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 |
|---|---|---|---|---|
3,694,719 | <p>source: BMO2 2004 q4</p>
<p>The real number <span class="math-container">$x$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span> has decimal representation <span class="math-container">$0.a_1a_2a_3a_4\dots$</span>
And enjoys the following property: the number of distinc... | QC_QAOA | 364,346 | <p>We prove a more general statement: if a real number <span class="math-container">$x$</span> between <span class="math-container">$0$</span> and <span class="math-container">$1$</span> has decimal representation <span class="math-container">$x=0.a_1a_2a_3a_4\dots$</span> (in base <span class="math-container">$b$</spa... |
4,011,999 | <p>Let <span class="math-container">$n$</span> be a positive integer, and consider a sequence <span class="math-container">$a_1 , a_2 , \cdots , a_n $</span> of positive integers. Extend it periodically to an infinite sequence <span class="math-container">$a_1 , a_2 , \cdots $</span> by defining <span class="math-conta... | Abhinandan Saha | 854,039 | <p>Notice that in the range <span class="math-container">$a_1$</span> to <span class="math-container">$a_1+n$</span> there are <span class="math-container">$(a_1+n)-(a_1)+1=n+1$</span> positive integers
Hence from the inequality <span class="math-container">$a_1\leq a_2 \leq a_3 \leq ... \leq a_n \leq a_1+n$</span> we ... |
453,502 | <p>Let $I$ be a generalized rectangle in $\Bbb R^n$ </p>
<p>Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$</p>
<p>Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$. </p>
<hr>
<p>My idea is that</p>
<p>For $(\impliedby)$</p... | Kunnysan | 84,764 | <p>Suppose $f(x)>0$ for some $x\in I$, as $f$ is continuous $\exists\ \delta>0$ such that $\forall y\in B_\delta(x)\cap I$, we will have $f(y)>0$. Then, as $f(x)\ge 0$</p>
<p>$$\displaystyle\int_If\ge\displaystyle\int_{B_\delta(x)\cap I}f>0 \implies\text{contradiction}$$.</p>
|
62,621 | <p>There are three elements: x, y, z and a relation C:</p>
<p>x C y, y C z, z C x, x C x, y C y, z C z.</p>
<p>Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e. </p>
<p>x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z </p>
... | Bjørn Kjos-Hanssen | 4,600 | <p>It sounds like you are describing a situation where $a$ is more true than $b$, $b$ is more true than $c$, but nevertheless $c$ is more true than $a$. I am not sure about the best starting point in looking for relevant references, but maybe <i><a href="http://en.wikipedia.org/wiki/Arrow%2527s_impossibility_theorem" r... |
2,166,847 | <p>Let $$v_n=\dfrac 1 {n+1}\sum_{k=0}^n \dfrac 1 {k+1}$$
We wanna study the sum $$S=\sum_{n=0}^{+\infty}(-1)^n v_n$$</p>
<p>The problem says we should first find $\omega(x)$ s.t.
$$v_n=\int_0^1 x^n\omega(x)dx$$
Then we'll have $S=\int_0^1\dfrac {\omega(x)} {1+x}dx$, but I can't find such $\omega(x)$. What's the idea ... | Simply Beautiful Art | 272,831 | <p>Recall the geometric series:</p>
<p>$$\sum_{k=0}^nx^k=\frac{1-x^{n+1}}{1-x}$$</p>
<p>Integrate both sides from zero to one,</p>
<p>$$\sum_{k=0}^n\frac1{k+1}=\int_0^1\frac{1-x^{n+1}}{1-x}\ dx$$</p>
<p>It thus follows that</p>
<p>$$v_n=\int_0^1\frac{1-x^{n+1}}{(n+1)(1-x)}\ dx$$</p>
<p>Apply integration by parts ... |
2,049,714 | <p>Does it make sense when people say "statistically impossible"?</p>
| Community | -1 | <p>A statistical impossibility is a probability that is so low as to not be worthy of mentioning. Sometimes it is quoted as $10^{-50}$ although the cutoff is inherently arbitrary. Although not truly impossible the probability is low enough so as to not bear mention in a rational, reasonable argument.</p>
<p>In some ca... |
2,049,714 | <p>Does it make sense when people say "statistically impossible"?</p>
| Vincent | 332,815 | <p>It can mathematically make sense. You look like you are thinking about discrete probabilities. In continuous probabilities, you define what is called a density function, and whenever it is finite in some value $x$, the probability of picking $x$ is null.</p>
<p>If you are not familiar with the concept, consider $[0... |
2,049,714 | <p>Does it make sense when people say "statistically impossible"?</p>
| Larry Hill | 570,604 | <p>Consider the exponential equation, which never touches the value 0, but at what point would it be considered impossible or improbable to occur? This equation describes radioactive decay and so at what point can one safely say there is no more radiation coming from the material? IF you knew the exact number of atom... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Jose Arnaldo Bebita Dris | 28,816 | <p>You can check out the following books for "detailed explanations and progressive practice problems":</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/B00WLI2AKW" rel="nofollow">Schaum's Outline of Review of Elementary Mathematics</a></p>
<p><a href="http://rads.stackoverflow.com/amzn/click/B00UYDF4UO" rel=... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Asinomás | 33,907 | <p>The books Stewart's pre-calculus, mathematical Ideas and Stewart's calculus contain all of the material that is assumed to be needed to enter college and more. I used all of those during elementary and high school.</p>
<p>However I also recommend that you go to practice contest type questions because they are good f... |
3,412,217 | <p>The category of abelian groups <span class="math-container">$\mathbb{Ab}$</span> has a monoidal (closed) structure <span class="math-container">$(\otimes, \mathbb{Z})$</span>. Moreover, it is monadic over the category of sets via the free abelian group monad <span class="math-container">$$\mathbb{Z}[\_]: \text{Set} ... | Manx | 483,923 | <blockquote>
<p>Theorem</p>
<p>Suppose diophanitine equation <span class="math-container">$ax+by=c$</span> has solution <span class="math-container">$\{x_0,y_0\}\subset\mathbb{Z}$</span></p>
<p><span class="math-container">$$\text{And } d:=\gcd(a,b)\mid c$$</span></p>
<p><span class="math-container">$$\text{Then }\fora... |
164,871 | <p>i want to choose optimal decision from following problem
Imagine having been bitten by an exotic, poisonous snake. Suppose the ER
physician estimates that the probability you will die is $1/3$ unless you receive
effective treatment immediately. At the moment, she can offer you a choice of
experimental antivenins fro... | Jack D'Aurizio | 44,121 | <p>An elementary proof, too: if $p>5$, at least one residue class among $\{2,5,10\}$ must be a quadratic residue, since the product of two quadratic non-residues is a quadratic residue. But every element of the set $\{2,5,10\}$ is a square-plus-one, giving at least a couple of consecutive quadratic residues among $(... |
3,288,534 | <p><span class="math-container">$7^{-1} \bmod 120 = 103$</span></p>
<p>I would like to know how <span class="math-container">$7^{-1} \bmod 120$</span> results in <span class="math-container">$103$</span>.</p>
| lab bhattacharjee | 33,337 | <p>Like </p>
<p><a href="https://math.stackexchange.com/questions/257127/how-to-convert-a-diophantine-equation-into-parametric-form">How to convert a diophantine equation into parametric form?</a> OR </p>
<p><a href="https://math.stackexchange.com/questions/407478/solving-a-linear-congruence/407486#407486">Solving a ... |
2,461,773 | <p>I would like some clarification about the Cantor Set:</p>
<ul>
<li>What are the elements in the Cantor Set?</li>
<li>How do I write the Cantor Set in mathematical terms (i.e in a summation)? I have seen online a formula but I do not understand how they got it so would be grateful if you could explain why it is this... | Community | -1 | <p>That formula isn't all that helpful. The most useful property is $\mathcal C= \frac{\mathcal C}3\cup\left(\frac23+\frac{\mathcal C}3\right)$, i.e. the Cantor set is the union of two subsets obtained by shrinking/shifting of the Cantor set. That's called self-similarity.</p>
|
29,601 | <p>I know the order of the group is the number of elements in the set. For example the group of $U_{10}$ (units of congruence class of 20) has order 4. </p>
<p>Major Edit, kinda changed the question.
Lets say my element $a$ has a finite order $n$. Then what is the order of $a, a^2, a^3...a^{11}$?</p>
| Bill Dubuque | 242 | <p>Suppose that <span class="math-container">$\:a\:$</span> has order <span class="math-container">$\:n\:.\:$</span> To compute the order of <span class="math-container">$\:a^j\:$</span> we may proceed efficiently as follows</p>
<p><span class="math-container">$$ a^{jk}\! = 1\iff n\mid jk \iff n\mid jk,nk\iff n\mid (jk... |
1,835,295 | <blockquote>
<p>What is $\gcd(12345,54321)$?</p>
</blockquote>
<p>I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an easy way to calculate such greatest common divisors.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\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{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
275,151 | <p><a href="https://i.stack.imgur.com/XxGxa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XxGxa.png" alt="enter image description here" /></a>I'm a beginner at Mathematica, and I'm trying to figure out how to fill between two lines horizontally. Consider the toy example</p>
<pre><code>Plot[{Conditi... | Daniel Huber | 46,318 | <p>Something like:</p>
<pre><code>dat1 = Table[ {i, RandomReal[{-1, 1}]}, {i, 10}];
dat2 = Table[ {i, 2 + RandomReal[{-1, 1}]}, {i, 2, 12}];
ListLinePlot[{dat1, dat2}, Filling -> {1 -> {2}}]
</code></pre>
<p><a href="https://i.stack.imgur.com/41l01.png" rel="noreferrer"><img src="https://i.stack.imgur.com/41l01.p... |
1,989,229 | <p>My approach : I tried using integral calculus and using infinite geometric series..however it didn't match..any trick?</p>
| kotomord | 382,886 | <p>F = 1 + x + x^2 + ... </p>
<p>Your sum is
F + x*F + x^2*F + x^3 * F ...</p>
|
2,668,538 | <p>Firstly I'm aware of the proofs/reasons regarding .999...=1. I'm not asking for anyone to reference or reiterate them but rather to look at my proof in isolation and help me understand my own mistakes and fallacies.</p>
<p>Another disclaimer I suppose; it's difficult to call this 'proof' my 'own' as it's extremely ... | law-of-fives | 440,408 | <p>There is not just one point at which you make some kind of error. You're exploring an idea and trying some notation out and it seems ok here and there. But the question is really one that starts even sooner. </p>
<blockquote>
<p>What do you mean by $0.999\cdots$?</p>
</blockquote>
<p>If you resolve this in one w... |
4,382,739 | <p>I’ve been given the joint density function: f<span class="math-container">$_X$$_,$$_Y$</span>(x,y)=C when (X,Y) is uniform over [-1,1]<span class="math-container">$^2$</span>. I’ve been tasked with finding P{|2X+Y|<span class="math-container">$\le$</span>1} and P{X=Y} however I’m stuck in my question, I’ve deduced a... | Graham Kemp | 135,106 | <p>You have a uniform distribution over a <span class="math-container">$2{\times}2$</span> square; specifically the <span class="math-container">$[-1,1]^2$</span> square. Probabilities of events within this space can be measured graphically; just compare the areas covered by the events.</p>
<p>Plot the lines <span cla... |
96,198 | <p>I'm trying to prove that the variance of a RV whose values are discrete 1's or 0's is greater than the variance of a RV who's values are 0's or continuous on the domain (0,1], where any "1" in the Bernoulli RV corresponds to a value on (0,1] in the other RV. Intuitively, I think this is the case, but I'm trying to d... | Dilip Sarwate | 15,941 | <p>If $A$ and $B$ are jointly normal random variables, then the conditional density of $A$ given $B = b$ is a normal density with mean and variance as follows:
$$E[A\mid B = b] = \mu_A + \left.\left.\frac{\text{cov}(A,B)}{\sigma_B^2}
\right(b - \mu_B\right), ~~
\text{var}(A\mid B = b) = \frac{\sigma_A^2\sigma_B^2
... |
203,627 | <p>I initially asked this question on <a href="https://math.stackexchange.com/q/1219052/39599">MSE</a> but I haven't had any luck.</p>
<hr>
<p>The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous... | Igor Rivin | 11,142 | <p>According to <a href="http://www.rac.es/ficheros/doc/00128.pdf" rel="noreferrer">this paper by Michael Langenbruch</a>, this was proved by none other than H. Whitney. The paper has lots of references.</p>
|
25,556 | <p>Why is the zero set in <span class="math-container">$\mathbb{C}\times\mathbb{C}$</span> of a polynomial <span class="math-container">$f(x,y)$</span> in two complex variables always non-discrete (no zero of <span class="math-container">$f$</span> is isolated)?</p>
| Zarrax | 3,035 | <p>This actually holds for any analytic function of two complex variables on an open set. Suppose $f(x_0,y_0) = 0$. Then viewed as a function of $y$, $f(x_0,y)$ must have an isolated zero of some order $k$ at $y = y_0$. By complex analysis (the residue theorem will work for example), for some small $r$ one has
$${1 \o... |
943,535 | <p>Let $E,F$ two $\mathbb{K}$ vector spaces, $u\in\mathcal{L}(E,F)$.</p>
<p>a) Show that there exists $v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$</p>
<p>b) Can we additionally have $v\circ u\circ v=v$ ?</p>
<hr>
<p>I have asked in the past <a href="https://math.stackexchange.com/questions/836438/existence-... | Ben Grossmann | 81,360 | <p>As with the last question you asked, the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse" rel="nofollow">Moore-Penrose pseudoinverse</a> will work here as long as $E$ and $F$ are finite dimensional inner product spaces.</p>
<p>Without using this, we could still make do, at least in the fin... |
3,600,065 | <p><strong>Problem</strong></p>
<p>Solve the system:</p>
<p><span class="math-container">$$
\begin{align}
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1982-t_0 \right ) \right )&=1.5 \\
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1984-t_0 \right ) \right )&=1 \\
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1985-t_0 \right ) \right ... | Blue | 409 | <p>To keep track of things, help spot errors, and avoid having to re-do everything if (when?) values change, it's often better to obscure distracting specific values (especially messy decimals that may not be exact). So, let's consider the system in this form:
<span class="math-container">$$\begin{align}
y + z \cos(\th... |
2,270,019 | <p>Is $\displaystyle \int _ a^ b \left(\int_c^d f_{X,Y}(x,y)\, dx\right)\, dy$</p>
<p>a) $\displaystyle \int _ a^ b dy \int_ c^ d f(x,y)\,dx$</p>
<p>or</p>
<p>b) $\displaystyle \int _ a^ b dx \int_ c^ d f(x,y)\,dy$</p>
<p>?</p>
| Jason | 195,308 | <p>A function is integrable if <em>both</em> positive and negative integrals are finite. A function can integrate to $\infty$ or $-\infty$, but we do not call such a function integrable.</p>
|
3,434,656 | <p>We are given a <span class="math-container">$3 \times 3$</span> real matrix <span class="math-container">$A$</span>, and we know it has three eigenvalues. One eigenvalue is <span class="math-container">$\lambda_1=-1$</span> with corresponding eigenvector <span class="math-container">$v_1=\left[\begin{matrix}
0 \... | Srivatsan Ramesh | 724,666 | <p>Sum of the Eigen values is equal to the trace(sum of the diagonal elements) of the matrix A.
Since you are aware of <strong>v1</strong> and <strong>v2</strong>, you can easily find the third Eigen value.</p>
<p>For the third Eigenvector,<span class="math-container">$Av_3=\lambda_3v_3$</span>
You are aware of <stron... |
4,482,285 | <p>I'm currently making a game and have run into a problem I'm not quite sure how to solve, I'll try to lay it out as a maths question. None of the values are fixed, so I'm looking for an equation that solves the below question:</p>
<p>A plane lies on the position vector <code>p0</code> <code><x0, y0, z0></code> ... | Cesareo | 397,348 | <p>Given</p>
<p><span class="math-container">$$R, \ \
\cases{
\vec p_0 = (x_0,y_0,z_0)\\
\vec n = (n_x,n_y,n_z)\\
\vec s = (s_x,u,s_z)\\
}
$$</span></p>
<p>determine <span class="math-container">$\vec t=(t_x,t_y,t_z),\ u,\ \lambda$</span> such that</p>
<p><span class="math-container">$$
\cases{
(\vec t-\vec p_0)\cdot \... |
1,468,070 | <p>Let <span class="math-container">$\{a_n\}_1^\infty$</span> and <span class="math-container">$\{b_n\}_1^\infty$</span> be two sequences in <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$\forall n \in \mathbb{N}$</span>, it is true that <span class="math-container">$a_n \leq b... | Learnmore | 294,365 | <p>It is clear that $a_n$ is a monotonically increasing sequence bounded above by $b_1$.Hence by Monotone Convergence Theorem $a_n\to r$ (say )</p>
<p>Since you have already proved that $a_n\leq b_n\forall n\in \mathbb N$ it follows that $r\leq b_n\forall n$ </p>
<p>Hence $a_n\leq r\leq b_n$</p>
|
1,468,070 | <p>Let <span class="math-container">$\{a_n\}_1^\infty$</span> and <span class="math-container">$\{b_n\}_1^\infty$</span> be two sequences in <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$\forall n \in \mathbb{N}$</span>, it is true that <span class="math-container">$a_n \leq b... | KHOOS | 637,368 | <p>First we note that <span class="math-container">$b_1$</span> is an upper bound for <span class="math-container">$(a_n)$</span>. Since <span class="math-container">$(a_n)$</span> is monotone increasing, the Monotone Convergence Theorem states that <span class="math-container">$(a_n)$</span> converges to its supremum ... |
27,759 | <p>Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?</p>
<p>Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can... | Did | 6,179 | <p>Remember that:</p>
<blockquote>
<p>Matrices act on vectors.</p>
</blockquote>
<p>Here the linearly independent vectors $u=\begin{pmatrix} 1\\ 1\end{pmatrix}$ and $v=\begin{pmatrix} 1\\ -1\end{pmatrix}$ are such that $Au=5u$ and $Av=-v$. </p>
<p>Hence a suitable $C=\begin{pmatrix} a & b\\ c& d\end{pmatri... |
3,946,998 | <p>I was doing a question. Suddenly I got stuck at this last part of the problem. It was to prove
<span class="math-container">$ 2^r +2 = a^2 +b^2$</span> where <span class="math-container">$r \neq 2$</span>, r is a prime and <span class="math-container">$ a \neq b$</span>. Also <span class="math-container">$r^2 -1$</s... | Jan Eerland | 226,665 | <blockquote>
<p>Not a 'real' answer, but it was too big for a comment.</p>
</blockquote>
<p>I wrote and ran some <a href="https://en.wikipedia.org/wiki/Wolfram_Mathematica" rel="nofollow noreferrer">Mathematica</a>-code:</p>
<pre><code>In[1]:=Clear["Global`*"];
n = 2;
ParallelTable[
If[TrueQ[2^r + 2 == a^2 ... |
3,946,998 | <p>I was doing a question. Suddenly I got stuck at this last part of the problem. It was to prove
<span class="math-container">$ 2^r +2 = a^2 +b^2$</span> where <span class="math-container">$r \neq 2$</span>, r is a prime and <span class="math-container">$ a \neq b$</span>. Also <span class="math-container">$r^2 -1$</s... | Neat Math | 843,178 | <p><span class="math-container">$2^{2k+1}+2=(2^k-1)^2+(2^k+1)^2$</span>.</p>
|
276,818 | <p>I am attempting to numerically optimize a function of only 2 parameters <code>k, \[Theta]</code>.</p>
<p>The function is well-defined and the constraints are simple. However, the optimisation keeps returning <code>Indeterminate</code> or <code>not a real number</code> comment over the parameter space, i.e. <code>fun... | userrandrand | 86,543 | <p>You can obtain the coefficients using <a href="https://reference.wolfram.com/language/ref/SeriesCoefficient.html" rel="nofollow noreferrer"><code>SeriesCoefficient</code></a> on the <a href="https://reference.wolfram.com/language/ref/DifferentialRoot.html" rel="nofollow noreferrer"><code>DifferentialRoot</code></a> ... |
2,488,218 | <p>I encountered this problem while practicing for a mathematics competition. </p>
<blockquote>
<p>A cube has a diagonal length of 10. What is the surface area of the cube? <strong>No Calculators Allowed.</strong></p>
</blockquote>
<p>(Emphasis mine)</p>
<p>I'm not even sure where to start with this, so I scribble... | Seyed | 362,378 | <p>It is just the pythagorean theorem:
<a href="https://i.stack.imgur.com/RYuBZ.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/RYuBZ.jpg" alt="enter image description here"></a></p>
|
31,539 | <p>I want to learn a bit about Linear Programming. </p>
<p>After some research, I decided to solve the <a href="http://en.wikipedia.org/wiki/Cutting_stock_problem" rel="nofollow">Cutting Stock</a> problem as an example to learn. After doing some more research, I feel like I finally understand Linear Programming enough... | Suresh Venkat | 972 | <p>I don't know if you're more interested in using solvers or understanding the basics, but from the latter perspective, the problem of using LPs to get integer solutions comes under a very fascinating area termed 'rounding'. In brief, there are ways to take a solution to an LP and systematically transform the variable... |
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | Frank | 332,250 | <p>You can derive a formula for <span class="math-container">$\sqrt{a+b\sqrt{c}}$</span>. You will have to assume that <span class="math-container">$\sqrt{a+b\sqrt{c}}$</span> can be rewritten as the sum of two surds (radicands). So <span class="math-container">$$\sqrt{a+b\sqrt{c}}=\sqrt{d}+\sqrt{e}$$</span></p>
<p>Sq... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | TonyK | 1,508 | <p>If $x$ is your number, is my brother's height in metres more than $10(x-2)+2$?</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | mau | 89 | <p>"Among all prime numbers except 3, is there a positive and finite number of couples whose difference is the number you are thinking of?"</p>
<p>$N = 1 \implies$ <strong>no</strong>, since there are none ( $\{2, 3\}$ is disallowed).</p>
<p>$N = 2 \implies$ <strong>I don't know</strong>, at least until the Twin Pri... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | John Gowers | 26,267 | <p>The mathematician's answer: </p>
<p>I am thinking of a function $f:\{1,2,3\}\to\{0,1\}$. $f$ takes $1$ to $0$, $2$ to $1$ and $3$ either to $0$ or $1$, but I'm not telling you which. </p>
<p>Let $n$ be your number. Is $f(n)$ equal to $1$?</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | sksallaj | 36,841 | <p>Without any math involved:</p>
<p>Boy: I'll map your numbers to your answers. 1 means "NO", 2 means "YES", 3 means "I don't know", the first word that comes out of your mouth, will you please say the phrase that is mapped to the number you are thinking of?</p>
<pre><code>Girl: ahhhhhhhhhh
Boy: No, you have to say ... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | peterwhy | 89,922 | <p>"Let your number be $n$, consider an equation
$$x^n=n^2$$
If I randomly choose one of the real root(s) of $x$, is that root equal to your number $n$?"</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | DanielV | 97,045 | <blockquote>
<p>Call your number is <span class="math-container">$n$</span>. Is <span class="math-container">$n=1$</span>, or, is your reasoning assuming <span class="math-container">$n=3$</span> consistent?</p>
</blockquote>
<p>If <span class="math-container">$n=1$</span> then trivially "yes".</p>
<p>If <span cla... |
174,339 | <p>This one is somewhat hard to explain, but I'll try my best to.<br>
I'm trying to generate a list of numbers (containing only pi/10, 0, -pi/10). Numbers are randomly selected from these 3, where the probability of getting 0 is always 70%. But the probability of getting pi/10 and -pi/10 depend on the previous outcomes... | OkkesDulgerci | 23,291 | <p>Here is different approach. </p>
<pre><code>weight = {0.25, 0.7, 0.05};
list = Table[
Which[val = First@RandomSample[weight -> {π/10, 0, -π/10},1];
(val == -(π/10)) || (val == π/10),
weight = Reverse@weight; val, True, val], 10000];
Count[list, #] & /@ {0, Pi/10, -Pi/10}
</code></pre>
<... |
3,768,836 | <blockquote>
<p><strong>Exercise 16 (Stein): The Borel-Cantelli Lemma: Suppose <span class="math-container">$\{E_k\}_{k=1}^\infty$</span> is a countable family of measurable subsets of <span class="math-container">$\mathbb{R}^d$</span> and that
<span class="math-container">\begin{equation*}
\sum_{k=1}^\infty m(E_k) &l... | Oliver Díaz | 121,671 | <p>Notice that <span class="math-container">$E=\bigcap_n\bigcup_{m\geq n}E_m$</span>. Measurability follows immediately.</p>
<p>If you meant to say "prove that <span class="math-container">$E$</span> has <span class="math-container">$\mu$</span> measure zero, then that follows from monotone convergence since
<spa... |
763,381 | <p>There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the ... | Hagen von Eitzen | 39,174 | <p>Clearly, if $x$ is a root then so is $x^2+1$.
Thus if $x_0$ is the largest real root, then we need $x_0\ge x_0^2+1$, that is $x_0^2-x_0+1\le0$. But $x_0^2-x_0+1=(x_0-\frac12)^2+\frac34>0$</p>
|
3,980,796 | <p>Given <span class="math-container">$f: R \to (0,\infty]$</span>, twice differentiable with <span class="math-container">$f(0) = 0, f'(0) = 1, 1 + f = 1/f''$</span></p>
<p>Prove that <span class="math-container">$f(1) < 3/2$</span></p>
<p>I found some useless (I believe) facts about <span class="math-container">$f... | trancelocation | 467,003 | <p>Using Taylor you get for a <span class="math-container">$\xi \in (0,1)$</span>:</p>
<p><span class="math-container">$$f(1) =f(0) + f'(0) + \frac{f''(\xi)}{2!}=1+\frac 12\cdot \frac 1{1+f(\xi)}<1+\frac 12$$</span></p>
|
15,013 | <p>I am doing a plot where I have multiple shaded regions, and I want the line that separates the two regions to be dashed with dashes being alternating colors (so the demarcation stands out from both regions).</p>
<p>For example, say I am plotting the two regions shown here</p>
<pre><code>Plot[{1, Abs[BesselJ[1, x]]... | Fabio | 10,294 | <p>Like this?</p>
<pre><code>Plot[Evaluate[{Sin[2 \[Pi] x], Sin[2 \[Pi] x]}], {x, 0, 1},
PlotStyle -> {{Dashing[{0.1, 0.1, 1*^-12, 1*^-12}],
Red}, {Dashing[{1*^-12, 0.1, 0.1, 1*^-12}], Black}},
ImageSize -> 500]
</code></pre>
<p><a href="https://i.stack.imgur.com/3iqoH.jpg" rel="nofollow noreferrer"><i... |
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | Christian Blatter | 1,303 | <p><em>Continuity</em> is the definitive concept formalizing the following idea: If $x$ is near $x_0$ then $f(x)$ is near $f(x_0)$. If this idea has to make sense for some particular instance then the minimum requirement for $f$ is that $f(x_0)$ be defined. When $f(x_0)$ is defined, but there are no points $x$ "near" $... |
810,514 | <p>How to compute the following series:</p>
<p>$$\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$$</p>
<p>I tried</p>
<p>$$\frac{n+1}{2^nn^2}=\frac{1}{2^nn}+\frac{1}{2^nn^2}$$</p>
<p>The idea is using Riemann zeta function</p>
<p>$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$</p>
<p>but the term $2^n$ makes complicated. I k... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
3,256,879 | <p>I've only been exposed to basic abstract algebra (Like Definition of a group + Subgroup lemma etc) and some first year linear algebra. (We have not seen lagranges theorem, incase that is required for this question).</p>
<p>I was hoping if someone could show an elementary way of doing this question:</p>
<blockquote... | Community | -1 | <p>Every element of <span class="math-container">$\langle A,B\rangle $</span> can be written as a word <span class="math-container">$A^{\alpha_1}B^{\beta_1}A^{\alpha_2}B^{\beta_2}\dots A^{\alpha_k}B^{\beta_k}$</span> for some <span class="math-container">$k$</span>, with <span class="math-container">$0\le\alpha_j\le3$<... |
3,256,879 | <p>I've only been exposed to basic abstract algebra (Like Definition of a group + Subgroup lemma etc) and some first year linear algebra. (We have not seen lagranges theorem, incase that is required for this question).</p>
<p>I was hoping if someone could show an elementary way of doing this question:</p>
<blockquote... | Alan Wang | 165,867 | <p>First, verify that <span class="math-container">$|A|=4$</span> and <span class="math-container">$|B|=2$</span>.<br>
Next compute <span class="math-container">$AB,A^2B,A^3B,BA,$</span> and so on.<br>
But from here you will get that <span class="math-container">$$BA=A^3B$$</span>
By using this relation, we obtain that... |
4,584,609 | <p>Which is bigger</p>
<p><span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\sin x}{1+x^2}dx$$</span> or <span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\cos x}{1+x^2}dx~?$$</span></p>
<p>I let <span class="math-container">$x=\frac{\pi}{2}-t$</span> in the second integral, and I obtain this
<span ... | Ryszard Szwarc | 715,896 | <p>Let <span class="math-container">$f$</span> and <span class="math-container">$g$</span> be strictly decreasing nonnegative continuous functions on the interval <span class="math-container">$[0,1].$</span> Then for <span class="math-container">$0\le x< {1\over 2}$</span> we have
<span class="math-container">$$[ f(... |
2,464,726 | <p>I have quite a basic question but I don't know how to do it.
EDIT: Sorry for writing it unclearly I hope I can clarify it. </p>
<p>I want to write an if-else statement as a vector in linear algebra. However, this vector is the result of an if-else-statement of the form: </p>
<pre><code>if b>1
then a=0
else a=... | Tancredi | 487,564 | <p>I guess you cannot get the if-then-else construct in linear algebra intended as choice of the type "if $x_i>a$ ..." because this would introduce discontinuities, but all the linear operator (on finite dimension real spaces) are continuous. A try could be to encode true and false as $(0,1)$ and $(1,0)$ or to use b... |
455,642 | <p>I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $16$ elements a unique number id. Can you suggest me where to start ? </p>
| Harish Kayarohanam | 30,423 | <p><strong>Pairing Function :</strong> </p>
<p>Use pairing function . This is suggested by mathematicians to be the best way to generate a unique id , given 2 natural numbers (in our case say position in matrix) .</p>
<p>refer . <a href="http://en.wikipedia.org/wiki/Pairing_function" rel="nofollow noreferrer">http://... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | Niel de Beaudrap | 439 | <p>Matrix multiplication — more specifically, powers of a given matrix <em>A</em> — are a useful tool in graph theory, where the matrix in question is the <a href="http://en.wikipedia.org/wiki/Adjacency_matrix" rel="noreferrer">adjacency matrix</a> of a graph or a directed graph.</p>
<p>More generally, one... |
3,355,570 | <p>In a recent lecture I attended the following limit was discussed:
<span class="math-container">$$\lim_{(x,y)\to (0,0)} \frac{x^2y^4}{(x^2+y^4)^2}$$</span>
Multiple solutions were used to try and find the limit and illustrate how one would attack similar problems.</p>
<p>In particular we rewrote the expression using... | hamam_Abdallah | 369,188 | <p>Put
<span class="math-container">$$y^2=Y$$</span> and
<span class="math-container">$$x=r\cos(t)\; \; , Y=r\sin(t)$$</span></p>
<p>the limit becomes
<span class="math-container">$$\lim_{r\to 0}\frac{r^4\cos^2(t)\sin^2(t)}{r^4}$$</span></p>
<p>which depends on the angle <span class="math-container">$t$</span></p>
... |
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | Behrouz Maleki | 343,616 | <p>let $x_i=i\,$, $\,i=1,2,\cdots,6$ and apply Lagrange's interpolation method
$${{L}_{i}}(x)=\frac{\prod\limits_{j\ne i,j=1}^{6}{(x-{{x}_{j}})}}{\prod\limits_{j\ne i,j=1}^{6}{({{x}_{i}}-{{x}_{j}})}}\,\,\,\,,\,\,\,i=1,2,\ldots ,7$$
$$P(x)=\sum\limits_{i=1}^{6}{{{L}_{i}}}(x)P({{x}_{i}})$$</p>
|
4,242,877 | <p>I was reading the book of Real Analysis by H.L.Royden and they have given the definition of Riemann sums and Riemann integral as follows:<a href="https://i.stack.imgur.com/cFfRc.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cFfRc.jpg" alt="enter image description here" /></a></p>
<p>After this d... | Koro | 266,435 | <p>Let <span class="math-container">$f$</span> be bounded on <span class="math-container">$[a,b]$</span>.</p>
<p>Let <span class="math-container">$ P=\{a=x_{0} < x_{1} < x_{2} < ...< x_{n} =b\}$</span> be any partition of <span class="math-container">$[ a,b]$</span>.</p>
<p>Define <span class="math-containe... |
847,316 | <p>Why is the solution to linear differential equations with constant coefficients sought in the form of $Ce^{kx}$ ?</p>
| Community | -1 | <p>Short answer: because it works, i.e., solutions of this form do exist (if we allow complex $k$). (Though they do not always form a basis of solutions, which is why we sometimes have to consider $x^j e^{kx}$ as well). </p>
<p>Perhaps you wanted to know how people came up with the idea of using exponential functions ... |
794,182 | <p>I am trying to find an example of when $(A \bigcup B)^\circ \supset A^\circ \bigcup B^\circ$. Where $^\circ$ denotes the interior of a set. It has been previously shown in <a href="http://web.pdx.edu/~erdman/PTAC/problemtext_pdf.pdf" rel="nofollow">the text</a> that:</p>
<blockquote>
<p>The interior of the set $\... | abstract | 149,507 | <p>(ln((7x-(x^2))/12))^(3/2)=1</p>
<p>or, (7x-(x^2))/12)=e (2
<p>or, (x^2)-7x+12e=0</p>
<p>discriminant=49-48e (<0)
no real roots of x could be found for f(x)=7.
thus, by theory of contradiction, f(x) could never be 1.</p>
|
3,118,282 | <p>I'm trying to do the following problem in my book, but I don't understand how the book got their answer.</p>
<p>The problem:
Determine whether the following relations are equivalence relations:<span class="math-container">$\newcommand{\relR}{\mathrel{R}}$</span></p>
<p>The relation <span class="math-container">$\... | zwim | 399,263 | <ul>
<li>Reflexivity means <span class="math-container">$xRx$</span> which is <span class="math-container">$|x-x|=0\le 1$</span> verified.</li>
<li>Symmetric means <span class="math-container">$xRy\implies yRx$</span> which is <span class="math-container">$|x-y|\le 1\implies |y-x|\le 1$</span> verified too.</li>
</ul>
... |
827,899 | <p>I came across this problem on a HackerRank challenge.</p>
<p>The function $f(n)$ is</p>
<ul>
<li>$1$ if $n = 0$</li>
<li>$2f(n - 1)$, if $n$ is odd</li>
<li>$f(n -1) + 1$, if $n$ is even</li>
</ul>
<p>I solved the problem using a recursive function and it worked just well. However, I am assuming that a program wo... | gar | 138,850 | <p>To get a closed form, we will first rewrite the recurrence as:
\begin{align*}
f_n &= 2\, f_{n-2}+1+ \left(n\mod 2\right)
\end{align*}
Next, we will use generating functions, suppose $G(x)= \sum_{n\ge 0}f_n\, x^n$
\begin{align*}
\sum_{n\ge 2} f_n x^n &= 2\, \sum_{n\ge 2}f_{n-2}x^n+ \sum_{n\ge 2}x^n+ \su... |
633,522 | <ol>
<li><p>$ \lim_{n\to \infty} \sqrt[n]{3^n+4^n} $ . I think the limit is $4$. I did :
$ \sqrt[n]{3^n+4^n} = 4 \sqrt[n]{(\frac{3}{4}) ^n+1}$ .Am I right?</p></li>
<li><p>$ \lim_{n\to \infty} \frac{1}{1\cdot 4 } + \frac{1}{4\cdot 7} +...+\frac{1}{(3n-2)(3n+1)} $. I know that for each $k$ , this sequence is the sum of... | John | 104,867 | <p>1- Your first limit is correct.</p>
<p>2- As I suggested in the comments, you have to partialize the fractions into two, as such:</p>
<p>$$\frac{1}{(3n+1)(3n-2)}= \frac{1}{3} \left(\dfrac{-1}{(3n+1)}+\dfrac{1}{(3n-2)}\right)$$</p>
<p>Using the fact that it is a telescoping sum, you now have to find the:</p>
<p>$... |
2,329,475 | <p>Google returns that response, not sure why is a complex number.</p>
| Jan Eerland | 226,665 | <p>Using <a href="https://en.wikipedia.org/wiki/Complex_logarithm#Definition_of_principal_value" rel="nofollow noreferrer">the principal value</a> of the complex logarithm:</p>
<p>$$\ln\left(\text{z}\right)=\ln\left|\text{z}\right|+\arg\left(\text{z}\right)i\space\space\space\to\space\space\space\ln\left(-1\right)=\ln... |
1,240,871 | <p>I have a feeling that the only invertible matrix - $A$ . that when it squared $A^2$ is still $A$ , is the Identity matrix.<br>
Am I right? and if so , could anybody show me the proof? </p>
| Ben Grossmann | 81,360 | <p>The answer is yes: multiply both sides by $A^{-1}$.</p>
|
1,240,871 | <p>I have a feeling that the only invertible matrix - $A$ . that when it squared $A^2$ is still $A$ , is the Identity matrix.<br>
Am I right? and if so , could anybody show me the proof? </p>
| marwalix | 441 | <p>If $A$ is invertible, multiplying $A^2=A$ by $A^{-1}$ you get $A=I$</p>
|
533,812 | <p>A field is quadratically closed if each of its elements is a square.</p>
<p>The field <span class="math-container">$\mathbb{F}_2$</span> with two elements is obviously quadratically closed.</p>
<p>However, testing some more finite fields with this property, I didn't find any more. Hence my question is:</p>
<blockquo... | Community | -1 | <p>What about $\mathbb{F}_4$?</p>
<p>The multiplicative group of nonzero elements of a finite field is always cyclic. For odd $p$, that group has even order. This means there are always elements that aren't squares in the finite field.</p>
<p>Are you sure about your definition of quadratically closed field? I would h... |
3,134,854 | <p>In a 2D environment, I have a circle with velocity <em>v</em>, a stationary point (infinite mass), and I am trying to calculate the velocity of the circle after a perfectly elastic collision with the point. </p>
<p>This is what I've came up with: </p>
<p><span class="math-container">$p$</span> is the position of t... | Robert Israel | 8,508 | <p>Draw a line from the point through the centre of the circle at the moment of contact.
The outward velocity vector of the circle after the collision is <span class="math-container">$-$</span> the reflection of
the initial velocity vector across this line. This allows kinetic energy and angular momentum about the po... |
1,239,560 | <p>I was told that the relation $\le$ is a total order on R, it is dense, and it has a least upper bound property. I actually have don't understand those 3 properties... :/</p>
| Sebastian Bechtel | 228,972 | <p>Total order means, that the order relation $\leq$ is antisymmetric (if $a\leq b$ and $b\leq a$ then $a=b$), transitive ($a\leq b$ and $b\leq c$ implies $a\leq c$) and total (for all $a,b$ holds $a\leq b$ or $b\leq a$, compare e.g. with the subset relation which satisfies the first two, but not totality).</p>
<p>Den... |
3,132,404 | <p>For the purpose of this question, <span class="math-container">$A'$</span> is the derived set of set <span class="math-container">$A$</span>, <span class="math-container">$A^\circ$</span> is the interior of set <span class="math-container">$A$</span>, and <span class="math-container">$A^{c}$</span> is the complement... | dyna | 649,675 | <p>No. These sets are different.
Plese think of <span class="math-container">$A$</span> which has a isolated point.
<span class="math-container">$(A^{'})^c$</span> have a isolated point, but <span class="math-container">$(A^c)^\circ$</span> doesn't have.</p>
<p><span class="math-container">$A^c$</span> doesn't have it... |
3,132,404 | <p>For the purpose of this question, <span class="math-container">$A'$</span> is the derived set of set <span class="math-container">$A$</span>, <span class="math-container">$A^\circ$</span> is the interior of set <span class="math-container">$A$</span>, and <span class="math-container">$A^{c}$</span> is the complement... | Henno Brandsma | 4,280 | <p>Your argument is invalid.</p>
<p>If <span class="math-container">$x \in (A^\complement)^\circ$</span> then <span class="math-container">$U(x,r) \subseteq A^\complement$</span> for some <span class="math-container">$r>0$</span>. This implies that <span class="math-container">$x \notin A'$</span> or <span class="m... |
2,915,786 | <p>I started writing a proof using the method of proof by contradiction and encountered a situation which was true. More specifically, the hypothesis that I set out to prove was:</p>
<p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the ... | Henry | 6,460 | <p>It means that your precise approach does not work, but since this does not provide a counterexample it means the question would still be open. </p>
<p>Seeing $170-165$ is so small, there may be a way to save your proof. Try $$a_i + a_{i+1} + a_{i+2} \le 16$$ </p>
<p>leading to $$3 \cdot (a_1 + a_2 + \dots + a_{1... |
2,296,968 | <p>So far I have this: Let P(n) be the statement "$n2^n \lt n!$". $k_{0}=6$. $(6)2^6 = 384 <720=6!$. $P(k_{0})$ is true. Let $n \geq 6$ and assume P(n) to be true. By the induction hypothesis, $(n+1)2^{n+1}=(n+1)(2)2^n ...$ Somehow this gets to be $n!(n+1)(2)2^n$. I am clearly missing a few steps in my proof. </p>
| Franklin Pezzuti Dyer | 438,055 | <p>Here's what you do. You start off by assuming that
$$n2^n \lt n!$$
for some number $n_0$. Then
$$n_02^{n_0} \lt n_0!$$
By multiplying both sides by $n_0+1$, we get
$$n_0(n_0+1)2^{n_0} \lt (n_0+1)!$$
Now notice that $n_0(n_0+1)2^{n_0} \gt (n_0+1)2^{n_0+1}$ whenever $n_0 \gt 2$, so
$$(n_0+1)2^{n_0+1} \lt (n_0+1)!$$
Wh... |
97,579 | <p>Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?</p>
| Federico Poloni | 1,898 | <p>You can use the surprising identity $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, and take the norms of the three factors separately.</p>
|
3,845,756 | <p>I am studying sequences in <span class="math-container">$\mathbb{R}$</span>. I would like to know if a sequence satisfies that
"For all <span class="math-container">$n \in \mathbb{N}, |a_{n+1} - a_{n}| \leq \frac{1}{3^{n}}$</span>" then sequence <span class="math-container">$\left\{a_{n}\right\}_{n \in \... | player3236 | 435,724 | <p>Another classic counterexample is the <a href="https://en.wikipedia.org/wiki/Harmonic_number" rel="nofollow noreferrer">Harmonic Series</a>.</p>
|
352,243 | <p>In the page 10 of the paper "Filling Riemannian manifolds" by Gromov <a href="https://projecteuclid.org/euclid.jdg/1214509283" rel="noreferrer">(ProjetEuclid link)</a>, the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
<span class="math-container">$$\operatorn... | RobPratt | 141,766 | <p>Computational experiments for <span class="math-container">$1 \le m \le n \le 20$</span> yield feasible solutions even if you impose the upper bound <span class="math-container">$a_j \le n$</span>. Here is an infinite family for <span class="math-container">$m \le \lceil n/2\rceil +1$</span>:
<span class="math-cont... |
3,891,168 | <p>Let f be the function on <span class="math-container">$[-\pi,\pi]$</span> given by <span class="math-container">$f(0)=9$</span> and <span class="math-container">$f\left( x \right) = \frac{{\sin \left( {\frac{{9x}}{2}} \right)}}{{\sin \left( {\frac{x}{2}} \right)}}$</span> for <span class="math-container">$x\ne 0$</s... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
103,164 | <p>I want to solve an equation which contains an infinite continued fraction $F(n)$. Then I must (obviously) truncate this continued fraction at $n=2000$.</p>
<p>The problem here is that <em>Mathematica</em> does not display the $2000$ terms of this fraction on the screen. The screen closes directly.</p>
<p>Please, h... | Mr.Wizard | 121 | <p>I don't know if this is really an answer as the question itself seems misguided. (Why would one want to "display" a <em>massive</em> expression even if this could be done?) Nevertheless I would like to comment on the code used by the Betatron. <code>Hold</code> was used for manual control of recursion, however it ... |
886,097 | <p>First of all, we notice that: $$(a,b)=\bigcup_{a<x<b} [x,b).$$ </p>
<p>Also, we notice that: $$(0,1)-\left\{\frac{1}{n} |\, n\in \mathbb{Z^+}\right\}=\left(\frac{1}{2},1\right)\cup \left(\frac{1}{3},\frac{1}{2}\right)\cup \left(\frac{1}{4},\frac{1}{3}\right)\cup \left(\frac{1}{5},\frac{1}{4}\right)\cup \ldots... | Crostul | 160,300 | <p>It's true that $(0,1) \setminus K$ can be written as a union of open intervals, but it's not true that you can do it for all sets of the form $(a,b) \setminus K$. For example
$$(-1, 1) \setminus K = (-1, 0] \cup \bigcup_{n\geq 1} (\frac{1}{n+1}, \frac{1}{n})$$</p>
<p>Note that $(-1, 0]$ is not open in $\mathbb{R}_L... |
3,658,545 | <p>I have a question similar to the one posed <a href="https://math.stackexchange.com/questions/2128719/how-many-ways-can-4-males-and-4-females-be-seated-in-a-row-with-no-same-sex-sitt">here.</a></p>
<p>How many ways can 4 things of type A, 4 things of type B, and 4 things of type C be arranged so no two things of the... | Christian Blatter | 1,303 | <p>The following is a generating function approach:</p>
<p>We have to produce certain <span class="math-container">$abc$</span>-strings of length <span class="math-container">$12$</span>. These strings are beginning with <span class="math-container">$a$</span> and contain exactly four <span class="math-container">$a$<... |
566,088 | <p>Prove the following identity.</p>
<p>$$\sum_{i=0}^{n}(-1)^n\binom{-1/2}{i}\binom{-1/2}{n-i} = 1$$</p>
| alexjo | 103,399 | <p>Using the Vandermonde's convolution formula
$$
\begin{align}
\sum_{i=0}^n(−1)^n\binom{−1/2}{i}\binom{−1/2}{n-i}&=(−1)^n\sum_{i=0}^n\binom{−1/2}{i}\binom{−1/2}{n-i}\\
&=(−1)^n\binom{-1}{n}\\
&=(−1)^n(−1)^n\\
&=1
\end{align}
$$</p>
|
2,293,934 | <p>I'm sorry for such a vague question, but I couldn't make it more plain. The other day i was working with a solution to a problem and out of curiosity i decomposed the number into its prime factors and noticed an interesting pattern. The number $216$ distribute in $2,3,2,3,2,3$ and when i grouped it, i had three pair... | wythagoras | 236,048 | <p>When you know the prime factorisation, you can find the amount of divisors. If you have a prime factorisation $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ with $p_1, p_2, \cdots, p_k$ different primes and $e_1, e_2, \cdots, e_k$ positive integers, you can find that the number of positive divisors equals $(e_1+1)(e_2+1)\cd... |
11,651 | <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p>
<p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Ca... | NiloCK | 308 | <p>The 2016 result about <a href="https://arxiv.org/abs/1603.03720" rel="noreferrer" title="Arxiv Paper">Unexpected biases in the distribution of consecutive primes</a>.</p>
<p>This is really pretty simple to understand - the distribution of primes had been supposed to be unconditionally random, but this interesting b... |
11,651 | <p>Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students.</p>
<p>This is <em>approximately</em> true for elementary and secondary mathematics, and for the standard progression of undergraduate courses (eg, Ca... | guest | 19,343 | <p>I think a lot of results are "accessible" in terms of a pop science-y explanation, rather than following a detailed proof. And since you're looking for emotional motivation, I actually think the former is the more important!</p>
<p>For example, see this (quite engaging) video about moving sofas around cor... |
2,879,041 | <blockquote>
<p>Let $f: [a,b] \to \mathbb R$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a) = 0$ and $|f'(x)|\le k|f(x)|$ for some $k$, then $f(x)$ is zero on $[a,b]$. </p>
</blockquote>
<p>I tried proving it using Legrange's Mean Value Theorem but couldn't get it.</p>
<p>$f(x)$ is differentiable ... | Dark Malthorp | 532,432 | <p>Seeing that there was some confusing about my suggestion in the comments, I'll explain more fully:</p>
<p>We have by assumption
$$
|\frac{d}{dx}\ln f(x)| \le k
$$
wherever this is well defined. Since $f(a) = 0$ and $|\ln 0| = \infty$, we also know $\ln f(x)$ must be unbounded on $(a,b)$.
Suppose $\exists c\in [a,b... |
132,415 | <p>Let $x \in \mathbb{R}^n$</p>
<p>What is</p>
<p>$$\frac{\partial}{\partial x} [ x^Tx ]$$</p>
<p>My guess is: $\frac{\partial}{\partial x} [ x^Tx ] = 0$, because $[x^Tx] \in \mathbb{R}^1$, hence a real number as is interpreted as scalar in this derivation.</p>
| Did | 6,179 | <p>Let $u:\mathbb R^n\to\mathbb R$, $x\mapsto u(x)=x^Tx$. There exists a linear application $\ell_x:\mathbb R^n\to\mathbb R$, called the gradient of $u$ at $x$, such that </p>
<blockquote>
<p>$$u(x+z)=u(x)+\ell_x(z)+o(\|z\|)$$ </p>
</blockquote>
<p>when $z\to0$. To compute $\ell_x$, note that
$$
u(x+z)=(x+z)^T(x+z)... |
982,259 | <p>Let $n\in\mathbb{N}$.</p>
<p>So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$.</p>
<p>Help?</p>
| Arthur | 15,500 | <p>You know that $n=9i + k$ and $5n = 9j+ k$. So $4n = 9(j-i)$ is divisible by $9$. But since $\gcd(4,9)=1$, we must have that $n$ is divisible by $9$.</p>
|
99,018 | <p>If $g$ is Lie algebra over field char(k)=0,
then the following facts are well-known:</p>
<p>1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).</p>
<p>2) S(g)^g and ZU(g)=U(g)^g are isomorphic as commutative algebras. (The <a href="http://en.wikipedia.org/wiki/Duflo_i... | Vladimir Dotsenko | 1,306 | <p>It seems that in some particular cases it is known, see e.g. <a href="http://www.wisdom.weizmann.ac.il/~dimagur/OzAbstract.pdf" rel="nofollow">this recent result</a>.</p>
<p>I was looking at that a while ago, and at least it became clear to me that one has to be very careful with lifting the $p$th powers from the P... |
99,018 | <p>If $g$ is Lie algebra over field char(k)=0,
then the following facts are well-known:</p>
<p>1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).</p>
<p>2) S(g)^g and ZU(g)=U(g)^g are isomorphic as commutative algebras. (The <a href="http://en.wikipedia.org/wiki/Duflo_i... | Christopher Drupieski | 7,932 | <p>Your question may be related to <a href="https://mathoverflow.net/questions/89795/symmetrization-for-hyperalgebras-in-positive-characteristic/89826#89826">this question</a> about hyperalgebras. In my answer there, I gave a reference to a paper of Friedlander and Parshall (<em>Rational actions associated with the adj... |
1,131,323 | <p>I am studying a book on proofs and there are two statements that I don't understand the difference:</p>
<ol>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that for each integer $m$, $m + x = m$, then $x = 0$.</p></li>
<li><p>Let $x$ belong to the set of integers. If $x$ has the property that... | Sammy | 212,118 | <p>2 has a weaker condition. In 2, the only thing we need to show that $x = 0$ is that there is a single integer $m$ such that $x + m = m$. For 1, however, it must hold for all integers, $m$, that $x + m = m$.</p>
|
2,386,864 | <p>Six persons P, Q, R, S, T and U play in a tournament called "High Rollers". Every game involved two players. Each of the participants played with every other participant exactly once. In the game both the players rolled an unbiased die each. The player who gets the larger number on the top surface of the die wins th... | Greg Martin | 16,078 | <p>Here are some heuristics. As Hans Engler defines, let $k(n)$ be the number of pairs $(a,b)$ with $a<b$ for which $a+b=n$ and $a^2+b^2$ is prime. In other words,
$$
k(n) = \#\{ 1\le a < \tfrac n2 \colon a^2 + (n-a)^2 = 2a^2 - 2an + n^2 \text{ is prime} \}.
$$
Ignoring issues of uniformity in $n$, the <a href="h... |
803,488 | <p>Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.</p>
<ul>
<li><p>If you win with Rock, you get \$9. Your opponent loses the \$9.</p>
</li>
<li><p>If you win with Paper, you get \$3. Your opponent loses the \$3.</p>
</li>
<li><p>If you win with Scissors, you get \$5. Your... | Brian Fitzpatrick | 56,960 | <p>This table summarizes the possible outcomes in playing this game once:
$$
\begin{array}{c|c|c|c|c}
\text{Hero Plays} & \text{Villain Plays} & \text{Hero's Earnings} \\ \hline
R & R & +\$0 \\
R & P & -\$3 \\
R & S & +\$9 ... |
758,135 | <p>A visiting speaker in Economics recently happened to mention that John Maynard Keynes' <a href="http://www.gutenberg.org/ebooks/32625">A Treatise on Probability</a> revolutionized probability theory. I have not heard any such claim before and it struck me as strange. The <a href="http://en.wikipedia.org/wiki/A_Treat... | Michael Emmett Brady | 160,180 | <p>Thank you for your comment.Unfortunately, Kolmogorov's axiom system assumes linearity and additivity. His precise or point estimate approach to probability gives correct answers only in those fields where the weight of the evidence, w, equals, approximates, or approaches 1 in the limit. It is in many of the areas o... |
140,639 | <p><strong>Bug introduced in 11.0 and persisting through 11.3</strong></p>
<hr />
<p>From <a href="https://mathematica.stackexchange.com/a/140460/21532">this answer</a>, I doubt the capability to work on single character. So I give some test to verify this possibility. You can get my test <code>imgs</code> by this code... | Alexey Popkov | 280 | <p>You can use <code>Dilation</code> with rectangular kernel to extend the bounding boxes vertically in order to connect closely related components:</p>
<pre><code>MorphologicalTransform[#, "BoundingBoxes", Infinity] & /@ imgs
Dilation[#, Table[1, {6}, {1}]] & /@ %
</code></pre>
<blockquote>
<p><a href="htt... |
640,680 | <p>Could somebody help me to solve these two unrelated questions?
I have to prove or disprove them. The first one is which I have to answer. The second one is just for me, to understand the topic better.</p>
<p>Prove or disprove the following statements:</p>
<ol>
<li><p>If the sequence $(a_n)_{n\in \mathbb{N}}$ is b... | ireallydonknow | 109,467 | <p>Do you mean</p>
<ol>
<li>If the sequence is bounded, then is it necessarily convergent?</li>
<li>If the sequence is convergent, then is it necessarily bounded?</li>
</ol>
<p>The first statement is false.</p>
<p>Counterexample: Consider the alternating sequence $a_n = (-1)^n$</p>
<p>The second statement is true.<... |
2,828,636 | <p>I wanted to show that both sets are equal. My Textbook says following:</p>
<p>$\in$ means "is element of", $\land$ is the and operator, $\lnot$ is the not operator, $\notin$ means "is not element of", $\lor$ is the or operator</p>
<p>$$\def\-{\setminus}\begin{split} x \in A\-A\-B &\iff\\
(x \in A) \land (\lnot... | Bernard | 202,857 | <p>The vector $(\cos a, \sin a)$ is normal to the line, which passes through the point $(x_0,y_0)=(d\cos a, d\sin a)$, hence the equation of the line is
$$\color{red}{x\cos a+y \sin a}=x_0\cos a+y_0 \sin a=d(\cos^2a+\sin^2a)\color{red}{=d}.$$</p>
|
2,392,411 | <p>I am in Adv. Algebra 2 and I have a question. Firstly, would like to say I haven't taken algebra in a year due to geometry (stupid order they do but oh well) and I have a question understanding this: $(x+5)^{0}$. That would be $x^{0} + 5^{0}$ which then, wouldn't that be $1 + 1$ since anything that has a power of $0... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>Solve the inequality
$$
\left|1-\frac{2}{x}+1 \right|<\epsilon
$$</p>
<p>and verify that the solution is a neighborough of $1$.</p>
|
2,392,411 | <p>I am in Adv. Algebra 2 and I have a question. Firstly, would like to say I haven't taken algebra in a year due to geometry (stupid order they do but oh well) and I have a question understanding this: $(x+5)^{0}$. That would be $x^{0} + 5^{0}$ which then, wouldn't that be $1 + 1$ since anything that has a power of $0... | hamam_Abdallah | 369,188 | <p>Given $\epsilon>0$, we look for $\delta $ such that</p>
<p>$$0 <|x-1|<\delta\implies |1-\frac {2}{x}+1|<\epsilon $$</p>
<p>or
$$0 <|x-1|<\delta\implies 2|\frac {x-1}{x}|<\epsilon $$</p>
<p>As $x $ goes to $1$, we can suppose that $x $ is not far from $1$, for example we can assume that $$|x-1... |
2,473,089 | <p>I have a question on combinatorics, related to the pigeonhole principle:</p>
<blockquote>
<p>Consider the set $S= \{1,2,3,...,100\}$. Let $T$ be any subset of $S$ with $69$ elements. Then prove that one can find four distinct integers $a,b,c,d$ from $T$ such that $a+b+c=d$. Is it possible for subsets of size $68$... | BAI | 448,487 | <p>Well, I just realized that I’ve seen this problem days ago... the solution goes like this:</p>
<p>Let the numbers in $T$ be $1\le a_1<a_2<...<a_{69}\le 100$. Clearly, $a_1\le 32$.</p>
<p>Now, consider the sequences
$$b_n:=a_n+a_1, 3\le n\le 69$$
$$c_n:=a_n-a_2, 3\le n\le 69 $$</p>
<p>Apparently, $1 \le b... |
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