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 |
|---|---|---|---|---|
1,174,359 | <p>I have the following problem where n is a positive integer $(n >= 1)$:</p>
<p>Prove that $\frac{1}{2n}\le\frac{1*3*5*...*(2n-1)}{2*4*...*2n}$</p>
<p>I know that I must start with the basic step showing that $P(1)$ is true as follows:
$1/(2*1) = 1/2$ so $P(1)$ is true.</p>
<p>Now follows the induction step wher... | abel | 9,252 | <p>look at what the function $$y=f(x) = (5x)^{1/2} \text{ does at } x = 16/5, y = 4.$$ the derivative of $f$ is $$f'(x) = \frac12 (5x)^{-1/2}\times 5,\quad
f'(16/5) = \frac 52 \times \frac 14= \frac58 $$</p>
<p>so far we have at $$x = {16}/{5}, y=f(16/5) = 4, f'(16/5) = \frac58.$$ therefore
$$ y = 4, f^{-1}(4) = 16... |
3,856,180 | <p><span class="math-container">$X = \{0 ,1\}^{\mathbb N}$</span> be the metric space. Can anyone please tell me how to define a continuous injective function from <span class="math-container">$X = \{0 ,1\}^{\mathbb N}$</span> to the cantor set ?</p>
<p>Can anyone please give an idea ?</p>
| Henno Brandsma | 4,280 | <p>The Cantor set is the intersection <span class="math-container">$$\bigcap_{n=1}^\infty C_n$$</span></p>
<p>where <span class="math-container">$C_0 = [0, 1]$</span> and <span class="math-container">$C_{n+1} = \frac{1}{3}C_n \cup \frac{2}{3}C_n$</span> for each <span class="math-container">$n\ge 0$</span>. Every <spa... |
667,293 | <p>We are given a $n \times m$ rectangular matrix in which every cell there is a light bulb, together with the information whether the bulb is ON or OFF.</p>
<p>Now i am required to switch OFF all the bulbs but i can perform only one operation that is as follows:</p>
<ul>
<li>I can simultaneously flip all the bulbs f... | J. J. | 3,776 | <p>I don't know of an efficient algorithm that produces the minimal number of moves, but there's an algorithm that gives pretty good results.</p>
<p>The algorithm is based on the following observation: From the original $m \times n$ matrix construct a derived matrix of size $m+1 \times n+1$ by considering all $2\times... |
2,426,892 | <blockquote>
<p>Between which two integers does <span class="math-container">$\sqrt{2017}$</span> fall? </p>
</blockquote>
<p>Since <span class="math-container">$2017$</span> is a prime, there's not much I can do with it. However, <span class="math-container">$2016$</span> (the number before it) and <span class="mat... | eyeballfrog | 395,748 | <p>You could use the root extraction algorithm to find it directly. It's sort of like long division.</p>
<ol>
<li>Starting from the decimal place, divide the number into pairs of digits. So $20\,17$.</li>
<li>Find the largest integer whose square is less than the first pair. $4^2 < 20 < 5^2$. This is the first ... |
2,040,293 | <p>I am trying to follow this tutorial: <a href="http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling" rel="nofollow noreferrer">http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling</a></p>
<p>I am stuck to understand how to make a sta... | Aleksejs Fomins | 250,854 | <p>Warning: very hand-wavy argument :D</p>
<p>1) Real number to the real power can be defined as the limit of a real number to the quotient power. If some law holds for quotient powers, it holds for real powers in the limit</p>
<p>$x^r = \lim_{\frac{a}{b} \rightarrow r} (x^{\frac{a}{b}}) $</p>
<p>2) Real number to q... |
4,038,392 | <p>This question is inspired by the problem <a href="https://projecteuler.net/problem=748" rel="nofollow noreferrer">https://projecteuler.net/problem=748</a></p>
<p>Consider the Diophantine equation
<span class="math-container">$$\frac{1}{x^2}+\frac{1}{y^2}=\frac{k}{z^2}$$</span>
<span class="math-container">$k$</span>... | piepie | 365,969 | <p>This is a partial answer. Consider the case <span class="math-container">$k=1$</span>. The parametric solutions are given by <span class="math-container">$(x,y,z)=\left(2ab(a^2+b^2),a^4-b^4,2ab(a^2-b^2)\right)$</span>, where <span class="math-container">$\gcd(a,b)=1$</span>, <span class="math-container">$a>b>0... |
216,421 | <p>How do I go about proving this? Do I have to show total boundedness (I don't know how to use the finiteness of the residue field, and this seems like something that it might pertain to).</p>
| Martin Brandenburg | 1,650 | <p>Since this was asked in the comments: Compactness refers to the topology on $R$, which is induced by the absolute value, which is again induced by the valuation.</p>
<p>Here is a hint for the solution: Think of power series with coefficients in the residue field and apply Tychonov's Theorem.</p>
|
255,660 | <p>I want to get a solution of a equation using NSolve.</p>
<p><span class="math-container">$BesselI[1,x]/(x*BesselI[0,x])=0.2$</span></p>
<p>So I plugged this equation to NSolve:</p>
<pre><code>NSolve[BesselI[1,x]/(x*BesselI[0,x])==0.2, x]
</code></pre>
<p>But when I use this, the Mathematica gives the same expression... | Bob Hanlon | 9,362 | <p>Bound the value for <code>x</code></p>
<pre><code>Solve[{BesselI[1, x]/(x*BesselI[0, x]) == 1/5, -5 < x < 5}, x]
(* {{x -> Root[{(-5) BesselI[1, #] +
BesselI[
0, #] #& , -4.38411711031472304526702680222165674734`18.}]}, {x ->
Root[{(-5) BesselI[1, #] + BesselI[0, #] #& ,
4... |
3,653,148 | <p>Let <span class="math-container">$w$</span> be a primitive 5th root of unity. Then the difference equation <span class="math-container">$$x_nx_{n+2}=x_n-(w^2+w^3)x_{n+1}+x_{n+2}$$</span> generates a cycle of period 5 for general initial values:
<span class="math-container">$$u,v,\frac{u-(w^2+w^3)v}{u-1},\frac{uv-(w... | Aravind | 4,959 | <p>Too long for a comment:
This is to show one periodic sequence that satisfies a similar recurrence in a natural way.</p>
<p>The recurrence <span class="math-container">$x_nx_{n+2}=x_n+tx_{n+1}+x_{n+2}-(1+t)$</span> is 6-periodic, for every <span class="math-container">$t \neq 0$</span> (and non-degenerate initial va... |
3,653,148 | <p>Let <span class="math-container">$w$</span> be a primitive 5th root of unity. Then the difference equation <span class="math-container">$$x_nx_{n+2}=x_n-(w^2+w^3)x_{n+1}+x_{n+2}$$</span> generates a cycle of period 5 for general initial values:
<span class="math-container">$$u,v,\frac{u-(w^2+w^3)v}{u-1},\frac{uv-(w... | Pavel Kozlov | 143,912 | <p>Let us prove that for all nonzero <span class="math-container">$a,b\in \mathbb{C}$</span> the difference equation <span class="math-container">$$x_nx_{n+2}=ax_n+bx_{n+1}+x_{n+2}$$</span>
never generates a cycle of length <span class="math-container">$8$</span>.
Let us denote <span class="math-container">$u=x_0, y=x... |
2,602,799 | <p>This is exactly what is written in Walter Rudin chapter 2, Theorem 2.41:</p>
<p>If $E$ is not closed, then there is a point $\mathbf{x}_o \in \mathbb{R}^k$ which is a limit point of $E$ but not a point of $E$. For $n = 1,2,3, \dots $ there are points $\mathbf{x}_n \in E$ such that $|\mathbf{x}_n-\mathbf{x}_o| < ... | Atbey | 327,944 | <p>If $S$ was finite, then you can take minimum of $|x_n - x_0|$, since there are finitely many $x_n\in S$. (minimum is not $0$ since $x_0\notin S$) But this contradicts with the fact that there are $x_n$'s satisfying $|x_n-x_0| < 1/n$ for all $n$.</p>
<p>For the last inequality;</p>
<p>\begin{equation}
|\mathbf{x... |
4,176,152 | <p>I tried to simplify it</p>
<p><span class="math-container">$$f(z)=\frac{3z^4 -2z^3 +8z^2 -2z +5}{z-i}$$</span>
<span class="math-container">$$f(z)=\frac{{3z^2 -2z +5}{z^2+1}}{z-i}$$</span>
<span class="math-container">$$f(z)=\frac{(3z^2 -2z +5)(z+i)(z-i)}{z-i}$$</span></p>
<p><span class="math-container">$$f(z)=(3z^... | mjw | 655,367 | <p><span class="math-container">$$f(z) = \left\{ \begin{aligned} (z+i)(3z-5)(z+1), \quad z & \ne i \\
\text{undefined}, \quad z & =i
\end{aligned} \right.$$</span></p>
<p>Since <span class="math-container">$f(z)$</span> is undefined at <span class="math-container">$z=i$</span>, it cannot be continuous there.</p... |
4,176,152 | <p>I tried to simplify it</p>
<p><span class="math-container">$$f(z)=\frac{3z^4 -2z^3 +8z^2 -2z +5}{z-i}$$</span>
<span class="math-container">$$f(z)=\frac{{3z^2 -2z +5}{z^2+1}}{z-i}$$</span>
<span class="math-container">$$f(z)=\frac{(3z^2 -2z +5)(z+i)(z-i)}{z-i}$$</span></p>
<p><span class="math-container">$$f(z)=(3z^... | ultralegend5385 | 818,304 | <p>Recall the limit definition of continuity.</p>
<blockquote>
<p>A function <span class="math-container">$f$</span> is continuous at <span class="math-container">$x=c$</span> iff <span class="math-container">$$\lim_{x\to c}f(x)=f(c)$$</span></p>
</blockquote>
<p>When <span class="math-container">$f(c)$</span> is not d... |
22,340 | <p>Prove that for all natural numbers statement n, statement is dividable by 7 </p>
<p>$$15^n+6$$</p>
<p><strong>Base.</strong> We prove the statement for $n = 1$</p>
<p>15 + 6 = 21 it is true</p>
<p><strong>Inductive step.</strong></p>
<p><em>Induction Hypothesis.</em> We assume the result holds for $k$. That is,... | lhf | 589 | <p>By induction hypothesis, you have $15^k=7t-6$.</p>
|
22,340 | <p>Prove that for all natural numbers statement n, statement is dividable by 7 </p>
<p>$$15^n+6$$</p>
<p><strong>Base.</strong> We prove the statement for $n = 1$</p>
<p>15 + 6 = 21 it is true</p>
<p><strong>Inductive step.</strong></p>
<p><em>Induction Hypothesis.</em> We assume the result holds for $k$. That is,... | Bill Dubuque | 242 | <p>Often textbook solutions to induction problems like this are magically "pulled out of a hat" - completely devoid of intuition. Below I explain the intuition behind the induction in this proof. Namely, I show that the proof easily reduces to the completely trivial induction that <span class="math-container">$\rm\ \co... |
1,713,778 | <p>Let $P=\{p_1,p_2,\ldots ,p_n\}$ the set of the first $n$ prime numbers and let $S\subseteq P$. Let
$$A=\prod_{p\in S}p$$ and $$B=\prod_{p\in P-S}p.$$ Show that if
$A+B<p_{n+1}^2$, then the number $A+B$ is prime.
Also, if
$$1<|A-B|<p_{n+1}^2,$$ then the number $|A-B|$ is prime.</p>
| Community | -1 | <p>I believe due to Bertrand's postulate there is a finite (and relatively small) number of pairs $(n, S)$ for which $A + B < p_{n+1}^2$. An interesting satellite problem is to identify all such pairs; then the proposed problem is solved by inspection.</p>
|
3,183,274 | <p>This is a reinterpretation of my old question <a href="https://math.stackexchange.com/questions/3177594/fit-data-to-function-gt-frac1001-alpha-e-beta-t-by-using-least-s">Fit data to function $g(t) = \frac{100}{1+\alpha e^{-\beta t}}$ by using least squares method (projection/orthogonal families of polynomials)</a>. ... | Max | 2,633 | <p>Linear regression <strong>is</strong> linear algebra in disguise.</p>
<p>You are searching for a function <span class="math-container">$$l(t)= c_1 +c_2t$$</span> (where in your case <span class="math-container">$c_1= \ln \alpha$</span> and <span class="math-container">$c_2=-\beta$</span>), that is a linear combina... |
3,637,085 | <p>I have found this limit in <a href="https://oeis.org/A019609" rel="nofollow noreferrer">https://oeis.org/A019609</a> and I was wondering how to prove it (if it is actually correct):
<span class="math-container">$$\lim_{n\to\infty} \frac{4n}{a^2_n}=\pi e$$</span>
where
<span class="math-container">$$a_1=0,a_2=1, a_... | md5 | 301,549 | <p>You can get the asymptotics of the coefficients of the generating function:</p>
<p><span class="math-container">$$f(z)=\frac{e^{-z/2} z^2}{(1-z)^{3/2}}$$</span></p>
<p>using standard tools of singularity analysis from analytic combinatorics (see e.g. section B.VI of <a href="http://algo.inria.fr/flajolet/Publicati... |
3,733,229 | <p>In a probability space, it is said that a set of events should be <span class="math-container">$\sigma$</span>-algebra, meaning:</p>
<p><a href="https://i.stack.imgur.com/BgP8O.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BgP8O.png" alt="definition of sigma-algebra" /></a>
<em>This is from <a h... | Philipp123 | 704,651 | <p><span class="math-container">$\Omega\in\mathcal{F}$</span> does not imply <span class="math-container">$A\in\mathcal{F}$</span> if <span class="math-container">$A \subset \Omega$</span>.
An algebra is a set of sets. For example, it can happen, that <span class="math-container">$\{1,2\}\in \mathcal{F}$</span> but <sp... |
833,143 | <p>Wolfram alpha solves $\sqrt{x+1}\ge\sqrt{x+2}+\sqrt{x+3}$ for $x$, and answers $x=-2/3(3+\sqrt{3})$. How did it do it? Thanks!</p>
| georg | 144,937 | <p>I'd say it's okay. That the number $X=-2/3(3+\sqrt{3})$ is the solution of the <strong>equation</strong> $\sqrt{x+1} >\color{red}{=} \sqrt{x+2}+\sqrt{x+3}:$</p>
<p>$\sqrt{X+1}=i\sqrt{1/3(3+2\sqrt{3})}$</p>
<p>$\sqrt{X+2}+\sqrt{X+3}=i\sqrt{1/3 (3+2\sqrt{3})}$</p>
<p>For the desired inequality <strong>not explic... |
1,399,935 | <p>I'm reading Kleene's introduction to logic and in the beginning he mentions something that I have thought about for a while. The question is how can we treat logic mathematically without using logic in the treatment? He mentions that in order to deal with this what we do is that we separate the logic we are studying... | mathreadler | 213,607 | <p>You can use arithmetics to do logic.</p>
<p>true is 1</p>
<p>false is 0</p>
<p>$a \cdot b \Leftrightarrow$ logical and</p>
<p>$1-a \Leftrightarrow$ logical not</p>
<p>$a+b>0 \Leftrightarrow$ logical inclusive or</p>
<p>$a+b=1 \Leftrightarrow$ logical exclusive or</p>
|
2,645,611 | <blockquote>
<p>Prove that:
<span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span></p>
</blockquote>
<h3>My work so far:</h3>
<p><span class="math-container">$$\frac{(n+0)!}{0!}+\frac{(n+1)!}{1!}+...+\frac{(n+n)!}{n!}=\frac{(2n+1)!}{(n+1)!}$$</span>
<spa... | Felix Marin | 85,343 | <p>$\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{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
1,837,220 | <p>In this post:
<a href="https://math.stackexchange.com/questions/1056058/computing-int-sqrt14x2-dx">Computing $\int \sqrt{1+4x^2} \, dx$</a>
someone mentioned Euler substitution to compute the following integral:</p>
<p>$$\int \sqrt{1+4x^2} \, dx$$</p>
<p>I tried to follow this advice and got very nice result, name... | Behrouz Maleki | 343,616 | <p><strong>Hint</strong>
$$x=\frac{1}{2}\tan\theta$$
$$I=\frac12\int\sec^3\theta\,d\theta $$</p>
|
1,528,235 | <p>Recall that <a href="http://en.wikipedia.org/wiki/Tetration" rel="noreferrer">tetration</a> ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. </p>
<p>Its inverse function with respect to $x$ is called <a href="http://en.wikipedia.org/wiki/Tetration#Super-root" rel="noreferrer">s... | mick | 39,261 | <p>$$\mathcal L=\lim\limits_{n\to\infty}\frac{\sqrt[n]2_s-\sqrt2}{(\ln2)^n}\tag1$$</p>
<p>This limit is only possible if</p>
<p>$$\lim\limits_{n\to\infty}\frac{\sqrt[n+1]2_s - \sqrt 2}{\sqrt[n]2_s - \sqrt 2}= \ln2$$</p>
<p>To show this , use l'hopital</p>
<p>We get with $f(x) = x^{f(x)}$ :</p>
<p>$ \frac{D x^{f(x)... |
4,587,657 | <p>The call function is defined as</p>
<p><span class="math-container">$$
\text{call}: \begin{cases}
(\mathbb{R}^{I}\times I) \to \mathbb{R} \\
(f,x) \mapsto f(x)
\end{cases}
$$</span></p>
<p>is "<span class="math-container">$\text{call}$</span>" a measurable function? In other words: for a random fie... | FShrike | 815,585 | <p>This post assumes <span class="math-container">$\Bbb R^I$</span> is taken with the compact-open topology, and is given the Borel sigma-algebra. <span class="math-container">$\Bbb R^I$</span> is used to denote the set of <em>continuous</em> functions <span class="math-container">$I\to\Bbb R$</span>, so only addresses... |
2,259,109 | <p>If the value of $f(z_0)$ or $f^\prime(z_0)$ is complex number then is $f(z)$ analytic at $z_0$?</p>
| PM. | 416,252 | <p>$f (z) $ is analytic at a point if it is <em>differentiable</em> at the point , <em>and</em> on some region containing the point. </p>
|
64,406 | <p>It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. </p>
<p>I've never understood why these are the only two. Is there a reference or proof walkthrough on how to show any nonabelian... | Arturo Magidin | 742 | <p>Here's the proof that there are exactly five nonisomorphic groups of order $p^3$ for every prime $p$, as it appears in Marshall Hall's <strong>Theory of Groups</strong>.</p>
<ol>
<li><p>The abelian case is easy: you have $C_{p^3}$, $C_{p^2}\times C_p$, and $C_{p}\times C_p\times C_p$.</p></li>
<li><p>The nonabelian... |
1,634,807 | <p>The full question:</p>
<p>Having an equivalence relation $\sim$ on $\Bbb N$ defined by: $a \sim b$ meaning $a,b\in\Bbb N$ such that $a=b*10^k,$ for some $k\in\Bbb Z$, give a complete set of equivalence class representatives.</p>
<p>I am having trouble visualising these. I'm thinking you would need the whole set $\... | Ethan Bolker | 72,858 | <p>Here's another answer that may be easier to "visualize". Your equivalence relation says that $a$ and $b$ are equivalent if you can get $a$ by appending zeroes to $b$ or deleting them from the end of $b$. That's the same as saying $a$ and $b$ start with the same sequence of digits before the trailing zeroes (if any) ... |
641,744 | <p>I am obliged to find rank of matrix $A$ depending on $ p $.</p>
<p>I know how to do this using Gauss elimination method but I would like to try solve this using minors. I know that the rank of matrix is equal to degree of the biggest non-zero minor.</p>
<p>$A=\left(
\begin{array}{ccc}
p-1&p-1&1&1\\
1&a... | Bman72 | 119,527 | <p>This is your matrix. You have already checked in a correct way that the rank of the matrix is 3 for $p \neq 2,3$</p>
<p>$$A=\left(
\begin{array}{ccc}
p-1&p-1&1&1\\
1&p^{2}-1&1&p-1\\
1&p-1&p-1& 1
\end{array}
\right)$$</p>
<p>Now you have to check what happens for p=2 or 3. For p ... |
4,347,174 | <p>In the book Superlinear Parabolic Problems Blow-up, Global Existence and Steady States, page 493 this equation appears in which the book says it uses Young's inequality</p>
<p><span class="math-container">$$ |\Omega| u^p \leq ku^q +\epsilon(k)u $$</span></p>
<p>where <span class="math-container">$\epsilon(k) = C(|\O... | Glitch | 74,045 | <p>I assume that <span class="math-container">$1 \le p$</span> in the problem, as I believe the inequality is false without this assumption. Let's assume this.</p>
<p>The estimate is trivial if <span class="math-container">$p=1$</span> since you're free to make the constant on the second term larger than <span class="... |
4,164,820 | <p>CGM = Continuous Geometric Mean. Heuristics and mathematics are described in:</p>
<ul><li><a href="https://math.stackexchange.com/questions/4162869/subset-as-arithmetic-mean-of-geometric-means-not-really">Subset as arithmetic mean of geometric means. Not really?</a></li></ul>
A shortcut to the question as presented ... | Jean Marie | 305,862 | <p>This is a direct consequence of the following formula:</p>
<p><span class="math-container">$$
\frac{1}{2 \pi}\int_0^{2\pi} \ln(a+ b\cos\theta)d\theta = \ln \tfrac12(a + \sqrt{a^2 - b^2})\tag{1}$$</span></p>
<p>(valid for <span class="math-container">$a>b$</span>) established for example <a href="https://math.stac... |
439,941 | <p>I ran into this question and I am finding it very difficult to solve:</p>
<blockquote>
<p>How many different expressions can you get by inserting parentheses into:
$$x_{1}-x_{2}-\cdots-x_{n}\quad ?$$</p>
</blockquote>
<p>For example:</p>
<p>$$\begin{align*}
x_{1}-(x_{2}-x_{3}) &= x_{1}-x_{2}+x_{3}\\
(x_{1... | Brian M. Scott | 12,042 | <p>HINT: No matter how you parenthesize the expression, when you clear the parentheses, the first two terms will be $x_1-x_2$. Show by induction that the remaining $n-2$ signs can be any combination of plus and minus signs, meaning that for $n\ge 2$ you get $2^{n-2}$ distinct expressions.</p>
|
3,621,223 | <p>I use a software called Substance Designer which has a Pixel Processor where I can assign to every pixel of a image a gray-scale value defined by a series of operations.</p>
<p>I am basically trying to generate a <a href="https://i.stack.imgur.com/6jhYa.jpg" rel="nofollow noreferrer">"normal gradient"</a> generated... | Anonymous Coward | 770,101 | <p>Consider an axis-aligned ellipse centered at origin, height <span class="math-container">$2$</span> (semimajor axis <span class="math-container">$1$</span>), width <span class="math-container">$2\chi$</span> (semiminor axis <span class="math-container">$\chi \lt 1$</span>), parametrized using angle variable <span cl... |
2,325,436 | <p>I was reading <em>Introduction to quantum mechanics</em> by David J. Griffiths and came across following paragraph:</p>
<blockquote>
<p><span class="math-container">$3$</span>. The eigenvectors of a hermitian transformation span the space.</p>
<p>As we have seen, this is equivalent to the statement that any hermitia... | Jonas Dahlbæk | 161,825 | <p>To be quite frank, while the textbook by Griffiths may be a decent introduction to quantum mechanics, you should not rely on it for anything mathematics related.</p>
<p>That hermitian matrices can be orthogonally diagonalized is not a 'rather technical fact'. It is a completely standard result (known to both mathem... |
1,661,439 | <p>I'm trying to take the Lagrange polynomial $P_3(x)$ that passes through four points- $(x_1,y_1), (x_2,y_2), (x_3,y_3)$, and $(x_4,y_4)$, and integrate it (maybe deriving Simpson's rule in the process!).
All the x-values are a distance h apart from each other.
Once I did some simplifications using h, here's what I ... | user5713492 | 316,404 | <p>Why does this keep getting bumped by Community? Well, let's try doing the integrals:
<span class="math-container">$$\begin{align}I_1&=\int_{x_1}^{x_4}(x-x_2)(x-x_3)(x-x_4)dx\\
&=\left[\frac12(x-x_2)(x-x_3)(x-x_4)^2-\frac16(2x-x_2-x_3)(x-x_4)^3+\frac1{24}(2)(x-x_4)^4\right]_{x_1}^{x_4}\\
&=-\left[9h^4-\fr... |
3,460,843 | <p>I understand that the way to calculate the cube root of <span class="math-container">$i$</span> is to use Euler's formula and divide <span class="math-container">$\frac{\pi}{2}$</span> by <span class="math-container">$3$</span> and find <span class="math-container">$\frac{\pi}{6}$</span> on the complex plane; howeve... | an4s | 533,556 | <p>Substitute <span class="math-container">$t = \dfrac x2\implies x = 2t,\mathrm dx = 2\mathrm dt$</span>.</p>
<p>Therefore,</p>
<p><span class="math-container">$$\int\dfrac {-3}{x^2 + 4}\,\mathrm dx = -\int\dfrac 6{4t^2 + 4}\,\mathrm dt = -\int\frac6{4(t^2 + 1)}\,\mathrm dt = -\frac 32\int\frac1{t^2 + 1}\,\mathrm dt... |
1,705,159 | <blockquote>
<p>Find necessary and sufficient conditions for a Mobius transformation <span class="math-container">$T(z)=\frac{az+b}{cz+d}$</span> to map the unit circle to itself. So if <span class="math-container">$\gamma$</span> is a circle, <span class="math-container">$T(\gamma)=\gamma$</span>.</p>
<p>I've worked o... | Alan Muniz | 289,217 | <p>The map $T(z) = \frac{az+b}{cz+d}$ sends the unit circle to itself if and only if for any $\zeta$ in the circle, $|T(\zeta)|=1$. Now you just have to translate this into conditions on the coefficients.</p>
<p>$|T(\zeta)|=1$ is equivalent to
$$
|a|^2 + |b|^2 + 2Re(a\bar{b} \zeta) = |c|^2 +|d|^2+ 2Re(c\bar{d}\zeta)... |
95,741 | <p>I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain the differences between mapping and func... | Gaussler | 129,649 | <p>To me, function and map mean two entirely different things. A function is just a set-theoretic construction, something that assigns to each object in a set some unique object of another set. A map, on the other hand, is a construction from <em>category theory</em> rather than <em>set theory</em>. It means more or le... |
3,987,718 | <p>Let <span class="math-container">$L \in \mathbb{R}$</span> and let <span class="math-container">$f$</span> be a function that is differentiable on a deleted neighborhood of <span class="math-container">$x_{0} \in \mathbb{R}$</span> such that <span class="math-container">$\lim_{x \to x_{0}}f'(x)=L$</span>.</p>
<p>Fin... | Community | -1 | <p>In general, you have the following theorem.</p>
<blockquote>
<p>Suppose <span class="math-container">$f:(a,b)\to\mathbb{R}$</span> is continuous and differentiable on <span class="math-container">$(a,b)\setminus\{c\}$</span> for some <span class="math-container">$c\in (a,b)$</span>. If <span class="math-container">$... |
3,625,069 | <p>How to solve <span class="math-container">$10\sqrt{10\sqrt[3]{10\sqrt[4]{10...}}}$</span>?</p>
<p>I tried to solve this problem by letting <span class="math-container">$x=10\sqrt{10\sqrt[3]{10\sqrt[4]{10...}}}$</span> to observe the pattern.</p>
<p>Based on the pattern, the result is</p>
<p><span class="math-cont... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$\lim_{n\to\infty}10^{\sum_{r=1}^n\dfrac1{r!}}=10^{\lim_{n\to\infty}\sum_{r=1}^n\dfrac1{r!}}$$</span></p>
<p>Now <span class="math-container">$\lim_{n\to\infty}\sum_{r=1}^n\dfrac1{r!}=e-1$</span></p>
|
4,084,517 | <p>In geometry of 2D and 3D, it's not uncommon for people to call a square or rectangle a <code>Box</code> in the field I work in. This makes naming things easier since it's clear what's in a folder of 'boxes'.</p>
<p>Does the similar name exist for a circle and a sphere? Interestingly we have circles in 3D that could ... | Jean Marie | 305,862 | <p>There is a general geometric framework called the "space of spheres" covering spheres in any dimension, in particular</p>
<ul>
<li><p>ordinary circles for dimension 2 or even</p>
</li>
<li><p>line segments <span class="math-container">$[a,b] \subset \mathbb{R}$</span> for dimension 1.</p>
</li>
</ul>
<p>Th... |
4,049,293 | <p>I am learning about the cross entropy, defined by Wikipedia as
<span class="math-container">$$H(P,Q)=-\text{E}_P[\log Q]$$</span>
for distributions <span class="math-container">$P,Q$</span>.</p>
<p>I'm not happy with that notation, because it implies symmetry, <span class="math-container">$H(X,Y)$</span> is often us... | mechanodroid | 144,766 | <p>You can cut some work by just looking at the remainder mod <span class="math-container">$3$</span>.</p>
<p>Let <span class="math-container">$n=3q+r$</span>. Easy calculation gives
<span class="math-container">$$\frac15(3q+r)^5+\frac13(3q+r)^3+\frac{7}{15}(3q+r) = \text{some integer} + \frac{243q^5+7q}{5} + \frac{r^5... |
4,049,293 | <p>I am learning about the cross entropy, defined by Wikipedia as
<span class="math-container">$$H(P,Q)=-\text{E}_P[\log Q]$$</span>
for distributions <span class="math-container">$P,Q$</span>.</p>
<p>I'm not happy with that notation, because it implies symmetry, <span class="math-container">$H(X,Y)$</span> is often us... | J. W. Tanner | 615,567 | <p>To show that <span class="math-container">$15$</span> divides <span class="math-container">$n(3n^4+5n^2+7)$</span>, show that <span class="math-container">$3$</span> and <span class="math-container">$5$</span> do.</p>
<p><span class="math-container">$3$</span> does because either <span class="math-container">$3$</sp... |
3,841,806 | <p>Using spherical coordinates I have to find the volume of a cone <span class="math-container">$z=\sqrt{x^2+y^2}$</span> inscribed in a sphere <span class="math-container">$(x-1)^2+y^2+z^2=4.$</span></p>
<p>I can`t find <span class="math-container">$\rho$</span> because the center of sphere is displaced from the origi... | zkutch | 775,801 | <ol>
<li><p>Another variant is to use little "shifted" spherical coordinates
<span class="math-container">$$\begin{array}{}
x-1=\rho\cos\theta\sin\phi\\
y=\rho\sin\theta\sin\phi\\
z=\rho\cos\phi
\end{array}$$</span>
Then your first equation goes to <span class="math-container">$\rho^2 = 4$</span>, Jacobian wi... |
1,534,694 | <p>I tried to solve for the following limit: </p>
<p>$$\lim_{x\rightarrow \infty} (e^{2x}+x)^{1/x}$$
and I reached to the indeterminate form:
$${4e^{2x}}\over {4e^{2x}}$$
if I plug in, I will get another indeterminate form! </p>
| user | 505,767 | <p>We simply have</p>
<p><span class="math-container">$$(e^{2x}+x)^{1/x}=e^{2x/x}\left(1+\frac{x}{e^{2x}}\right)^{1/x}\to e^2\cdot1^0=e^2$$</span></p>
|
1,563,518 | <p>Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$.</p>
<p>I am trying to understand how units work in polynomial rings. My book doesn't really define it and I need a bit of help with this.</p>
| AnotherPerson | 185,237 | <p>Your solution is correct. What you have determined is that your solution is an element in the equivalence class of $-32$ mod $77$. Since these are just elements of the form $77k-32$, for $k\in \mathbb {Z}$, we can just let $k=1$ and obtain the smallest positive solution, namely $45$.</p>
<p>Note that typically when... |
2,250,339 | <p>The Mad Hatter sets up what he believes is a zero-knowledge protocol. The integer n is the product of two large primes p and q and he wants to prove to the March Hare that he knows the factorization of n without revealing to anyone the actual factors $p$, $q$. He devises the following procedure: </p>
<p>March Hare ... | miracle173 | 11,206 | <p>The answer to the second question is yes. The March hare could use this information to factor.</p>
<p>We have</p>
<p>$$x^2-y^2 \equiv (x+y)(x-y) \equiv 0 \pmod{ pq}$$</p>
<p>There is a chance that $p|(x+y)$ but $q\nmid (x+y)$ or that $q|(x-y)$ but $p\nmid (x-y)$. If one has some multiples of p one can calculate t... |
1,652,929 | <p>The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ </p>
<p>I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. Multiplying the equation by the integrating factor, I get $(y^{-5}-4)+(-y^{-2}-5xy^{-6})y'=0$ and then I solved $\in... | mickep | 97,236 | <p>The comments do not seem to lead to anything fruitful. I give you the first step, and then you confirm that you have the same in your solution, OK?</p>
<p>Write the differential equation as
$$
x'-\frac{5}{y-4y^6}x=\frac{y^3}{1-4y^5}
$$
Thus, an integrating factor is
$$
\exp\Bigl(\int-\frac{5}{y-4y^6}\,dy\Bigr).
$$
... |
2,818,427 | <p>Let $f \in \mathrm{End} (\mathbb{C^2})$ be defined by its image on the standard basis $(e_1,e_2)$: </p>
<p>$f(e_1)=e_1+e_2$</p>
<p>$f(e_2)=e_2-e_1$</p>
<p>I want to determine all eigenvalues of f and the bases of the associated eigenspaces.</p>
<p>First of all how does the transformation matrix of $f$ look like?... | Zarrax | 3,035 | <p>If you are looking for an equation of the form $y'' + p(x)y' + q(x)y = 0$, then you can plug in your two solutions $y = x^2$ and $y = e^{-x}$, giving two equations in two unknowns $p(x)$ and $q(x)$. You can then use standard linear algebra techniques to find $p(x)$ and $q(x)$.</p>
|
3,367,588 | <p>I've been studying Numerical Linear Algebra, Lloyd, 1997. I've came across the below incomprehensible paragraph.</p>
<blockquote>
<p>"Methods like Householder reflections and Gaussian elimination would
solve linear systems of equations exactly in a finite number of steps
if they could be implemented in exact ... | Theo Bendit | 248,286 | <p>An iterative method involves creating a sequence of estimates that converge to the desired result, often employing some kind of recursively-defined sequence. A classic example is <a href="https://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow noreferrer">Newton's method</a>, which takes a differentiable func... |
124,280 | <p>Show that the sequence ($x_n$) defined by $$x_1=1\quad \text{and}\quad x_{n+1}=\frac{1}{x_n+3} \quad (n=1,2,\ldots)$$ converges and determine its limit ? </p>
<p>I try to show ($x_n$) is a Cauchy sequence or ($x_n$) is decreasing (or increasing) and bounded sequence but I fail every step of all.</p>
| Unoqualunque | 17,703 | <p>You can also find the explicit form of $a_n.$ The following argument is taken from Kaczor-Novak, Problems in Mathematical Analysis, vol.1, AMS pub. p.228</p>
<p>The equation $x^2+3x-1=0$ has two solutions $a > 0 >b.$ It is easy to observe
$$
\frac{x_{n+1}-a}{x_{n+1}-b}=\frac{a}{b} \frac{x_n-a}{x_n-b}.
$$
W... |
1,707,853 | <p>To be more precise than the title, the function is actually piecewise</p>
<p>$$
f(x,y) = \begin{cases}
\frac{x^3+y^3}{x^2+y^2} & (x,y) \ne (0,0) \\
0 & (x,y) = (0,0) \\
\end{cases}
$$</p>
<p>I checked that the function is continuous at $(0,0)$, so I then calculated the partial derivative with respect to $x... | Community | -1 | <p>By definition of $f_1(x_0,y_0)$ you have :
$$f_1(0,0)=\lim_{h\to 0}\frac{f(h,0)-f(0,0)}{h}$$<br>
So
$$f_1(0,0)=\lim_{h\to 0}\frac{\frac{h^3+0}{h^2+0}}{h}=1$$</p>
|
47,974 | <p>I am interested in the following question:</p>
<p>Is it known that <span class="math-container">$2$</span> is a primitive root modulo <span class="math-container">$p$</span> for infinitely many primes <span class="math-container">$p$</span>?</p>
<p>There is some information about Artin's conjecture in <a href="https... | Igor Potapov | 98,281 | <p>I do not know direct answer to your problem, but I will think about this
as it is close to the area of problems which we are currently working on.</p>
<p>Although I would like to comment that not all questions are "reasonably efficiently decidable" for SL(2,Z) as most of known to me problems for
SL(2,Z) are at lea... |
478,566 | <p>I'm reading a book about combinatorics. Even though the book is about combinatorics there is a problem in the book that I can think of no solutions to it except by using number theory.</p>
<p>Problem: Is it possible to put $+$ or $-$ signs in such a way that $\pm 1 \pm 2 \pm \cdots \pm 100 = 101$?</p>
<p>My proof... | Brian M. Scott | 12,042 | <p>You can rephrase essentially the same argument in the following terms:</p>
<p>Suppose that there were such a pattern of plus and minus signs. Let $P$ be the set of positive terms, and let $N$ be the set of negative terms together with the number $101$. Then $\sum P-\sum N=0$, so $\sum P=\sum N$, and $\{P,N\}$ is a ... |
1,479,095 | <blockquote>
<p>For $f:[0,1]\to \mathbb{R}$ let $E\subset\left\{x \mid f'(x) \text{exists}\right\}$. Prove that if $|E|=0$, then $|f(E)|=0$.</p>
</blockquote>
<p>My attempt:</p>
<p>Let $E_{nk}=\left\{x\in [0,1]|\frac{|f(x+h)-f(x)|}{h}\leq n, |h|< \frac{1}{k} \right\}$. </p>
<p>I am not sure where to go from her... | Giovanni | 263,115 | <blockquote>
<p><strong>Claim:</strong> Let $E_n = \{x \in E : |f'(x)|\le n\}$, then $\lambda(f(E_n)) \le n\lambda(E_n)$.</p>
</blockquote>
<p>Notice that $E = \bigcup E_n$, hence the desired result follows from the fact that</p>
<p>$$\lambda(f(E_n)) \le n\lambda(E_n) \le n\lambda(E) = 0.$$ Indeed, to conclude from... |
232,424 | <p>Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods? </p>
<p>I'd also appreciate any mentions relating to sciences, social sciences or ordinary l... | Robert Israel | 8,508 | <p>"Common sense" in mathematics is not very common.<br>
Many things seem very anti-intuitive, at least until you train your intuition properly.
The untrained intuition is lost when dealing with, for example, infinite sets, or geometry in more than $3$ dimensions. However, one example of "common sense" that does come ... |
232,424 | <p>Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods? </p>
<p>I'd also appreciate any mentions relating to sciences, social sciences or ordinary l... | Godot | 38,875 | <p>The common sense is the backbone of whole mathematics.</p>
<p>It is fair to say that nowadays all branches of mathematics are axiomatic theories.
To start building an axiomatic theory you must decide what are your axioms, what are your axiom schemes, what are your rules of inference. When you finished setting up th... |
232,424 | <p>Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods? </p>
<p>I'd also appreciate any mentions relating to sciences, social sciences or ordinary l... | Ronnie Brown | 28,586 | <p>I would like to write about the problem of "expression", from my own experience. In the 1960s, it seemed to me that, from a commonsensical viewpoint, there should be some way of expressing that
<img src="https://i.stack.imgur.com/7a8Ig.jpg" alt="array"></p>
<p>in the above diagram the big square should be the "com... |
2,321,667 | <p>Patrick Suppes in his book <a href="http://rads.stackoverflow.com/amzn/click/0486406873" rel="nofollow noreferrer">Introduction to Logic</a> on page 63 asks a reader to proof a statement
$$\forall x\forall y\forall z(xPy\land yPz\to xPz)$$ from the theory which he calls "Theory of rational behavior". The statement i... | User4407 | 81,495 | <p>I came to the same conclusion using your approach, namely, that the <em>axiom of order</em> wasn't enough to prove $\neg zQx$. However, I found that it could be accomplished by using the <em>transitive property</em> along with <em>modus tollens</em>. That leads to the following idea: $\neg yQx \to (yQz \to \neg zQx)... |
38,193 | <p>For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:</p>
<p>ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.</p>
<p>(Here ZFC can be replac... | Andreas Blass | 6,794 | <p>For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the consistency of ZFC, then a proof of Con(ZFC) carried out in ZFC would indeed be of little use. But a proof of Con(ZFC... |
4,196,583 | <p>More precisely:</p>
<blockquote>
<p><strong>Definition.</strong><br />
A subset <span class="math-container">$S \subset \Bbb R$</span> is called <em>good</em> if the following hold:</p>
<ol>
<li>if <span class="math-container">$x, y \in S$</span>, then <span class="math-container">$x + y \in S,$</span> and</li>
<li>... | qualcuno | 362,866 | <p>Not quite what you want, but too long for a comment:</p>
<blockquote>
<p><strong>Proposition.</strong> If <span class="math-container">$S$</span> is a subgroup, then <span class="math-container">$S$</span> is good iff it is closed.</p>
</blockquote>
<p><em>Proof.</em> Suppose that <span class="math-container">$S$</s... |
4,117,409 | <blockquote>
<p>Prove or disprove: if for every <span class="math-container">$n\in\Bbb{N}, |a_{n+1}-a_n|<\frac{1}{n^2}$</span> then <span class="math-container">$a_n$</span> converges.</p>
</blockquote>
<p>I think this is true, and tried using Cauchy's theorem - I take some <span class="math-container">$\varepsilon ... | TheSilverDoe | 594,484 | <p>No need to use Cauchy criterion here.</p>
<p>The fact that <span class="math-container">$|a_{n+1}-a_n| \leq 1/n^2$</span> implies that the <em>series</em> <span class="math-container">$\sum (a_{n+1}-a_n)$</span> is absolutely convergent, hence convergent, hence the sequence <span class="math-container">$(a_n)$</span... |
3,866,285 | <p>Suppose I'd like to find the coefficient of <span class="math-container">$x^{l}$</span> in the expansion of <span class="math-container">$(1+x+x^{2}+...+x^{n})^{m}$</span>, where <span class="math-container">$n$</span> and <span class="math-container">$m$</span> are given positive integers, for some given integer <... | user158293 | 158,293 | <p>Using the ideas of a linked post such as <span class="math-container">$\left(\sum_0^n x^s\right)^m=(1-x^{n+1})^m (1-x)^{(-m)} $</span> found a much more concise answer.
Again using <span class="math-container">$L$</span> in place of <span class="math-container">$l\;$</span> the coeff of <span class="math-container">... |
854,438 | <p>I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I suppose the ideas of Church, Godel, and Turing were either in their infancy, or not well-known, but I still hear this kind... | Kile Kasmir Asmussen | 72,934 | <p>Mathematics is a science of discovery. It's domain is The Mathematical Principle a peculiar entity or feature of the universe we inhabit; one might formulate the principle as follows:</p>
<blockquote>
<p>Given a string of symbols and well defined symbolic transformations, applying a well specified series of trans... |
4,249,794 | <p>i have to determine whether this graph is bipartite or not:</p>
<p><img src="https://i.stack.imgur.com/AjxQzl.png" alt="" /></p>
<p>I have found an answer but i am not sure about it. If we divide the vertices set into <span class="math-container">$\{a,d,c,h\}$</span> and <span class="math-container">$\{b,f,e,g\}$</s... | Holy Moly | 959,258 | <p>Note that a graph is bipartite iff all its cycles have even length. Thus your graph is bipartite. To check that all cycles of your graph are actually of even length, we may apply the following result that can be proved by induction. Given a graph <span class="math-container">$G$</span> built up from even cycles conn... |
59,954 | <p>I can rather easily imagine that some mathematician/logician had the idea to symbolize "it <strong>E</strong> xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for <strong>A</strong> ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it w... | GEdgar | 442 | <p>The four types of propositions used in the classical Greek syllogisms were called A, E, I, O. Statements of type A were "All p are q". Statement of type E were "Some p are q". So of course a millennium later, mathematicians (who had a classical education) used A and E for these quantifiers, then later turned them... |
2,089,502 | <blockquote>
<p>How many numbers are there from $1$ to $1400$ which maintain these conditions:
when divided by $5$ the remainder is $3$ and when divided by $7$ the remainder is $2$?</p>
</blockquote>
<p>How can I start? I am newbie in modular arithmetics. I can just figure out that the number $= 5k_1+3 = 7k_2+2$. ... | Bernard | 202,857 | <p><em>Mean value theorem</em>: there exists $c$ between $x$ and $y$ such that $\;\cos x-\cos y=-\sin c(x-y)$, hence
$$\lvert\cos x-\cos y\rvert=\lvert\sin c\rvert\lvert x-y\rvert\le\lvert x-y\rvert. $$</p>
|
1,612,808 | <p>Suppose that $X$ is a finite $G$-set. A group $G$ is of prime power if $|G|=p^n$ for $p$ prime.</p>
<p>The fixed point set $X_G=\{x\in X : gx=x$ $\forall g\in G\}$.</p>
<p>I'm asked to prove that $|X|=|X_G|$ (mod $p$), but I'm unsure of how I should start.</p>
| Tsemo Aristide | 280,301 | <p>Hint: To show this, you have to remark that if $x$ is not fixed by $G$, the cardinal of $Orb_x$ the orbit of $x$ is $\mid Orb_x\mid =\mid G\mid/\mid G_x\mid$ Where $G_x$ is the subgroup of elements of $G$ which fix $x$. Lagrange implies that $\mid G_x\mid$ divides $\mid G\mid$ so $\mid Orb_x\mid =0$ mod p since $G_... |
30,220 | <p>Jeremy Avigad and Erich Reck claim that one factor leading to abstract mathematics in the late 19th century (as opposed to concrete mathematics or hard analysis) was <em>the use of more abstract notions to obtain the same results with fewer calculations.</em></p>
<p>Let me quote them from their remarkable historical... | Daniel Litt | 6,950 | <p>A wonderful example is the proof of the Poincare Lemma I sketch <a href="http://www.thehcmr.org/issue1_2/poincare_lemma.pdf" rel="nofollow noreferrer">here</a> (<a href="https://web.archive.org/web/20141017035427/http://www.thehcmr.org/issue1_2/poincare_lemma.pdf" rel="nofollow noreferrer">Wayback Machine</a>), as c... |
1,414,316 | <p>I am trying to optimize distance from point to plane using Lagrange multiplier.</p>
<p>Usually for such problems you are given specific point like (1,2,3) in 3D, and then an exact plane which is just the subject of Lagrange. But what I have here doesn't specify values for point and plane.</p>
<p>It says problem ha... | Noah Schweber | 28,111 | <p>Note that in Asaf's answer, the elementary embedding $j: M\rightarrow N$ does not "live" (that is, is not definable in) $M$.</p>
<p>By contrast, if we have an elementary embedding $j: M\rightarrow N$ which is definable in $M$ (from parameters in $M$), then $crit(j)$ <strong>is</strong> inaccessible, in fact measura... |
432,208 | <p>I want to grasp the moving frames method but I find some obstacles. I don't know if this question is suitable for MO, if it is not the case please let me know and I will move it.<br />
I am aware there are other related questions here like <a href="https://mathoverflow.net/questions/337294/moving-frames-method-for-n... | Robert Bryant | 13,972 | <p>I think you might want to read a couple of articles on the moving frame that carefully discuss this issue (and show that it is more subtle than most people realize).</p>
<p>The first is a paper by Mark Green, <em>The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces</em> (D... |
238,128 | <p>Let $G$ be an abelian group. <br/>
Show that $\{x\in{G} | |x| < \infty\}$ is a subgroup of $G$. Give an example of a non-abelian group where this fails to be a subgroup.</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Here is one of many ways of constructing an example. Let $G$ be the group of permutations of the integers. Let $f$ be the permutation that takes any integer $x$ to $-x$, and $g$ the permutation that takes any integer $x$ to $1-x$. </p>
<p>Both $f$ and $g$ have order $2$. Now consider the pe... |
4,275,780 | <blockquote>
<p>If <span class="math-container">$0<a<b$</span> and <span class="math-container">$0<c<d$</span> then <span class="math-container">$\frac{c+a}{d+a} <\frac{c+b}{d+b}.$</span></p>
</blockquote>
<p>I get to <span class="math-container">$$d+a<d+b \Longrightarrow \frac{1}{d+b} < \frac{1}{d... | Aryaman Maithani | 427,810 | <p><span class="math-container">$\renewcommand{\iff}{\Leftrightarrow}$</span>Here's a dumb way which requires no clever insight:<br />
Since <span class="math-container">$a, b, c, d > 0$</span>, we see that
<span class="math-container">$$\frac{c+a}{d+a} < \frac{c+b}{d+b} \iff(c + a)(d + b) < (d + a)(c + b).$$<... |
4,275,780 | <blockquote>
<p>If <span class="math-container">$0<a<b$</span> and <span class="math-container">$0<c<d$</span> then <span class="math-container">$\frac{c+a}{d+a} <\frac{c+b}{d+b}.$</span></p>
</blockquote>
<p>I get to <span class="math-container">$$d+a<d+b \Longrightarrow \frac{1}{d+b} < \frac{1}{d... | zwim | 399,263 | <p>There is this important general inequality (working for positive denominators), i.e. the compound fraction comes between the bounds:</p>
<p><span class="math-container">$$\frac AB<\frac CD\implies \frac AB<\frac{A+C}{B+D}<\frac CD$$</span></p>
<p>See here for a proof <a href="https://math.stackexchange.com/... |
4,275,780 | <blockquote>
<p>If <span class="math-container">$0<a<b$</span> and <span class="math-container">$0<c<d$</span> then <span class="math-container">$\frac{c+a}{d+a} <\frac{c+b}{d+b}.$</span></p>
</blockquote>
<p>I get to <span class="math-container">$$d+a<d+b \Longrightarrow \frac{1}{d+b} < \frac{1}{d... | David | 979,748 | <p>By the rearrangement inequality, <span class="math-container">$bc+ad\lt ac+bd.$</span> Thus, <span class="math-container">$(c+a)(d+b)\lt(c+b)(d+a).$</span></p>
|
344,166 | <p>I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of elementary(no measures) probability. The book is rigorous "hard" math but doesn't escape from giving a solid intuiti... | Community | -1 | <p>Here is a link to a free down-load of virtually verbatim lecture notes for a real analysis course taught by Fields Medal winner Vaughan Jones. They were my first introduction to real math - beautiful presentation, lots of motivation:</p>
<p><a href="https://sites.google.com/site/math104sp2011/lecture-notes" rel="no... |
344,166 | <p>I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of elementary(no measures) probability. The book is rigorous "hard" math but doesn't escape from giving a solid intuiti... | goblin GONE | 42,339 | <p>Goldrei's <a href="http://rads.stackoverflow.com/amzn/click/0412606100" rel="nofollow">Classic Set Theory For Guided Independent Study</a>. I don't necessarily think he's the greatest expositor, but his educational philosophy is spot on. For instance, he starts with the real number system and asks: how do we know th... |
1,653,106 | <p>I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$.</p>
<p>Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$? </p>
| Jay Lee | 166,706 | <p>That is, since $(x^2+4)(x+2)(x-2)$ is the simplest form of the equation $x^4-16$, rather than $(x^2-4)(x^2+4)$.</p>
|
1,653,106 | <p>I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$.</p>
<p>Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$? </p>
| hkBst | 261,514 | <p>Actually you could factor all the way to complex roots:</p>
<p>$$x^4-16 = (x^2-4)(x^2+4) = (x-2)(x+2)(x-2i)(x+2i).$$</p>
|
1,630,655 | <p>Not sure what to do / how to start this... I have equcation of 504 is: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$</p>
| lhf | 589 | <p>Write $504 = 2^3\cdot3^2\cdot7$ and
$n^9-n^3=n^3(n^6-1)=n^3(n^2-1)(n^4+n^2+1)$.</p>
<p>Then:</p>
<ul>
<li><p>For $m=2^3$, we have $n^2-1 \equiv 0$ for $n$ odd and $n^3 \equiv 0$ for $n$ even.</p></li>
<li><p>For $m=3^2$, we have $\phi(m)=6$ and so $n^6-1 \equiv 0$ when $n$ is not a multiple of $3$ and $n^3 \equiv... |
322,858 | <p>Let <span class="math-container">$G$</span> be a split semisimple real Lie group in characteristic zero, and let <span class="math-container">$B=TU$</span> be a Borel subgroup with unipotent radical <span class="math-container">$U$</span> and Levi <span class="math-container">$T$</span>. Fix an ordering on the root... | Jim Humphreys | 4,231 | <p>Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of sim... |
2,579,156 | <p>I found the solution of series on Wolfram Alpha
<a href="http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n" rel="nofollow noreferrer">http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n</a></p>
<p>$ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\righ... | Dr. Wolfgang Hintze | 198,592 | <p>This is not an answer to the specific question but a long comment which gives a derivation of a much simpler formula for the sum in question. </p>
<p>Let</p>
<p>$$s = \sum _{k=1}^n \left(\frac{1}{2 k+1}-\frac{1}{2 k+2}\right)$$</p>
<p>Adding and subtracting a sum of even terms we get for $s$</p>
<p>$$\begin{arra... |
113,854 | <p>I'm importing a *.pdb file containing a single protein. <em>Mathematica</em> automatically produces a plot of the protein.</p>
<p>I want to specify the color of each residue independently, in this plot. Is this possible?</p>
<p>Additionally, I would like to change the type of plot to "cartoon". How can I do this?<... | J. M.'s persistent exhaustion | 50 | <p>Here is a very hack-ish way of coloring each residue sequentially. The trick is in constructing a <code>Blend[]</code> function where the color corresponding to each residue appears twice in the first argument. A slight shift is apparently needed to match up the colors, even if only approximately.</p>
<pre><code>n ... |
2,807,356 | <blockquote>
<p><strong>If $z_1,z_2$ are two complex numbers such that $\vert
z_1+z_2\vert=\vert z_1\vert+\vert z_2\vert$,then it is necessary that</strong> </p>
<p>$1)$$z_1=z_2$</p>
<p>$2)$$z_2=0$</p>
<p>$3)$$z_1=\lambda z_2$for some real number $\lambda.$</p>
<p>$4)$$z_1z_2=0$ or $z_1=\lambda z... | Peter Szilas | 408,605 | <p>A bit of geometry:</p>
<p>Consider $z_1, z_2$ vectors in the complex plane.</p>
<p>Vector addition: </p>
<p>$\vec {OA} =z_1$; $\vec {AB} =z_2$; and $\vec {OB} = z_1+z_2.$</p>
<p>For $OAB$ to be a triangle we must have:</p>
<p>$|z_1|+|z_2|> |z_1+z_2|$, i.e. the sum of the lengths of 2 sides is greater than th... |
2,476,717 | <p>Let $f: \mathbb R^n \rightarrow \mathbb R^n$ with arbitrary norm $\|\cdot\|$. It exists a $x_0 \in \mathbb R^n$ and a number $r \gt 0 $ with </p>
<p>$(1)$ on $B_r(x_0)=$ {$x\in \mathbb R^n: \|x-x_0\| \leq r$} $f$ is a contraction with Lipschitz constant L</p>
<p>$(2)$ it applies $\|f(x_0)-x_0\| \le (1-L)r$</p>
<... | Surb | 154,545 | <p>For $x\in B_r(x_0)$, we have
$$\|f(x)-x_0\|=\|f(x)-f(x_0)+f(x_0)-x_0\|\leq \|f(x)-f(x_0)\|+\|f(x_0)-x_0\|\\
\leq L\|x-x_0\|+(1-L)r \leq Lr+(1-L)r=r$$
thus $f(x)\in B_r(x_0)$.</p>
<p>To show that $\lim_{k\to\infty}x_k=x_f$ and $f$ has a unique fixed point in $B_r(x_0)$, you can indeed use the Banach fixed point theo... |
4,605,888 | <p>Let's say I have a uniformly distributed random number sequence whose values are in the range [1, <strong>m</strong>]. Each value has a chance of <strong>p</strong> = 1/<strong>m</strong> appearing. Take a sample of size <strong>s</strong> from that sequence. For a given value in the sample, let <strong>n</strong> b... | user51547 | 51,547 | <p>Let <span class="math-container">$X_i$</span> denote the number of values in the sequence that take on value <span class="math-container">$i$</span>, where <span class="math-container">$1 \leq i \leq m$</span>, and let <span class="math-container">$Z_n$</span> the number of values in the sequence that are the duplic... |
1,494,022 | <p>Find a closed expression in terms of $n$.
$$\sum_{k=1}^n(k!)(k^2+k+1); n=1,2,3...$$<br>
Any idea about how to do this.. I'm a new to this so a little explanation would be helpful. Thanks in advance!</p>
| SchrodingersCat | 278,967 | <p>$$\sum_{k=1}^n(k!)(k^2+k+1)=\sum_{k=1}^n(k!)(k^2+2k+1-k)$$
$$=\sum_{k=1}^n(k!)[(k+1)^2-k]$$
$$=\sum_{k=1}^n(k+1)!(k+1)-k(k)!$$
$$=\sum_{k=1}^n(k+1)!(k+2-1)-(k+1-1)(k)!$$
$$=\sum_{k=1}^n[(k+2)!-(k+1)!-(k+1)!+(k)!]$$
$$=\sum_{k=1}^n[(k+2)!-(k+1)!]-\sum_{k=1}^n[(k+1)!-(k)!]$$
$$=(n+2)!-2-[(n+1)!-1]$$
$$=(n+2)!-(n+1)!-1... |
713,104 | <p>Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$?</p>
<p>Finding examples of games of order $2$ is easy (for example any impartial game), but I have not been able to think up an example with finite order where the order did not come from ... | Mark Bennet | 2,906 | <p>There is a general formula $$\sum_{k=1}^n\frac {k(k+1)\dots(k+r-1)}{r!}=\frac {n(n+1)\dots(n+r)}{(r+1)!}$$</p>
<p>Which can be proved by induction - base case $n=1$, both sides of the equation are equal to $1$.</p>
<p>Then $$\frac {n\left[(n+1)\dots(n+r)\right]}{(r+1)!}+\frac {\left[(n+1)\dots(n+r)\right]}{r!}=\fr... |
1,234,726 | <p>How many lattice paths are there from $(0, 0)$ to $(10, 10)$ that do not pass to the point $(5, 5)$ but do pass to $(3, 3)$?</p>
<p>What I have so far:</p>
<p>The number of lattice paths from $(0,0)$ to $(n,k)$ is equal to the binomial coefficient $\binom{n+k}n$ (according to Wikipedia). So the number of lattic... | Hemant Rupani | 230,966 | <p>From (0,0) to (3,3) there are $\binom{6}{3}=20$ paths.</p>
<p>From (3,3) to (10,10) there are $\binom{14}{7}=3432$ paths.</p>
<p>From (5,5) to (10,10) there are $\binom{10}{5}=252$ paths.(the paths you don't need)
6*252 paths that from (3,3) to (5,5) to (10,10)</p>
<p>Hence, total conditional lattice path will b... |
4,234,095 | <p>I need to show that <span class="math-container">$[\mathbb{Q}(2^{1/4},2^{1/6}):\mathbb{Q}]$</span> is a field extension of degree <span class="math-container">$12$</span>. It is possible to show that the degree is at least <span class="math-container">$12$</span> because it is divisible by <span class="math-containe... | Alex Wertheim | 73,817 | <p>You've already done the hard work. Let <span class="math-container">$K = \mathbb{Q}(2^{1/4}, 2^{1/6})$</span>, let <span class="math-container">$\alpha = 2^{1/12}$</span>, and let <span class="math-container">$L = \mathbb{Q}(\alpha)$</span>. Clearly, <span class="math-container">$K \subset L$</span>, since <span cla... |
376,861 | <p>A knot can be represented with a <a href="http://katlas.math.toronto.edu/wiki/MorseLink_Presentations" rel="nofollow noreferrer">Morse link presentation</a>, as a combination of cups, caps and crossings (which is not uniquely determined by the knot, of course):</p>
<p><a href="https://i.stack.imgur.com/47mxu.png" re... | M. Ozawa | 46,903 | <p>I think this was Question 3.5 in "Thin position in the theory of classical knots" by Martin Scharlemann, and a counterexample was given by Zuapn.</p>
<p><em>Zupan, Alexander</em>, <a href="http://dx.doi.org/10.2140/agt.2011.11.1097" rel="nofollow noreferrer"><strong>Unexpected local minima in the width com... |
4,521,774 | <p>In many posts on MSE, it is discussed that Cauchy sequences can't be defined in General topological spaces and in a typical topology book it is discussed what converging sequences are, but, what I don't understand is, why, on an abstract level, does convergence generalize even without a metric while cauchy-ness does... | jacktrnr | 901,848 | <p>Here's my attempt to explain.</p>
<p>Consider this definition of convergence that does not require a notion of metric:</p>
<p><em>A sequence <span class="math-container">$\{x_n\}_{n\in\mathbb N}$</span> converges to <span class="math-container">$x$</span> if and only if for each open neighborhood <span class="math-c... |
2,209,034 | <blockquote>
<p>The question asks to compute:
<span class="math-container">$$\sum_{k=0}^{n-1}\dfrac{\alpha_k}{2-\alpha_k}$$</span>
where <span class="math-container">$\alpha_0, \alpha_1, \ldots, \alpha_{n-1}$</span> are the <span class="math-container">$n$</span>-th roots of unity.</p>
</blockquote>
<p>I started... | Jaideep Khare | 421,580 | <p>Since <span class="math-container">$\alpha_0,\alpha_1,\alpha_2, \dots , \alpha_{n-1}$</span> are roots of the equation</p>
<p><span class="math-container">$$x^n-1=0 ~~~~~~~~~~~~~ \cdots ~(1)$$</span> </p>
<p>You can apply <a href="https://en.wikipedia.org/wiki/Polynomial_transformation" rel="nofollow noreferrer">T... |
20,784 | <p>I have a set of 3-space coordinates for the atoms of a molecule (I could also transform them into spheres with radii corresponding to the atoms they represent). I would like to place this molecule into the tightest possible 3D rectangular bounding box and determine the coordinates for the box vertices. Is there a ... | Daniel Lichtblau | 51 | <p>Here is an idea for an approximate method. Center the data, compute the singular value decomposition, and use the right factor rotation matrix to align the singular values axes with the coordinate axes (I'm not saying that very well but it gives the rotation matrix we want). Now take min and max values along coordin... |
20,784 | <p>I have a set of 3-space coordinates for the atoms of a molecule (I could also transform them into spheres with radii corresponding to the atoms they represent). I would like to place this molecule into the tightest possible 3D rectangular bounding box and determine the coordinates for the box vertices. Is there a ... | Carl Woll | 45,431 | <p>In versions M10.4+ you can use the function <a href="http://reference.wolfram.com/language/ref/BoundingRegion" rel="nofollow noreferrer"><code>BoundingRegion</code></a>. Taking @DanielLichtblau's example:</p>
<pre><code>SeedRandom[11111];
dmat=DiagonalMatrix[{4,7,13}];
rmat=RandomReal[{-1,1},{3,3}];
mat=dmat+rmat+T... |
3,491,978 | <blockquote>
<p>Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.</p>
</blockquote>
<p>My attempt:</p>
<p>(X,d) is compact. Therefore there exists a finite subcover of X.</p>
<p>Any element x in X must lie ... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$\mathcal{A}$</span> be an open cover of <span class="math-container">$X$</span>. For each <span class="math-container">$x \in X$</span> we can find <span class="math-container">$A_x \in \mathcal{A}$</span> such that <span class="math-container">$x \in A_x$</span>, and as <span class... |
3,491,978 | <blockquote>
<p>Let (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that for every x ∈ X, B(x,ε) is contained in some member of the cover.</p>
</blockquote>
<p>My attempt:</p>
<p>(X,d) is compact. Therefore there exists a finite subcover of X.</p>
<p>Any element x in X must lie ... | Milo Brandt | 174,927 | <p>You have proved that for all <span class="math-container">$x$</span>, there exists a <span class="math-container">$\varepsilon$</span> so that <span class="math-container">$B(x,\varepsilon)$</span> is contained in some member of the cover - which does not rely on the fact that the chosen subcover is finite or on com... |
2,379,955 | <p>Assume I want to minimise this:
$$ \min_{x,y} \|A - x y^T\|_F^2$$
then I am finding best rank-1 approximation of A in the squared-error sense and this can be done via the SVD, selecting $x$ and $y$ as left and right singular vectors corresponding to the largest singular value of A.</p>
<p>Now instead, is possible t... | Royi | 33 | <p>This is a Convex Optimization Problem and you can easily solve it using CVX:</p>
<pre><code>numRows = 5;
mA = randn([numRows, numRows]);
vB = randn([numRows, 1]);
cvx_begin('quiet')
cvx_precision('best');
variable vX(numRows)
minimize( norm(mA - vB * vX.', 'fro') )
cvx_end
disp([' ']);
disp(['CVX So... |
3,555,084 | <blockquote>
<p>Let
<span class="math-container">$$f(z) = e^z (1+\cos\sqrt{z} ) $$</span>
<span class="math-container">$\Omega=\{z\in\Bbb C: |z|\gt r\}$</span>, <span class="math-container">$r\gt 0$</span>. What is <span class="math-container">$f(\Omega)$</span>?</p>
<p>where <span class="math-container">$... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$x^2(x^2+6)=(x^2+3)^2-3^2$$</span></p>
<p>WLOG <span class="math-container">$x^2+3=3\csc t,t\to0^+$</span></p>
<p><span class="math-container">$$\lim_{t\to0^+}3(\csc t-1)(3\csc t-3\cot t)$$</span></p>
<p><span class="math-container">$$=\lim\dfrac{9(1-\sin t)(1-\cos t)}{\sin^2t}$$</sp... |
408,128 | <p>Suppose a vector $y$ and a <em>symmetric</em> matrix $M$ are given.</p>
<p>\begin{equation}
\forall x; \quad x^Ty=0 \implies x^TMx \ge 0
\end{equation}</p>
<p>Prove that $M$ has at most one negative eigenvalue.</p>
| user1551 | 1,551 | <p><strong>Hint.</strong> As $M$ is real symmetric, it has a orthonormal eigenbasis. Suppose $(\lambda_1,v_1),(\lambda_2,v_2)$ are two eigenpairs of $M$ with $\lambda_1,\lambda_2<0$ and $v_1\perp v_2$. Use a dimension argument to show that $\mathrm{span}\{y\}^\perp\cap\mathrm{span}\{v_1,v_2\}\neq0$. That is, there e... |
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