,q \in \mathbb{Z}^+$</span>.</p>
<p><strong>My attempts:</strong> I initially thought to make the expression <span class="mat... | jjagmath | 571,433 | <p>Assume BOTH are true: <span class="math-container">$p \equiv 1 \pmod {q-1}$</span> AND <span class="math-container">$q \equiv 1 \pmod{p-1}$</span>. Then your argument shows that <span class="math-container">$q<p$</span> AND <span class="math-container">$p<q$</span>. This contradiction proves that one of the co... |
53,332 | <p>I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is particularly fond of a certain reference and why. </p>
<p>Also, I am very interested in ms/phd thesis (particularly ms thesis... | Henrique Tyrrell | 17,153 | <p>It's a more recent book, but I've read An introduction to quantum group and duality from Thomas Timmermann. I love how this book is written, although a lot of the proofs are missing. It's more of a year, year-and-a-half course. He closes the book with a treatment of Baaj-skandalis's beautiful result on duality of ac... |
368,332 | <p>Find the value of $\displaystyle\lim_{n \rightarrow \infty} \dfrac 1{n^4}\left[1\left(\sum^n_{k=1}k\right)+2\left(\sum^{n-1}_{k=1}k\right)+3\left(\sum^{n-2}_{k=1}k\right)+\dots+n.1\right]$</p>
<p>Please guide how to proceed in this case .....Thanks..</p>
<p>Here answer is : </p>
<p>The required limit is $\dfrac{1... | Stefan Hansen | 25,632 | <p>The sum of the first $n$ natural numbers is well-known to be
$$
\sum_{k=1}^nk=\frac{n(n+1)}{2}.
$$
Thus the expression within the brackets can be written as
$$
\left[\frac{n(n+1)}{2}+2\frac{(n-1)n}{2}+3\frac{(n-2)(n-1)}{2}+\cdots +n\right]
$$
or in sum-notation as
$$
\sum_{k=1}^n\left(k\frac{(n+1-k)(n+2-k)}{2}\right... |
368,332 | <p>Find the value of $\displaystyle\lim_{n \rightarrow \infty} \dfrac 1{n^4}\left[1\left(\sum^n_{k=1}k\right)+2\left(\sum^{n-1}_{k=1}k\right)+3\left(\sum^{n-2}_{k=1}k\right)+\dots+n.1\right]$</p>
<p>Please guide how to proceed in this case .....Thanks..</p>
<p>Here answer is : </p>
<p>The required limit is $\dfrac{1... | Matt L. | 70,664 | <p>You can write your expression as</p>
<p>$$\frac{1}{n^4}\sum_{l=1}^{n}l\sum_{k=1}^{n+1-l}k=
\frac{1}{n^4}\sum_{l=1}^{n}l\frac{(n+1-l)(n+2-l)}{2} = \\
= \frac{1}{2n^4}\sum_{l=1}^{n}l(n^2+3n-2nl+2-3l +l^2)=\\
=\frac{1}{2n^4}\left\{ n^2\sum_{l=1}^{n}l+ 3n\sum_{l=1}^{n}l-
2n\sum_{l=1}^{n}l^2 +2\sum_{l=1}^{n}l-3\sum_{l=1... |
234,153 | <p>Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)</p>
| Amr | 29,267 | <p>The statement you mentioned does not hold when d is allowed to be zero. Consider $x^3$ as a counterexample.I will assume that d is different from zeroLet p(x) be the polynomial.thus p(0)<0. Since p(x) tends to infinity as x goes to infinity. Therefore there exists r>0 such that p(r)>0. Now apply the intermediate ... |
3,807,293 | <p>I do apologise if this question has been asked before, but I am having trouble trying to get from <code>!A & B || A & !B</code> to <code>A XOR B</code>.</p>
<p>I've tried doing it in reverse by starting with <code>A XOR B</code> (<code>(A || B) & !(A & B)</code>), and I reached <code>(A || B) & (... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,045,222 | <blockquote>
<p>A pyramid <span class="math-container">$OABC$</span> has vectors <span class="math-container">$\vec{OA}=a$</span>, <span class="math-container">$\vec{OB}=b$</span> and <span class="math-container">$\vec{OC}=c$</span>.</p>
<p>The vectors <span class="math-container">$v_1,v_2,v_3$</span> and <span class="... | StackTD | 159,845 | <p>If the system of equations is written in matrix form, <span class="math-container">$A\mathbf{x}=\mathbf{b}$</span>, then <span class="math-container">$A$</span> is the coefficient matrix and <span class="math-container">$[A|\mathbf{b}]$</span> is the so called <em>augmentend matrix</em>: the matrix formed by adding ... |
3,045,222 | <blockquote>
<p>A pyramid <span class="math-container">$OABC$</span> has vectors <span class="math-container">$\vec{OA}=a$</span>, <span class="math-container">$\vec{OB}=b$</span> and <span class="math-container">$\vec{OC}=c$</span>.</p>
<p>The vectors <span class="math-container">$v_1,v_2,v_3$</span> and <span class="... | Theo Bendit | 248,286 | <p>I'm guessing that <span class="math-container">$[A|b]$</span> refers to the augmented matrix formed by augmenting the column vector <span class="math-container">$b$</span> onto the matrix <span class="math-container">$A$</span>. That is, it's a matrix with one extra column: <span class="math-container">$b$</span>.</... |
975,398 | <p>I want to know about the convergence or divergence of the following series:</p>
<p>$$\sum \sin (a_n) $$</p>
<p>where </p>
<p>$$a_n=\frac{(-1)^n}{n}$$</p>
<p>The tests that I tried were inconclusive. Is it possible to know?</p>
<p>Or, more generally, does this series converge for all conditionally convergent ser... | drhab | 75,923 | <p>$\sin(a_n)=(-1)^n\sin(\frac{1}{n})$ so you are dealing with an <a href="http://en.wikipedia.org/wiki/Alternating_series" rel="nofollow">alternating series</a>. Then the fact that $\sin(\frac{1}{n})$ is monotonically decreasing and converges to $0$ justifies the conclusion that the series is converging. The so-called... |
938,791 | <p>I want to check, whether $$\sum\limits_{n=0}^{\infty }{\frac{n!}{(a+1)(a+2)...(a+n)}}$$</p>
<p>converges or diverges.</p>
<p>$a$ is a constant number</p>
<p>Ratio test</p>
<p>$$\begin{align}
& \frac{a_{n}}{a_{n-1}}=\frac{n!}{(a+1)(a+2)...(a+(n-1))(a+n)}\cdot \frac{(a+1)(a+2)...(a+(n-1))}{(n-1)!}=\frac{n}{a... | anonymous67 | 161,212 | <p>I don't know much about Gamma Function and summation under the integral so here is a more elementary proof.</p>
<ul>
<li>If $a>0$ we have $\ln a_n=-(\ln (1+\frac{a}{1})+\ln (1+\frac{a}{2})+\dots+\ln (1+\frac{a}{n}))$ and using</li>
</ul>
<p>$$x\ge\ln(1+x)\ge x-\frac{x^2}{2}$$</p>
<p>(for $x>0$ proven by usi... |
1,931,075 | <p>How many distinct ways can the letters of the word UNDETERMINED be arranged so that all the vowels are in alphabetical order?
Do I need to consider the positions of the vowels?</p>
| Ben Grossmann | 81,360 | <p><strong>Hint:</strong> This is the same as finding the number of arrangements of the word XNDXTXRMXNXD. Given any arrangment of this word, we can just put the vowels in alphabetical order wherever X's appear. </p>
<p>So, for example, the arrangement XXXNDTDRMNXX of XNDXTXRMXNXD corresponds only to the <em>actual<... |
3,335,850 | <p>I know that if <span class="math-container">$x$</span> is an element of A by definition of subsets then it is an element of <span class="math-container">$B$</span>, but I don't know how to use the complement to prove the latter half.</p>
| Feng | 624,428 | <p>For <span class="math-container">$x\in C-B$</span>, <span class="math-container">$x\notin B$</span> so <span class="math-container">$x\notin A$</span> for otherwise suppose <span class="math-container">$x\in A$</span> then <span class="math-container">$x\in B$</span> which is a contradiction. Moreover, <span class="... |
4,593,996 | <p>I am looking for someone to explain how the images of basis vectors are calculated. There are a ton of problems like this in my textbook, but there is no explanation on how to do the calculation.</p>
<p><a href="https://i.stack.imgur.com/o7csZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/o7csZ.... | Seeker | 1,050,393 | <p>The basis vectors are calculated by evaluating the expression presented for the linear map in question. It's exactly like how you would calculate the function <span class="math-container">$$f(x)=x^2$$</span> for <span class="math-container">$x=1, 2, 3$</span>. Namely, by evaluating the expression <span class="math-c... |
1,723,707 | <p>I have proven that Martingales have orthogonal increments. </p>
<p>From this I need to show that $\operatorname{Cov}[M(t),M(s)]$ relies only on $\min\{s,t\}$.</p>
<p>I used the expected value definition of Covariance and eliminated one piece using the orthogonal increments bit. Also, I introduced conditioning on $... | Cavents | 322,026 | <p>What is the mathematical meaning of 'rely on'? Typically, to compute covariances of martingales you can do the following. Suppose $s<t$
\begin{align}
\mbox{Cov}[M(t),M(s)] &= \mbox{Cov}[M(t)-M(s)+M(s),M(s)] \\
&=\mbox{Cov}[M(t)-M(s),M(s)] + \mbox{Var}[M(s)] \\
&= \mathbb{E}[M(s)(M(t)-M(s))] - \mathbb{... |
901,549 | <p>I am struggling with part c of this question. Could someone please tell me how to approach and solve this type of questions?</p>
<p><img src="https://i.stack.imgur.com/SIpJ8.png" alt="enter image description here"></p>
| user137481 | 137,481 | <p>The marginal probability of Y means you sum across the columns for a given value of Y. </p>
<p>a) P(Y = -2) = 0.1 + 0.2 + 0.2 = 0.5<br>
$\quad$ P(Y = 1) = 0.1 + 0.2 + 0 = 0.3<br>
$\quad$ P(Y = 2) = 0.2 + 0 + 0 = 0.2 </p>
<p>b) The possible values of XY are 2, 0, 4, -1, 0, 2, -2, 0, 4<br>
E(XY) is simply t... |
2,155,201 | <p>By a real ODE I mean an ordinary differential equation with only real coefficients and the resulting function is a function of a real argument. If such a solution exists, can you give an example?</p>
<p>Edit: To add to this, is it still possible if the initial conditions must also be real?</p>
| BobaFret | 43,760 | <p>$y'' + y = 0$ has solutions $e^{ix}$ and $e^{-ix}$.</p>
<p><strong>Update</strong>: $y = ix$ is a solution to $(y')^2 + 1 = 0$ with $y(0)=0$</p>
|
46,663 | <p>suppose $X$ is a topological space and $G$ and $H$ are groups acting on it.</p>
<p>1) if $G$ is isomorphic to $H$ do we have necessarely $X/G$ is homeomorphic to $X/H$</p>
<p>2) suppose $G$ and $H$ are two conjugate subgroups of a group $K$ acting on $X$, in this case do we have $X/G$ is homeomorphic to $X/H$</p>
... | Luffy | 12,342 | <p>First a question. When you talk about `action of a group on a topological space' is it assumed that the action is continuous?</p>
<ol>
<li>No. You could let $\mathbf{Z}$ act on $\mathbf{R}$ by translation. Then $\mathbf{R}/\mathbf{Z}$ is homeo to the circle. If we let $\mathbf{Z}$ act on $\mathbf{R}$ trivially, we... |
2,899,422 | <p>I need some hints for solving $yy''-(y')^2=xy^2$. </p>
<p>I noticed that the left hand side is close to $(yy')'$:</p>
<p>$yy''-(y')^2=xy^2\ \Leftrightarrow\ yy''+(y')^2-2(y')^2=xy^2\ \Leftrightarrow\ (yy')'-2(y')^2=xy^2$.</p>
<p>But I don't know how to continue expressing the terms as derivatives of some function... | Kavi Rama Murthy | 142,385 | <p>Her is the correct way of doing this. $\int x(xe^{x^{2}})\, dx=\frac 1 2 x(e^{x^{2}})-\int \frac 1 2 e^{x^{2}}$ since $xe^{x^{2}}$ is the derivative of $e^{x^{2}}$. $\int e^{x^{2}}$ cannot be evaluated explicitly.</p>
|
3,230,922 | <p>I was working through some textbook problems for my Number Theory class and needed some help with the following question:</p>
<p>Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are associates.</p>
<p>I'd really like some help with this problem. Thank you!</p>
| Berci | 41,488 | <p><strong>Hint:</strong> You need to find complex numbers (for the preimages of) <span class="math-container">$t, u$</span> such that
<span class="math-container">$$tu=t+u\\
t^2=t-u\\
u^2=-t+u$$</span></p>
|
1,328,297 | <p>I have been just trying to start working with the books <em>Topics in Algebra</em> by I.N. Herstein, and I am having a bit of trouble understanding a definition.</p>
<p>It is,</p>
<blockquote>
<p>Definition: If $S$ is a nonempty set then $A(S)$ is the set of all one-to-one mappings of $S$ onto itself.</p>
</bloc... | Kevin Arlin | 31,228 | <p>If $S=\{1,2\}$, then $A(S)$ consists of two functions $f,g$ where $f(1)=1,f(2)=2,g(1)=2,g(2)=1$. For other choices of $S$, construct $A(S)$ similarly as the set of all such functions. For instance, $A(\{1,2,3\})$ has six elements and $A(\mathbb{N})$ has uncountably many. A simple exercise to check your understanding... |
4,417,018 | <p>Consider</p>
<p><span class="math-container">$$n \rightarrow ... \rightarrow an+b$$</span></p>
<p>to be a sequence of natural numbers to which we apply <span class="math-container">$3n+1$</span> or <span class="math-container">$\frac{n}{2}$</span> operations.</p>
<p>This sequence is called a cycle, if and only if <s... | Gottfried Helms | 1,714 | <p>In case you start your question from simplest state, then I can show you a "progress" in proving of nonexistence of (nontrivial) 2-step and 3-step cycle, and from this method an extension to a handful of more <span class="math-container">$n$</span>.</p>
<p>I write the transformation on <em><strong>odd</str... |
485,402 | <p>I read earlier that if $A$ is a real $3\times 3$ matrix satsifying $A^3=I$, then $A$ is similar to a matrix of form
$$
\begin{bmatrix}
1 & 0 & 0\\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta & \cos\theta
\end{bmatrix}
$$</p>
<p>From the structure theorems for normal operators, I know that ... | W. Cadegan-Schlieper | 2,957 | <p>This is not true. That $A^3=I$ implies that $A$ is diagonalizable over $\mathbf{C}$ and that its eigenvalues are cube roots of unity and thus on the unit circle.</p>
<p>And while being diagonalizable with eigenvalues on the unit circle is necessary for being unitary (over $\mathbf{R}$, orthogonal), it is not suffi... |
485,402 | <p>I read earlier that if $A$ is a real $3\times 3$ matrix satsifying $A^3=I$, then $A$ is similar to a matrix of form
$$
\begin{bmatrix}
1 & 0 & 0\\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta & \cos\theta
\end{bmatrix}
$$</p>
<p>From the structure theorems for normal operators, I know that ... | copper.hat | 27,978 | <p>Let $A=\begin{bmatrix} 1 & -\frac{3}{2} & \frac{\sqrt{3}}{2} \\ 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{bmatrix}$. Then $A^3=I$, but $A$ is not orthogonally
similar to the matrix above since the matrix above is normal whereas $A$ is not.</p>
<p><... |
1,779,314 | <p><strong>harmonic numbers</strong>
$H_n=\sum_{i=1}^{n}\frac{1}{i}$</p>
<p><strong>Euler's constant</strong></p>
<p>$\gamma=\lim_{n\to \infty} [H_n-\ln(n)]$</p>
<p><strong>Factorial</strong></p>
<p>$n!=n(n-1)(n-2)\cdots2\cdot1$; valid for all non-negative integers</p>
<p>Show that,</p>
<p>$$(s+1)\gamma-\int_0^1\... | windircurse | 331,921 | <p>First you have to choose 5 fruits for the first basket, you have $15\choose5$ combinations.
When you choose fruits for the second basket, now you have 10 fruits and you need to choose 5 again, so you have $10\choose5$ combinations.
Finally, you are left with 5 fruits and 1 basket, so there is $5\choose5$ which is 1 ... |
298,706 | <p>Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have.</p>
<p>I read on "G. Björck: Linear partial differential operators and generalized distributions" that you can construct an $f$ for whi... | Michael Renardy | 12,120 | <p>Regarding the first question, the answer is basically no. You cannot have exponential decay, because that would imply $\hat f$ is analytic, which is inconsistent with compact support.</p>
|
298,706 | <p>Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have.</p>
<p>I read on "G. Björck: Linear partial differential operators and generalized distributions" that you can construct an $f$ for whi... | Alexandre Eremenko | 25,510 | <p>Fourier transform of a function with bounded support is an entire function
of exponential type. Even in dimension 1, such function cannot have decay
as $e^{-\epsilon x}$. In fact,
$$\int_{-\infty}^\infty\frac{\log|f(x)|}{1+|x|^2}dx<\infty$$
for every such $f\in L^2$.
This condition is almost optimal, at least in ... |
22,720 | <p>Let the heat kernel on $(0,\infty)\times \mathbb R^n$ be given by </p>
<p>$\Psi(x,t) = (4\pi t )^{-\frac{n}{2}} e^{ -\dfrac{|x|^2}{4t} }$</p>
<p>for $t>0$, otherwise $0$ except at the origin of space-time.</p>
<p>It is clear that the heat kernel is smooth everywhere except at the origin. Let $\delta > 0$. H... | Zarrax | 3,035 | <p>The Fourier transform in the $x$-variables of ${\displaystyle h(x,t) = {1 \over (4\pi t)^{n \over 2}}e^{-{\vert x\vert ^2 \over t}}}$ is given by ${\displaystyle 2^{-n}e^{-\pi|\xi|^2t}}$ (The $2^{-n}$ factor may not be there depending on your normalization).</p>
<p>Taking an $x_i$-derivative of $h(x,t)$ corresponds... |
191,618 | <p>Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then
$$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$
I can prove these identies under the condition $\mathbb{E}T<+\infty$. But what about this one?</p>
| fedja | 1,131 | <p>OK, since I slipped again (there is no way I will believe nowadays that after 40 one's abilities do not decline sharply, though when I was younger I was more optimistic about the lifetime of a mathematician), let's try to figure it out.</p>
<p>We are going to use the standard trick of considering the truncated stop... |
1,002,203 | <p>Show that $\int_0^{\infty}\dfrac{\ln x}{1+x^2}\hspace{1mm}dx=0$</p>
<p>The problem I am facing is with the hint.</p>
<p>The hint says, use the substitution $u = 1/x$, it makes no sense to me.</p>
<p>Why would we use the substitution u = 1/x, its makes the integral strange.</p>
<p>Can anyone explain</p>
| lab bhattacharjee | 33,337 | <p>Using the hint setting $x=\dfrac1y,dx=-\dfrac{dy}{y^2}$</p>
<p>$$I=\int_0^\infty\frac{\ln x}{1+x^2}dx=\int_\infty^0\frac{\ln(1/y)}{1+(1/y)^2}\left(-\frac{dy}{y^2}\right)=-\int_\infty^0\frac{-\ln y}{1+y^2}dy$$</p>
<p>$$=\int_\infty^0\frac{\ln y}{1+y^2}dy$$</p>
<p>As $\displaystyle\int_a^bf(x)dx=-\int_b^af(x)dx,$</... |
496,958 | <p>I'm new to formal languages. I'm stuck with the following question. Any help is appreciated.</p>
<p>Find finite automata for </p>
<blockquote>
<p>$$L = \{a^i b^j \mid i, j\text{ are even, }j\ge0\}$$</p>
</blockquote>
<p>Thank you</p>
| J.-E. Pin | 89,374 | <p><strong>Hint</strong>. Your language is $(aa)^*(bb)^*$. Now you may apply standard algorithms to get a finite automaton accepting $L$.</p>
|
496,958 | <p>I'm new to formal languages. I'm stuck with the following question. Any help is appreciated.</p>
<p>Find finite automata for </p>
<blockquote>
<p>$$L = \{a^i b^j \mid i, j\text{ are even, }j\ge0\}$$</p>
</blockquote>
<p>Thank you</p>
| Rick Decker | 36,993 | <p>Start by making a FA to accept the $a$s: you'll need two states, the start state will be a final state and the other won't be final. Then make another FA to handle the $b$s: you'll need a machine similar to the first one. Then link your first FA to the second with an appropriate transition on input $b$. Finally, fig... |
980,875 | <p>I want to minimize the function </p>
<p>$$ \frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} $$ subject to the constraint $$x^2 + y^2 + z^2 = 1 \space\text{and} \space x,y,z > 0$$</p>
<p>Wolfram Alpha tells me that the minimum occurs at $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{3\sqrt{... | mookid | 131,738 | <p><strong>Hint:</strong> Try with polar coordinates:
$$\begin{align}
x &= \sin a \sin b\\
y &= \sin a \cos b\\
z &= \cos a\\
0&\le a,b \le \frac\pi 2
\end{align}$$</p>
|
417,855 | <p>Let <span class="math-container">$a>1$</span> and define <span class="math-container">$G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$</span> where <span class="math-container">$\Gamma$</span> is the Gamma function. This series is convergent on <span class="ma... | Carlo Beenakker | 11,260 | <p>There are closed-form expressions for all integer values of <span class="math-container">$a$</span>; for <span class="math-container">$a=\{1,2,3,4\}$</span> these read
<sub>
<span class="math-container">$$\left\{\frac{1}{1-x},e^{x/4},\frac{1}{6\Gamma(1/3)}\left[3 x \, _1F_4\left(1;\frac{2}{3},\frac{5}{6},\frac{7}{6}... |
2,973,278 | <p>In the book of <em>The theory of Functions</em> by Titchmarsh, at page 83, in the proof of the fact that analytic functions can be expressed as power series, it is stated that</p>
<p><span class="math-container">$$\frac{1}{w-z} = \frac{1}{w-a} + \frac{z-a}{(w-a)^2} + \dots \frac{(z-a)^n}{(w-a)^{n+1}} \dots ,$$</s... | José Carlos Santos | 446,262 | <p><span class="math-container">\begin{align}\frac1{w-a}+\frac{z-a}{(w-a)^2}+\frac{(z-a)^2}{(w-a)^3}+\cdots&=\frac1{w-a}\sum_{n=0}^\infty\left(\frac{z-a}{w-a}\right)^n\\&=\frac{\frac1{w-a}}{1-\frac{z-a}{w-a}}\\&=\frac1{w-a-z+a}\\&=\frac1{w-z}.\end{align}</span></p>
|
1,512,377 | <p>As usual, this is probably a lot easier than I'm making it out to be. I just don't know... where to start. I know that if I have one generator, $g$, of a cyclic group of order $m$, then all the other generators are $g^k$, where $(k,n)=1$.</p>
<p>Is the order of $\mathbb{Z}_9 \times \mathbb{Z}_{10}$... 90? Does $(1,... | Hamed | 191,425 | <p>OK, let's just check hands on! Let $(n,m)$ be an element of $\mathbb{Z}_9\times \mathbb{Z}_{10}$. We define the orbit of this element as $O(n,m)=\{(kn, km)|k\in \mathbb{Z}\}$, more intuitively just $(n,m), (2n,2m), (3m,3m), \cdots$. The orbit of $(n,m)$ is the whole group if and only if $(n,m)$ is a generator.</p>
... |
2,050,078 | <p>Is there a efficient formula to calculate this sum ? $$S = 1^1+2^2+3^3+...+n^n$$ , or it simply diverges ?</p>
| clark | 33,325 | <p>Claim $$\lim_{n\rightarrow \infty} \frac{1^1+2^2+3^3+\cdots+n^n}{(n+1)^n}= \frac{1}{e}$$</p>
<p>We will do that by using <a href="https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="nofollow noreferrer">Cesaro-Stolz</a> the discrete version of the L'Hospital's rule.</p>
<p>\begin{align*}
\lim_{n\... |
574,536 | <p>I tried this problem by counting the number of 5-digit numbers and then dividing it by 2 just like we do for three digits. Please help. Thank you.</p>
| AG. | 80,733 | <p>In how many ways can the five digit number be chosen so that the sum of digits is even? The first digit can be chosen in 9 ways (since it can't be a 0), the second in 10 ways, the third in 10 ways, the fourth in 10 ways, and the fifth in five ways (the last digit must be one of the five even digits if the first four... |
574,536 | <p>I tried this problem by counting the number of 5-digit numbers and then dividing it by 2 just like we do for three digits. Please help. Thank you.</p>
| Smriti | 135,091 | <p>Can you see how many $5$ digit nos. can be formed with nos. $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$? </p>
<p>Clearly $1$st place can't be occupied by $0$, so the $1$st place can be filled from $9$ different numbers. The rest of the places can be filled from $10$ different numbers, so the total number of five digit numbers... |
3,997,676 | <p>Let <span class="math-container">$1\leq n\in \mathbb{N}$</span>, <span class="math-container">$E=\left (e_1, \ldots , e_n\right )$</span> the standardbasis of <span class="math-container">$\mathbb{R}^n$</span>, <span class="math-container">$B=\left (b_1, \ldots , b_n\right )$</span> a basis of <span class="math-cont... | José Carlos Santos | 446,262 | <p>You can take, for instance, <span class="math-container">$f(n)=(-1)^n\left\lfloor\dfrac n2\right\rfloor$</span> (assuming, as you did, that <span class="math-container">$0\in\Bbb N$</span>).</p>
|
2,910,643 | <p>This question was in my maths paper . I tried to prove that the series converges but failed to do so. Any hint are welcomed .</p>
| Brevan Ellefsen | 269,764 | <p>$$\sum_{k = 1}^\infty\frac{1}{(n+1)(n+2)(n+3)…(n+k)} = n!\sum_{k = 1}^\infty\frac{1}{(n+k)!}$$
Which clearly converges super-exponentially. We can express this in terms of generalized Gamma function is desired.</p>
|
293,505 | <p>It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. I was wondering about the similar classification for a general compact four-manifolds possibly with boundaries or eve... | Paul Siegel | 4,362 | <p>Suppose you can classify all open 4-manifolds. In particular you can classify all manifolds of the form $M^4 - pt$ where $M^4$ is a closed 4-manifold, and consequently you can classify all closed 4-manifolds. But this classification problem reduces to the word problem on a finitely presented group (every such grou... |
3,173,766 | <p>If you have the option to roll 2 separate 10-sided dice or a single 10-sided die that you add 2 to the result with a goal of at least one die getting a specific number or higher (say 6 for the purposes of this question) is one of these options inherently better?</p>
<p>It seems that the single die with a modifier i... | Ross Millikan | 1,827 | <p>To get <span class="math-container">$6$</span> from <span class="math-container">$d10+2$</span> you need to roll at least a <span class="math-container">$4$</span>, which is <span class="math-container">$7$</span> of the <span class="math-container">$10$</span> faces. Your chances are <span class="math-container">$... |
32,091 | <p>I have a <code>list</code> of points (e.g. <code>{1.2,0.4}</code>) which I export to a file using</p>
<pre><code>Export[fileName, list];
</code></pre>
<p>I want to add a line before the end of file. For instance instead of ending the file with </p>
<pre><code>1.236677 4.64567567
1.236677 4.64567567
1.236676 4.645... | Alexey Popkov | 280 | <p>Try</p>
<pre><code>Export[fileName, Append[list, ""], "Table"]
</code></pre>
<p>This should be OS-independent although I cannot check this under OS X.</p>
|
2,861,825 | <p>The question specifies to solve this problem by maximizing $f(x,y)=xy(s-x-y)$ on an appropriate set. I admittedly do not have much of an attempt to post here. My initial thought was to try LaGrange multipliers, but I don't see how that would relate to the choice of "an appropriate set". Any advice on how to approach... | fleablood | 280,126 | <p>So if $f(x,y)=xy(s-x-y)$</p>
<p>Then if we solve for $\frac{f(x,y)}{dx}= 0$ and $\frac {f(x,y)}{dy} = 0$ we get</p>
<p>$\frac{f(x,y)}{dx} = y(s - x-y) -xy = 0$ so $ys - y^2 = 2xy$. The occurs if $y=0$ or if $s-y = 2x$</p>
<p>$\frac{f(x,y)}{dy} = x(s-x-y) - xy = 0$ so $sx-x^2 = 2xy$. This occurs if $x=0$ or $s- ... |
201,103 | <p>When comparing datasets using <code>DistributionChart</code>, is it possible to add a horizontal line to the chart that lies over all of the individual distributions within the chart?</p>
| Albert Retey | 169 | <p>Just for completness, and in case anyone has need for even more general additions: most chart and plot functions have the <code>Prolog</code> and <code>Epilog</code> options for exactly that. They are in some sense just a shortcut to the <code>Show</code> as in MeloaGo's answer, but they are quite handy to add litte... |
1,140,339 | <p>Steps I took:</p>
<p>I drew out a circle with a radius of 1 and drew a trapezoid inscribed in the top portion of it. I outlined the rectangle within the trapezoid and the two right triangles within it. This allowed me to come to the conclusion, using the pythagorean theorem, that the height of the trapezoid is $h=\... | Hagen von Eitzen | 39,174 | <p>You may even ask</p>
<blockquote>
<p>What is the largest quadrilateral inscribed in a semicircle, given that one side is the diameter?</p>
</blockquote>
<p>If this maximal quadrilateral happens to be a trapezoid (which will be the case), this also answers the original question.
Complete the figure by refletcing ... |
57,976 | <p>I'm trying to show:</p>
<p>If $K\subset \mathbb{R}^n$ is connected and $x\in \mathbb{R}^n$ then $x+K=\{x+y\in \mathbb{R}^n: y\in K\}$ is connected.</p>
<p>Thanks for your help.</p>
| Brian M. Scott | 12,042 | <p>Fix $y \in \mathbb{R}^n$ and let $f:\mathbb{R}^n \to \mathbb{R}^n:x \mapsto x+y$; it’s easy to check that $f$ is continuous and that $f[K] = y + K$. Continuous functions preserve connectedness, so $y + K$ is connected. (For that matter, it’s easy enough to check that $f$ is a homeomorphism, which makes the conclusio... |
1,869,843 | <p>We know that:
$$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ </p>
<p>My question is there other examples that satisfies:
$$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$</p>
| abnry | 34,692 | <p>We can write</p>
<p>$$\sum^n_{k=1} k^a = \frac{1}{a+1} n^{a+1} + O(n^a).$$</p>
<p>The easiest way to convince yourself of this is to use a comparison test with an integral.</p>
<p>Thus, for the following to hold
$$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b,$$
we need the highest order terms in the polynomia... |
4,235,927 | <p>There is a game where you are asked to roll two fair six-sided dice. If the sum of the values equals 7, then win £21. However, must pay £5 to play each time both dice are rolled. Do you play this game?</p>
<p>One way to think about this is that getting a 7 comes with 1/6 chance, and to make money we need to get 7 at... | Milo Brandt | 174,927 | <p>You are correct about the <em>probabilities</em> involved here. The probability of rolling a total of <span class="math-container">$7$</span> on a pair of fair dice is indeed <span class="math-container">$1/6$</span>. Likewise, the probability of rolling a <span class="math-container">$7$</span> <em>at least once</e... |
1,101,520 | <blockquote>
<p>Evaluate the integral
$$\int\frac{5x^3+3x-1}{(x^3+3x+1)^3}\,dx$$</p>
</blockquote>
<p><strong>My Attempt:</strong></p>
<p>Let $f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now differentiate both side with respect to $x$, and we get</p>
<p>$$
\begin{align}
f'(x) &= \frac{(x^3+3x+1)^2\cdot a-2\cdot (x^3+... | user149844 | 149,844 | <p>$$
\begin{aligned}\int\frac{5x^3 + 3x - 1}{\left(x^3 + 3x + 1\right)^3}\,\mathrm{d}x &=\int\frac{5x^3 + 3x-1}{\left(\sqrt{x}\left(x^{5/2}+3\sqrt{x}+x^{-1/2}\right)\right)^3}\,\mathrm{d}x\\&=\int\frac{5x^3 + 3x-1}{x^{3/2}\left(x^{5/2}+3\sqrt{x}+x^{-1/2}\right)^3}\,\mathrm{d}x\\&=\int\frac{5x^{3/2}+3x^{-1/... |
1,505,859 | <p>Some days ago I presented some exercises to a bunch of engineers. One of the exercises involved was: Use $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}$ to show that $\lim_{n \to \infty} \left(1 + \frac{\sqrt{n}}{n}\right)^{-n} = 0$. I've done this by showing that $\left(1 + \frac{\sqrt{n}}{n}\right)^... | Jochen | 38,982 | <p>If $X$ and $Y$ are Banach spaces continuity follows from the closed graph theorem. In the normed case you have to show that $f$ maps bounded sets to bounded sets and this follows from the weak continuity since the norm topology and the weak topology on $Y$ have the same bounded sets.</p>
|
2,595,650 | <p>Lets say we want to try to show $f(n) \in O(g(n))$, which may or may not be true.</p>
<p>Then by definition we find a large enough $n$ and a constant $k > 0$ such that $|f(n)| \le k|g(n)|$, but lets say $$\lim_{n \to \infty}\frac{|f(n)|}{|g(n)|} = \infty$$ what can I conclude from this? That $f(n) \not\in O(g(n)... | operatorerror | 210,391 | <p>Note that the statement
$$
\lim_{n\to \infty}\frac{|f(n)|}{|g(n)|}=\infty
$$
means that for any $M>0$ we can find a large enough $n$ such that
$$
\frac{|f(n)|}{|g(n)|}>M\implies|f(n)|>M|g(n)|
$$
Can you see why this is a problem for a claim that $f\in O(g)$?</p>
|
4,363,566 | <p>I am trying to prove Euler's equation for the case of <span class="math-container">$4x$</span>, that is:
<span class="math-container">$$e^{i4x} = \cos 4x+i \sin 4x.$$</span>
I can get the imaginary part in a form that is equivalent to <span class="math-container">$\sin 4x$</span>, but I am struggling with the real p... | Henry | 6,460 | <p>Use <span class="math-container">$\cos(2x)= \cos^2(x)-\sin^2(x)$</span> twice:</p>
<p><span class="math-container">$$\cos(4x) \\= (\cos(2x))^2-(\sin(2x))^2 \\= (\cos^2(x)-\sin^2(x))^2-(2\cos(x)\sin(x))^2 \\= \cos^4(x) - 2\cos^2(x)\sin^2(x)+\sin^4(x) - 4\cos^2(x)\sin^2(x) \\= \cos^4(x) - 6\cos^2(x)\sin^2(x)+\sin^4(x)... |
3,276,351 | <blockquote>
<p>A particle is allowed to move in the <span class="math-container">$\mathbb{Z}\times \mathbb{Z}$</span> grid by choosing any of the two jumps:</p>
<p>1) Move two units to right and one unit up</p>
<p>2) Move two units up and one unit to right.</p>
<p>Let <span class="math-container">$P=(... | Parcly Taxel | 357,390 | <p>Both possible moves increase the sum of coordinates by <span class="math-container">$3$</span>, so any reachable position must have the sum of coordinates a multiple of <span class="math-container">$3$</span>. This means <span class="math-container">$Q$</span> is not reachable, since its sum of coordinates is <span ... |
203,955 | <p>MMA 12 apparently has a new package, <code>HDF5Tools</code>, which includes desperately needed functions that allow one to open an HDF5 file, get data from it, then close it when done. I don't really <em>know</em> this to be true, but I think it is, because when I type , for instance, "HDF5O" MMA provides me with a... | Jason B. | 9,490 | <p>I believe this package contains internal functions, used by <code>Import</code> when working with HDF5 files. These functions have no documentation, and aren't meant to be called from top level.</p>
<p>If you evaluate </p>
<pre><code>Needs["HDF5Tools`"];
?HDF5Tools`*
</code></pre>
<p>and click any of the symbols... |
203,955 | <p>MMA 12 apparently has a new package, <code>HDF5Tools</code>, which includes desperately needed functions that allow one to open an HDF5 file, get data from it, then close it when done. I don't really <em>know</em> this to be true, but I think it is, because when I type , for instance, "HDF5O" MMA provides me with a... | Albert Retey | 169 | <p>Considering the question about the documentation Jasons answer makes clear that there basically is none (except for the usage messages) and the <code>HDF5Tools` </code> package at this time (2019) unfortunately seems to be meant for internal use only.</p>
<h1>Documented Import/Export Functionality</h1>
<p>You have b... |
65,260 | <p>Trying to learn how <code>Opacity</code> works, I found a possible issue.
Generating points in a square spiral with:</p>
<pre><code>sqrSpiral[rev_, inc_] := Module[{x, y, z, dx, dy, list = {}},
x = y = z = 0;
dx = {0, 1, 0, -1}; dy = {1, 0, -1, 0};
Do[
Do[
list = Append[list, {x += dx[[i]], y += dy[[i... | MathLind | 12,845 | <p>I got an answer from Wolfram Technical Support today. They had the following to say so far:</p>
<p><strong>"Our developers are aware of the problem and working to correct it."</strong></p>
<p>Short and concise.</p>
|
1,582,742 | <blockquote>
<p>I have this so easy question, yet it is giving me a headache</p>
<p>Consider the following two subsets of <span class="math-container">$R^3$</span></p>
<p><span class="math-container">$S_2 = [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]$</span></p>
<p><span class="math-container">$S_3 = [(1,0,-1),(1,0,1)]$</span... | Cameron Buie | 28,900 | <p>A set of vectors can be infinite and <em>still</em> not span $\Bbb R^n.$ For example, consider the set of all vectors in $\Bbb R^2$ of the form $(x,0)$.</p>
<p>n order to determine whether $S_2$ spans $\Bbb R^3,$ you should determine the span of $S_2,$ or try to find three linearly-independent vectors in $S_2.$</p>... |
29,344 | <p>I am working through Ireland/Rosen at the moment, and I cannot solve a (probably) simple exercise. Any nudges you can give me in the right direction are appreciated.</p>
<p>How does one show that $x^2 \equiv 1 \pmod{p^{\ell}}$ has only two solutions (namely $\pm 1$)? Here, $\ell$ is a positive integer (which we c... | André Nicolas | 6,312 | <p>Let $p$ be an odd prime. Then $p^{\ell}$ can divide at most one of $x-1$ and $x+1$. (Note that the argument breaks down if $p=2$, and indeed in that case there can be more than 2 solutions.)</p>
|
29,344 | <p>I am working through Ireland/Rosen at the moment, and I cannot solve a (probably) simple exercise. Any nudges you can give me in the right direction are appreciated.</p>
<p>How does one show that $x^2 \equiv 1 \pmod{p^{\ell}}$ has only two solutions (namely $\pm 1$)? Here, $\ell$ is a positive integer (which we c... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\,\ p\,$</span> prime, <span class="math-container">$\, (p,a,b) = 1,\ p^n\mid ab\ \Rightarrow\ p^n\ |\ a\ $</span> or <span class="math-container">$\ p^n\ |\ b\ $</span> by iterating Euclid's Lemma.</p>
<p>Note that <span class="math-container">$\ (p,x\!-\!1,... |
1,619,208 | <blockquote>
<p>A person walks downhill at 10 km/h, uphill at 6 km/h and on the plane at 7.5 km/h. If the person takes 3 hours to go from a place A to another place B, and 1 hour on the way back, the distance between A and B is
$\begin{array}1 1. 15 km && 2. 23.5 km \\ 3. 16 km && 4. \text{Given dat... | Em. | 290,196 | <p>Completing the square gives,
$$\frac{k!}{(k-2)!} = k(k-1) = k^2-k+\frac{1}{4} = 2!(45)+\frac{1}{4}.$$</p>
<p>This gives
\begin{align*}
\left(k-\frac{1}{2}\right)^2 &= \frac{361}{4}\\
k-\frac{1}{2}&=\sqrt{\frac{361}{4}}\\
k &= \frac{19}{2}+\frac{1}{2}\\
k&=10
\end{align*}</p>
|
778,253 | <p>A satellite system consists of 4 components and the system can function if at
least 2 components are working. If each component independently works with probability
0.8, what is the probability the system will function? </p>
<p>Can anyone help?</p>
| Leif Sabellek | 109,119 | <p>The system will not work, if 3 or 4 components are not working. We can calculate the probability for that:</p>
<p>$P($all components are broken$) = 0.2^4 = 0.0016$</p>
<p>$P($3 components are broken$) = {4 \choose 3} \cdot 0.2^3 \cdot 0.8^1 = 0.0256$</p>
<p>So the chance that the system is not running is $0.0016 ... |
436,316 | <p>Let <span class="math-container">$f\colon\mathbb R^2\to\mathbb R$</span> be a measurable function such that
<span class="math-container">\begin{equation*}
F(t):=\int_{\mathbb R}dx\,f(t,x)
\end{equation*}</span>
exists and is finite for all real <span class="math-container">$t$</span>. Suppose that
<span class="... | Kostya_I | 56,624 | <p>A simple example is given by
<span class="math-container">$$
f(t,x)=\cases{\exp(-(x-t^{-2})^2),&$t\neq 0$,\\0,&$t=0.$}
$$</span>
For each fixed <span class="math-container">$t\neq 0$</span>, <span class="math-container">$\int f(t,x)\,dx$</span> is a Gaussian integral equal to <span class="math-container">$\s... |
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | mathmandan | 198,422 | <p>Imagine that you're one of the $10$ players. In both cases you need to determine who your $4$ teammates will be, out of the other $9$ players. This can be done in $\binom{9}{4} = 126$ ways.</p>
<p>Now, if the two teams are treated differently, like one team gets to play while the other team has to sit on the benc... |
3,458,467 | <p>Given a matrix <span class="math-container">$A \in M_{2}(\mathbb{Q}) = R$</span>. I don't believe that <span class="math-container">$A$</span> is similar to <span class="math-container">$-A$</span>, but I just want to check. Also, is the only similarity class of matrices in <span class="math-container">$R$</span> wi... | user100101212 | 709,466 | <p>Actually I am wrong, the claim is not necessarily true, a counter example is <span class="math-container">$t = \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right)$</span>, and <span class="math-container">$x = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right)$</span>. Here <span class... |
1,905,397 | <p>I want to know if the popular Sudoku puzzle is a Cayley table for a group. </p>
<p>Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of elements, rows and columns, the solutions are bigger than th... | Jyrki Lahtonen | 11,619 | <p>It is easy to construct a Sudoku that cannot be a Cayley table. <strong>Not even if you label the rows and columns independently from each other.</strong> Consider the following.
$$
\begin{array}{ccc|ccc|ccc}
1&2&3&4&5&6&7&8&9\\
4&5&6&7&8&9&1&3&2\\
7&am... |
3,930,878 | <p>Can someone show me how to solve <span class="math-container">$$\arg z^3 = 0$$</span>
I don't know even how to start because of the zero.</p>
| user2661923 | 464,411 | <p>Hint:</p>
<p>First prove that given non-zero complex numbers <span class="math-container">$z,w$</span> that Arg<span class="math-container">$(zw)$</span> = Arg<span class="math-container">$(z) +~$</span> Arg<span class="math-container">$(w),~$</span> within a modulus of <span class="math-container">$(2\pi)$</span>.<... |
273,020 | <p>I want to reproduce the effect in the figure below, approximating a surface with tangent planes,
<a href="https://i.stack.imgur.com/sfjgz.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/sfjgz.gif" alt="enter image description here" /></a><br />
Now I can only use polygons to approximate inside the surface,... | Daniel Huber | 46,318 | <p>To get an approximation to your surface we first need the function and its gradient;</p>
<pre><code>f[{x_, y_}] = (1 - x^2 - y^2);
grad[{x_, y_}] = Grad[f[{x, y}], {x, y}];
</code></pre>
<p>Then we create a 2D grid of points {x,y}, where the tangent planes will touch the surface:</p>
<pre><code>centers = Flatten[Tab... |
3,328,969 | <blockquote>
<p>Let <span class="math-container">$a,b \geq 0$</span> with <span class="math-container">$a+b \leq c$</span> for a small constant <span class="math-container">$c$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>.</p>
<p>What is the minimum of <span class="... | C.F.G | 272,127 | <p>This is not an answer but it strengthen your guess. Suppose <span class="math-container">$a+b=2k\leq c$</span> and let <span class="math-container">$a=k-\varepsilon$</span> and <span class="math-container">$b=k+\varepsilon$</span> then:
<span class="math-container">$$f(a,b) :=\cos(a)+\cos(b)=2\cos(k)\cos(\varepsilon... |
3,328,969 | <blockquote>
<p>Let <span class="math-container">$a,b \geq 0$</span> with <span class="math-container">$a+b \leq c$</span> for a small constant <span class="math-container">$c$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>.</p>
<p>What is the minimum of <span class="... | DanielWainfleet | 254,665 | <p><span class="math-container">$\cos x$</span> is strictly decreasing and positive for <span class="math-container">$x\in [0,1].$</span> So <span class="math-container">$\cos a +\cos b$</span> cannot be a minimum if <span class="math-container">$a<c-b$</span> because then <span class="math-container">$\cos a +\cos ... |
3,372,488 | <p>Variance is</p>
<p><span class="math-container">$$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$</span></p>
<p>But why square the difference? Why not cube it, or any other exponent?</p>
<p>Related <a href="https://math.stackexchange.com/questions/717339/why-is-variance-squared/717355">question</a>.</p>
| Asad | 715,396 | <p>Still, I am not convinced by this answer.
The naming convention can be separate topics to discuss.
Here what is the issue if we take cubes or quad etc?</p>
|
3,620,409 | <blockquote>
<p>If a matrix and its transpose both have the same eigenvectors, is it necessarily symmetric?</p>
</blockquote>
<p>It's clear to see how if <span class="math-container">$A$</span> = <span class="math-container">$A^T$</span>, they would have the same eigenvectors, but is it the only way? And how would y... | quasi | 400,434 | <p>Suppose <span class="math-container">$A$</span> is a nonzero square matrix such that <span class="math-container">$A^T=-A$</span>.
<p>
Then <span class="math-container">$A$</span> and <span class="math-container">$A^T$</span> have the same eigenvectors, but <span class="math-container">$A$</span> is not symmetric.
<... |
3,620,409 | <blockquote>
<p>If a matrix and its transpose both have the same eigenvectors, is it necessarily symmetric?</p>
</blockquote>
<p>It's clear to see how if <span class="math-container">$A$</span> = <span class="math-container">$A^T$</span>, they would have the same eigenvectors, but is it the only way? And how would y... | user1551 | 1,551 | <p>If you mean that <span class="math-container">$A\in M_n(\mathbb R)$</span> and its transpose share a common eigenbasis <span class="math-container">$\mathcal B\subset\mathbb R^n$</span>, then <span class="math-container">$A$</span> must be symmetric.</p>
<p>By definition, for any <span class="math-container">$v\in\... |
146,054 | <p>Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries. </p>
<p>We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$. </p>
<p>I would like to obtain a lower bound for $\min_i (\max_j x_{ij})$ that holds with probability at least $1-\del... | Henry | 6,460 | <p>You know that </p>
<p>$$\Pr(\max_j x_{ij} \le k) = \Phi(k)^p$$ </p>
<p>so </p>
<p>$$\Pr(\max_j x_{ij} \gt k) = 1 - \Phi(k)^p$$ </p>
<p>so </p>
<p>$$\Pr(\min_i (\max_j x_{ij}) \gt k) = \left(1 - \Phi(k)^p\right)^n $$</p>
<p>so </p>
<p>$$\Pr(\min_i (\max_j x_{ij}) \le k) = 1- \left(1 - \Phi(k)^p\right)^n $$</p>... |
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