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 |
|---|---|---|---|---|
1,354,015 | <blockquote>
<p>If <span class="math-container">$a, b, c, d, e, f, g, h$</span> are positive numbers satisfying <span class="math-container">$\frac{a}{b}<\frac{c}{d}$</span> and <span class="math-container">$\frac{e}{f}<\frac{g}{h}$</span> and <span class="math-container">$b+f>d+h$</span>, then <span class="... | Klaus Draeger | 65,787 | <p>The updated question (with the additional constraint $b+f>d+h$) is also false. For example,</p>
<p>$\frac{1}{1}<\frac{3}{2}$ and $\frac{9}{4}<\frac{5}{2}$, but $\frac{1+9}{1+4} = \frac{10}{5} = \frac{8}{4} = \frac{3+5}{2+2}$.</p>
|
739,301 | <p>Show that a Dirac delta measure on a topological space is a Radon measure.</p>
<p>Show that the sum of two Radon measures is also a Radon measure.
Please help me.</p>
| George | 54,962 | <p>Dirac measure is a measure $\delta_{x}$ on a topological space $X$ (with a Borel $\sigma$-algebra $B(X)$ of subsets of $X$) defined for a given $x \in X$ and any Borel set $A \subseteq X$ by $\delta_{x}(A)=1$ iff $x \in A$. Notice that $\{ x\}$ like $\emptyset$ is compact subset of $X$. For each $A \in B(X)$ and ... |
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | DonAntonio | 31,254 | <p>$$\lim_{x\to\ 0}\frac{\log\cos 2x}{x\sin x}\stackrel{\text{L'Hospital}}=\lim_{x\to 0}\frac{-2\tan 2x}{\sin x+x\cos x}{}\stackrel{\text{L'H}}=\lim_{x\to 0}-\frac{4\sec 2x}{2\cos x-x\sin x}=-\frac{4}{2}=-2$$</p>
|
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Community | -1 | <p>One of my favourite books for harmonic analysis is Fourier analysis by Javier Duoandikoetxea. In fact, I would recommend that as your first port of call for learning harmonic analysis if you already have some background (such as an undergraduate/postgraduate course in harmonic analysis). That is a terrific reference... |
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Thomas Kojar | 99,863 | <p>Here are a few relevant books that appeared since this question was asked.</p>
<ul>
<li>Barry Simon's <a href="https://www.amazon.ca/Harmonic-Analysis-Barry-Simon/dp/1470411024/ref=sr_1_2?keywords=simon%20harmonic%20analysis&qid=1567982213&s=books&sr=1-2" rel="nofollow noreferrer">Harmonic analysis volu... |
2,704,955 | <p>In my test on complex analysis I encountered following problem:</p>
<blockquote>
<p>Find $\oint\limits_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z$</p>
</blockquote>
<p>So first I observed that function $z\text{Im}(z)$ is not holomorphic at least on real axis. Therefore we have to intgrate using parametrization.... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
3,298,311 | <p>How can I prove (by definition) that, if <span class="math-container">$a, b \in \mathbb{R}$</span> and <span class="math-container">$a<b$</span>, then <span class="math-container">$[a, b]$</span> is equal to the set of accumulation (limit) points?</p>
<p>Let <span class="math-container">$(E, d)$</span> a metric ... | Peter Szilas | 408,605 | <p>Attempt:</p>
<p><span class="math-container">$U:= $</span>{<span class="math-container">$x| x <a$</span>}; <span class="math-container">$V:=$</span>{<span class="math-container">$x|x >b$</span>};</p>
<p><span class="math-container">$U,V$</span> are open sets, </p>
<p>1) <span class="math-container">$U \cup... |
536,805 | <p>Three cards are drawn sequentially from a deck that contains 16 cards numbered 1 to 16 in an arbitrary order. Suppose the first card drawn is a 6.</p>
<p>Define the event of interest, A, as the set of all increasing 3-card sequences, i.e. A={(x1,x2,x3)|x1 < x2 < x3}, where x1,x2,x3∈{1,⋯,16}. Define event B as... | drhab | 75,923 | <p>If I understand you well then $A\cap B=\left\{ \left(x_{1},x_{2},x_{3}\right)\mid x_{1}<x_{2}<x_{3}\wedge x_{1}=6\right\} =\left\{ \left(6,x_{2},x_{3}\right)\mid6<x_{2}<x_{3}\right\} $
so $\left|A\cap B\right|=\left({10\atop 2}\right)=45$</p>
<p>Two distinct numbers are to be chosen from $\left\{ 7,\cdo... |
2,159,083 | <p>I am currently translating a research paper from French (which I do not speak well). I have made good progress with copious use of google translate & switching between French & English versions of articles on wikipedia, coupled with knowledge in the given field. However, I am stuck on the following ($M$ is a... | Derek Elkins left SE | 305,738 | <p>Before focusing on the specific question, I'd like to provide some context. First, every (non-trivial) topos has a Boolean subtopos. This is essentially what you say, and it roughly corresponds to the double negation interpretation of classical logic into intuitionistic logic. However, one of the things that makes... |
1,690,092 | <p>$a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$</p>
<p>and I need to check whether this sequence converges to a limit without finding the limit itself. I think about using the squeeze theorem that converges to something (I suspect '$1$').</p>
<p>But I wrote $a_{n+1}$ ... | Dac0 | 291,786 | <p>You're right, you just have to consider that
$$\frac{n}{\sqrt{n^2+n}}\ge a_{n} \ge \frac{n}{\sqrt{n^2+2n-1}}$$
Then take the limit and you're done. </p>
|
626,928 | <p>I took linear algebra course this semester (as you've probably noticed looking at my previously asked questions!). We had a session on preconditioning, what are they good for and how to construct them for matrices with special properties. It was a really short introduction for such an important research topic, so I'... | Han de Bruijn | 96,057 | <p>Look what I have here in my bookshelves: the original thesis by Henk (H.A.) van der Vorst where it all started with , <A HREF="http://books.google.nl/books/about/Preconditioning_by_Incomplete_Decomposit.html?id=9Ad7NwAACAAJ&redir_esc=y" rel="nofollow">titled</A> : <I>Preconditioning by Incomplete Decompositions<... |
40,500 | <blockquote>
<p>What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?</p>
</blockquote>
<p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a conf... | The Bridge | 2,642 | <p>Hi, </p>
<p>Regarding Martingales you can see them as fair games
This means that if the (martingale) process represents your (random) wealth, you should not be able to design a strategy to increase your current wealth, no matter what the outcome of the sample space is.</p>
<p>Brownian Motion can be seen as a limi... |
2,372,743 | <p>Show that $$\lim_{x\to\infty} \int_{1}^x e^{-y^2} dy$$
exist and is in $[e^{-4},1]$</p>
<p>I cannot find a good minoration, I must show it whitout using the double integration, I totally don't know how to solve it</p>
| ty. | 360,063 | <p>I have a funny answer, if you find Fourier transform without integrating fun. :D</p>
<p>Note that $e^{-y^2}$ is monotonically decreasing as $|y|\rightarrow\infty$ and that the function is always positive. That means
\begin{equation}
0<\int_{1}^{\infty}e^{-y^2}dy<\int_{0}^{\infty}e^{-y^2}dy.
\end{equation}
If... |
1,893,609 | <p>I am trying to show that $A=\{(x,y) \in \Bbb{R} \mid -1 < x < 1, -1< y < 1 \}$ is an open set algebraically. </p>
<p>Let $a_0 = (x_o,y_o) \in A$. Suppose that $r = \min\{1-|x_o|, 1-|y_o|\}$ then choose $a = (x,y) \in D_r(a_0)$. Then</p>
<p>Edit: I am looking for the proof of the algebraic implication t... | Narasimham | 95,860 | <p>Unless you <em>delete/discard</em> one out of the four given points and you do not have a unique circle with a center and radius.This is over-determined but consistent. </p>
|
4,133,760 | <p>Dr Strang in his book linear algebra and it's applications, pg 108 says ,when talking about the left inverse of a matrix( <span class="math-container">$m$</span> by <span class="math-container">$n$</span>)</p>
<blockquote>
<p><strong>UNIQUENESS:</strong> For a full column rank <span class="math-container">$r=n . A x... | hm2020 | 858,083 | <p><strong>Question:</strong> "I understand why there can be at most one solution for a full column rank but how does that lead to A having a left inverse? I'd be grateful if someone could help or hint at the answer."</p>
<p><strong>Answer:</strong> Let <span class="math-container">$k$</span> be a real number... |
4,480,570 | <p>I am reading over the proof of <strong>Lemma 10.32 (Local Frame Criterion for Subbundles)</strong> in Lee's <em>Introduction to Smooth Manifolds</em>.</p>
<p>The lemma says</p>
<blockquote>
<p>Let <span class="math-container">$\pi: E \rightarrow M$</span> be a smooth vector bundle and suppose that for each <span cla... | Leonhard Euler | 998,037 | <p>Unfortunately, Theorem 5.8 only works for smooth manifolds without boundary. But here <span class="math-container">$E$</span> probably has boundary. A better theorem is Theorem 5.51，but it also requires that <span class="math-container">$M$</span> is a smooth manifold without boundary there.</p>
|
2,362,942 | <p>How could I notate a matrix rotation?</p>
<p>Example:
$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\:\:\: A_{\text{rotated}} = \begin{pmatrix} c & a \\ d & b\end{pmatrix}$. </p>
<p>Notice the whole matrix is "rotated" clockwise. Is there any notation for this, and anyway to compute it genera... | Angina Seng | 436,618 | <p>Your rotated matrix is
$$A^t\pmatrix{0&1\\1&0}.$$</p>
|
46 | <p>I have solved a couple of questions myself in the past, and I think some of them are interesting to the public and will most likely appear in the future. One example for this is the question how to enable antialiasing in the Linux frontend, for which there is no native support right now. My question would now be whe... | Andy Ross | 43 | <p>Most of these answers say, "If it is interesting, ask it." But what do we mean by "interesting"? Or more appropriately, interesting to whom? </p>
<p>If we want only Mathematica experts to visit the site then we should probably ask questions only an expert would ask. I'm of the thinking that we probably want som... |
133,936 | <p>I am trying to understand a part of the following theorem:</p>
<blockquote>
<p><strong>Theorem.</strong> Assume that $f:[a,b]\to\mathbb{R}$ is bounded, and let $c\in(a,b)$. Then, $f$ is integrable on $[a,b]$ if and only if $f$ is integrable on $[a,c]$ and $[c,b]$. In this case, we have
$$\int_a^bf=\int_a^cf+\in... | Community | -1 | <p>(Too long for a comment.)</p>
<p>That the answer has been spelt out, I wanted to clarify this thing about refinements:</p>
<blockquote>
<ol>
<li>Neither of <span class="math-container">$P_i$</span> is a refinement of <span class="math-container">$P$</span>.</li>
</ol>
</blockquote>
<p>Yes. You're right. In fact, the... |
1,318,934 | <blockquote>
<p>Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by
$$f(x)=\alpha x^q+\alpha^q x$$
for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 \}$.
Show that $f$ is a bijection.</p>
</blockquote>
<p>Hint: If $\beta \in \ker(f) \setminus \{ 0 \}$ c... | Jyrki Lahtonen | 11,619 | <p>I like Adam's solution a lot. I arrived at the scene late, so I'm just adding this as my best guess as to what the hint means.</p>
<p>The mapping $f$ is linear over the subfield $\Bbb{F}_q$, so it suffices to show that its kernel is trivial. Assume that there exists a $\beta$ such that $f(\beta)=0$. This implies th... |
1,688,184 | <blockquote>
<p>Prove that $2\sqrt 5$ is irrational</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Suppose $$2\sqrt 5=\frac p q\quad\bigg/()^2$$ </p>
<p>$$\Longrightarrow 4\cdot 5=\frac{p^2}{q^2}$$</p>
<p>$$\Longrightarrow 20\cdot q^2=p^2$$</p>
<p>$$\Longrightarrow q\mid p^2$$</p>
<p>$$\text{gcd}(p,q)... | Galc127 | 111,334 | <p>You have $20|p^2$, thus $2|p^2\wedge 5|p^2$, hence by <a href="https://en.wikipedia.org/wiki/Euclid's_lemma" rel="nofollow">Euclid's lemma</a> $2|p\wedge 5|p$, hence $10|p$, so we can write $p=10k$ for $k\in\mathbb{Z}$ and then $20q^2=100k^2\Rightarrow q^2=5k^2$, hence $5|q^2$, and by the same lemma $5|q$, thus ... |
195,006 | <p>I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.</p>
<p>Since I may even be using incorrect terminology, I'll try to explain what the terms I'm using mean in my mind. Please correct my term... | Ross Millikan | 1,827 | <p>It sounds like you are thinking of your set as a tree and want to know if there is an infinite path. The path through the tree would be the ordering you are talking about. <a href="http://en.wikipedia.org/wiki/K%C3%B6nig%27s_lemma" rel="nofollow">König's lemma</a> assures you that if the tree is finitely branching... |
1,147,373 | <p>I need to prove that
<span class="math-container">$$I = \int^{\infty}_{-\infty}u(x,y) \,dy$$</span>
is independent of <span class="math-container">$x$</span> and find its value, where
<span class="math-container">$$u(x,y) = \frac{1}{2\pi}\exp\left(+x^2/2-y^2/2\right)K_0\left(\sqrt{(x-y)^2+(-x^2/2+y^2/2)^2}\right)$$<... | JJacquelin | 108,514 | <p>Are you sure that there is no typo in your equation ?</p>
<p>With :
\begin{eqnarray}
u(x,y) = \frac{1}{2\pi}\exp\left(+x^2/2-y^2/2\right)K_0\left(\sqrt{(x-y)^2+(-x^2/2+y^2/2)^2}\right)
\end{eqnarray}</p>
<p>I do not find $I=\int^{\infty}_{-\infty}u(x,y) \,d y \simeq 2.38$ and $I(x)$ is not constant.</p>
<p>For ex... |
1,225,122 | <p>So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when working with these ideals is when we reduce the generating set to something much simpler. For e.g.</p>
<p>Consider the I... | Bill Dubuque | 242 | <p>The tuple notation is used both for gcds and ideals because they share many of the same laws, e.g. those below that are familiar from wide use in the Euclidean algorithm and related results:</p>
<p>$$\quad a(b,c)\, =\, (ab,ac)\qquad \rm [Distributive\ Law]\qquad $$</p>
<p>$$ (a,b)\, =\, (a,b')\ \ \ {\rm if}\ \ \ b... |
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | sirfoga | 83,083 | <p><strong>Hint</strong>: Simply try to square both sides of the equation (since they are both positive numbers).</p>
|
1,547,972 | <p>Why is $\sin(x^2)$ similar of $\ x \sin(x)$? </p>
<p>I graphed it using desmos and when I look at it, the behavior as x approaches zero seems to be to oscillate less. </p>
<p>Yet as x approaches infinity and negative infinity $\sin(x^2)$ oscillates between y=1 and y=-1 while $\ x *sin(x)$ oscillates between y=x a... | Thomas Andrews | 7,933 | <p>When $x$ gets close to zero, $\sin x \approx x-\frac{x^3}{6}$. So $$\sin(x^2)\approx x^2-\frac{x^6}{6}\\x\sin(x)\approx x^2-\frac{x^4}{6}$$</p>
<p>Now, when $x$ is small, $x^4$ and $x^6$ are "very small." So the functions are dominated by $x^2$ near $x=0$. Indeed, if you graphed $y=x^2$ alongside, you'd see that bo... |
1,547,972 | <p>Why is $\sin(x^2)$ similar of $\ x \sin(x)$? </p>
<p>I graphed it using desmos and when I look at it, the behavior as x approaches zero seems to be to oscillate less. </p>
<p>Yet as x approaches infinity and negative infinity $\sin(x^2)$ oscillates between y=1 and y=-1 while $\ x *sin(x)$ oscillates between y=x a... | MKBG | 293,645 | <p>Try finding the $ \lim_{x \rightarrow 0} \frac{\sin(x^2)}{x \sin(x)}$, you will see it is finite, thus they behave simiraly near $0$, which is important for approximations near $0$.
Although $\sin(x^2)$ is bounded, and $x \sin(x)$ is not, they have local minima and maxima at the same points.</p>
|
905,672 | <p>Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.Prove that $S$ a is a subgroup of $G$.</p>
<p>My try:</p>
<p>For $h$ in $S$, If I show that $hS=S$, then that would imply that $S$ is closed. </p>
<p>Now $hS$ is a partiton o... | Hamou | 165,000 | <p>Let $x,y\in S$, and consider the sets $x^{-1}S$ and $y^{-1}S$, now remark that $1\in x^{-1}S\cap y^{-1}S$ as $\{aS\}_{a\in G}$ is a partition of $G$, then $x^{-1}S=y^{-1}S$, since $x^{-1}\in x^{-1}S$ we get $x^{-1}\in y^{-1}S$, there exists $s\in S$ such that $x^{-1}=y^{-1}s$, so $yx^{-1}=s\in S$. It follow that $S$... |
2,494,069 | <p>All the calculators I've tried overflow at 2^1024. I'm not sure how to go about calculating this number and potentially even larger ones.</p>
| Peter | 82,961 | <p>Calculate $$2048\cdot \ln(2)$$ which is equal to $\ln(2^{2048})$</p>
|
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Brian M. Scott | 12,042 | <p>The probability is $\frac14$. The argument given in <a href="https://math.stackexchange.com/a/172368/12042">this answer</a> to the corresponding question for $n$-gons applies equally well to circles. <a href="https://math.stackexchange.com/a/1407/12042">This answer</a> to the generalization of this question to arbit... |
3,775,554 | <p><strong>Identity for a set X:</strong></p>
<p><em>The set X has an identity under the operation if there is
an element <strong>j</strong> in set X such that <strong>j * a = a * j = a</strong> for all elements <strong>a</strong> in set X.</em></p>
<p>According to my college book the counting numbers don' t have an ... | Klaus | 635,596 | <p>The operation is crucial here. <span class="math-container">$(\mathbb{N},\cdot)$</span> does have an identity, namely <span class="math-container">$1$</span> as you observed. For <span class="math-container">$(\mathbb{N},+)$</span> the identity would be <span class="math-container">$0$</span> and so it depends on yo... |
11,724 | <p>What is the compelling need for introducing a theory of $p$-adic integration?</p>
<p>Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a Lebesgue measure on $p$-adic spaces, and just integrate real or complex valued functions on $p$-adic spaces, or ... | Matt E | 221 | <p>I would normally take $p$-adic integration to mean "integration of $p$-adic valued functions" or "integration of differential forms with some kind of $p$-adic valued functions as coefficients", where the integration is also taking place over some kind of $p$-adic space or manifold.</p>
<p>The reason for wanting suc... |
4,170,150 | <p>I have the following two definitions:</p>
<p>Let <span class="math-container">$S$</span> be a subset of a Banach space <span class="math-container">$X$</span>.</p>
<ol>
<li><p>We say that <span class="math-container">$S$</span> is <em>weakly bounded</em> if <span class="math-container">$l\in X^*$</span>, the dual sp... | alepopoulo110 | 351,240 | <p>Assume that <span class="math-container">$1$</span> holds. For <span class="math-container">$s\in S$</span> define <span class="math-container">$\phi_s:X^*\to\mathbb{C}$</span> by <span class="math-container">$\phi_s(f)=f(s)$</span>. Then <span class="math-container">$\phi_s$</span> is a bounded linear functional, s... |
2,459,493 | <p>Here's the <span class="math-container">$C^1$</span> norm : </p>
<p><span class="math-container">$|| f || = \sup | f | + \sup | f '|$</span></p>
<p>where the supremum is taken on <span class="math-container">$[a, b]$</span>. </p>
<p>Please, justify your answer (proofs or counterexamples are needed). </p>
| Donald Splutterwit | 404,247 | <p>Hint:
How many sequences are there in total ?</p>
<p>How many sequences are there that do not contains both $a$ and $b$ ?</p>
<p>So ... </p>
<p>Alternatively :</p>
<p>The sure fire bet to get the answer is to list the $10$ possibilities and calculate their multiplicities. ($S$ = sequence upto permutation, $M$= ... |
1,302,891 | <p>I would like to ask you a question about the following question.</p>
<p>Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow \ c-}{f(x)}$ and $\lim_{x \ \rightarrow \ c+}{f(x)}$ both exist.</p>
<p>If $f$ is continuous at $c$ then obv... | gnometorule | 21,386 | <p>The function's amplitude on $(a,b)$ is bounded by $m:= f(b)-f(a) \lt \infty$. Look at the right-side limit at a point of discontinuity $c$ (existence of the left-side limit follows in the same way). </p>
<p>Fix $m \gt \epsilon \gt 0$. Pick a point $x_1$ in the interval to the right of $c$. If $y_1 := f(x_1) - f(c) ... |
4,618,975 | <p>I've read the definition of the ring homomorphism:<br />
<strong>Definition</strong>. Let <span class="math-container">$R$</span> and <span class="math-container">$S$</span> be rings. A ring homomorphism is a function <span class="math-container">$f : R → S$</span> such that:<br />
(a) For all <span class="math-cont... | user 1 | 133,030 | <p><span class="math-container">$f : R \to R[x]$</span>, where <span class="math-container">$x$</span> is indeterminate.<br />
<span class="math-container">$f : R \to S^{-1} R$</span>, where <span class="math-container">$S$</span> is a multiplicatively closed set (localization).</p>
|
3,956,392 | <p>So the question is as follows:</p>
<blockquote>
<p>An urn contains m red balls and n blue balls. Two balls are drawn uniformly at random
from the urn, without replacement.</p>
</blockquote>
<blockquote>
<p>(a) What is the probability that the first ball drawn is red?</p>
</blockquote>
<blockquote>
<p>(b) What is the... | herb steinberg | 501,262 | <p>The secon ball probability is<span class="math-container">$\frac{m}{m+n}\frac{m-1}{m+n-1}+\frac{n}{n+m}\frac{m}{m+n-1}=\frac{m}{n+m}$</span> This is the same as the first, which, by symmetry, is what it should be.</p>
|
2,555,811 | <p>Is there a name for the following type of ordering on some set $S$ {$a,b,c$} that includes only $>$ and $=$ for example: </p>
<p>$$a>b>c$$
$$a>b=c$$
$$a=b>c$$
$$a=b=c$$</p>
<p>Is there some name for these orderings? </p>
<hr>
<p>I know that all these <em>satisfy</em> a <em>total preorder</em> on $... | Guy Fsone | 385,707 | <p>Let $$f(h)= \ln\left(\frac{1}{n}\sum\limits_{k=1}^{n} k^{h}\right)\implies f'(h) =\frac{\left(\frac{1}{n}\sum\limits_{k=1}^{n} k^{h}\ln k \right)}{\left(\frac{1}{n}\sum\limits_{k=1}^{n} k^{h} \right)}$$
$$f(0)= 0 ~~~and~~~~f'(0)=\left(\frac{1}{n}\sum\limits_{k=1}^{n} \ln k\right)=\color{blue}{ \frac{1}{n}\ln \left(... |
83,764 | <p>I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out <em>why</em> this is happening (for quite some time now actually). Any help is greatly appreciated. Here goes then:</p>
<p>Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^... | Issa | 32,602 | <p>This is not a surprising fact for driftless systems to have symmetries. In our previous work we considered only nonlinear systems with drift and classified the symmetries accordingly. In case of a driftless system
\dot x=F(x)u you can interpret $F$ (that's what it is) as a distribution of vector fields $f_1, \dots, ... |
602,248 | <p>This is the system of equations:
$$\sqrt { x } +y=7$$
$$\sqrt { y } +x=11$$</p>
<p>Its pretty visible that the solution is $(x,y)=(9,4)$</p>
<p>For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$.
Then I subtracted one equation from the another such that I got $4$ on RHS and factorized LHS to get two factors in te... | rajb245 | 72,919 | <p>If you move things around and square the equations, you have the following two starting equations.
$$x=(7-y)^2$$
$$y=(11-x)^2$$</p>
<p>If you plug one of these into the other and factor out a term, you get
$$
(x-9)(x^3-35x^2+397x-1444)=0.
$$</p>
<p>This is a quartic equation, with four solutions in general. There ... |
3,173,349 | <p>The question:</p>
<p>Suppose G is a finite group of Isometries in the plane. Suppose p is fixed by G. Prove that G is conjugate into O(2).</p>
<p>What exactly does "conjugate into O(2)" mean and how would I proceed?</p>
| AnalysisStudent0414 | 97,327 | <p>By reducing mod <span class="math-container">$2$</span> and <span class="math-container">$3$</span> you can show that <span class="math-container">$2$</span> and <span class="math-container">$3$</span> divide <span class="math-container">$m$</span>, hence <span class="math-container">$m^2$</span> has actually to be ... |
4,590,713 | <p>If I know the McLaurin series of
<span class="math-container">$$f(x)=\sum_{n=0}^\infty a_n x^n,$$</span>
can I say something about the McLaurin series of
<span class="math-container">$$e^{f(x)}=\sum_{n=0}^\infty b_n x^n?$$</span>
In other words, what is the relation between <span class="math-container">$a_n$</span> ... | KCd | 619 | <p>Let's aspire to do better than an <span class="math-container">$\Omega$</span>-estimate by getting an asymptotic estimate.</p>
<p>Here is a very useful fact for estimating partial sums of sequences: if <span class="math-container">$\{a_n\}$</span> and <span class="math-container">$\{b_n\}$</span> are sequences of po... |
1,969,479 | <blockquote>
<p>Prove that the symmetric point of orthocenter $H$ of a triangle with respect to the midpoint of any side resides on the triangle's circumcircle.
<a href="https://i.stack.imgur.com/tkh4c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tkh4c.png" alt="enter image description here"><... | Jack D'Aurizio | 44,121 | <p>The symmetric of $H$ with respect to the $BC$-side, say $J$, lies on the circumcircle of $ABC$ since $\widehat{BHC}+\widehat{BAC}=\pi$. If we consider the symmetric of $J$ with respect to the perpendicular bisector of $BC$ we get $K$, hence $K$ lies on the circumcircle of $ABC$, too, since the circumcircle of $ABC$ ... |
1,969,479 | <blockquote>
<p>Prove that the symmetric point of orthocenter $H$ of a triangle with respect to the midpoint of any side resides on the triangle's circumcircle.
<a href="https://i.stack.imgur.com/tkh4c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tkh4c.png" alt="enter image description here"><... | Robert Z | 299,698 | <p>A complex number proof. Without loss of generality we may assume that <span class="math-container">$A,B,C$</span> are complex numbers with the unit circle of <span class="math-container">$\mathbb{C}$</span> centered at the origin <span class="math-container">$O$</span>. Then <span class="math-container">$K=-C$</spa... |
1,395,093 | <blockquote>
<p>Prove that if $f\in L^1([0,1],\lambda)$ is not constant almost
everywhere then there exists an interval so that
$\int_I\!f\,\mathrm{d}\lambda\neq 0$. Here $\lambda$ is the Lebesgue
measure.</p>
</blockquote>
<p>Since this is obviously true for continuous functions, I've been trying to use the f... | Christopher Carl Heckman | 261,187 | <p>To find critical points, inflection points, etc., just do what you do when you don't have any parameters. The only difference is that the points will involve the value of $a$.</p>
<p>For instance,
$$f'(x) = \frac{-a}{x^2} + \frac{-2a}{x^3} = \frac{-ax-2a}{x^3}.$$
Now, for which values of $x$ is this expression zer... |
172,157 | <p>Today I was asked if you can determine the divergence of $$\int_0^\infty \frac{e^x}{x}dx$$ using the limit comparison test.</p>
<p>I've tried things like $e^x$, $\frac{1}{x}$, I even tried changing bounds by picking $x=\ln u$, then $dx=\frac{1}{u}du$. Then the integral, with bounds changed becomes $\int_1^\infty \f... | Keith Anker | 126,605 | <p>∫ ε to 1 e^x / x dx > ∫ ε to 1 1/x dx, → ∞ as ε → 0; and</p>
<p>∫ 1 to N e^x / x dx > ∫ 1 to N 1/x dx, → ∞ as N → ∞;</p>
<p>and so we are done.</p>
|
38,594 | <p>One of the definitions of a Galois extension is that $E/K$ is Galois iff $E$ is the splitting field of some <strong>separable</strong> polynomial $f(x) \in K[x]$, yes?</p>
<p>I want to understand why the following is true:</p>
<blockquote>
<p>Let $f(x) \in \mathbb{Q}[x]$ be a polynomial and let $F$ be a splittin... | Patrick Da Silva | 10,704 | <p>That is because there is a theorem that states that in every perfect field ($\mathbb Q$ and $\mathbb F_p$ are examples), every irreducible polynomial is separable (i.e. all their roots have multiplicity 1). Your polynomial f(x) in your assertion is not necessarily separable, but if you write
$$
f(x) = (p_1(x))^{a_... |
3,963,517 | <blockquote>
<p>Write to power series of <span class="math-container">$$f(x)=\frac{\sin(3x)}{2x}, \quad x\neq 0.$$</span></p>
</blockquote>
<p>I try to get a series for <span class="math-container">$\sin(3x)$</span> with <span class="math-container">$x=0$</span> and multiplying the series with <span class="math-contain... | K.defaoite | 553,081 | <p>Yes, that's completely fine. The Maclaurin series for <span class="math-container">$\sin$</span> is
<span class="math-container">$$\sin(t)=t-\frac{t^3}{3!}+\frac{t^5}{5!}+...$$</span>
So letting <span class="math-container">$t=3x$</span>,
<span class="math-container">$$\frac{1}{2x}\sin(3x)=\frac{1}{2x}\left(3x-\frac... |
3,486,456 | <p>Suppose <span class="math-container">$f:X\rightarrow Y$</span> is a homeomorphism. Show that if <span class="math-container">$X$</span> is hausdorff then so is <span class="math-container">$Y$</span>.</p>
<p>My attempt:
Let <span class="math-container">$y_1,y_2\in Y$</span> be distinct, by bijectivity of <span clas... | Yves Stalder | 722,843 | <p>There is a little problem with the sentence 'hence <span class="math-container">$f(V_1)\cap f(V_2)$</span> is the union of two open sets', because you should write <span class="math-container">$f(V_1)\cup f(V_2)$</span> instead of <span class="math-container">$f(V_1)\cap f(V_2)$</span>, and because you should prec... |
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Leucippus | 148,155 | <p>Using
$$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$
then
\begin{align}
D \left[ 10^{x} \, \log_{10}(x) \right] &= D \left[ e^{x \, \ln(10)} \, \frac{\ln(x)}{\ln(10)} \right] \\
&= \frac{1}{\ln(10)} \, \left[ \frac{10^{x}}{x} + \ln(10) \, 10^{x} \, \ln(x) \right] \\
&= 10^{x} \, \left[ \frac{1}{ x \, \ln(10... |
858,250 | <p>I would like to evaluate the following alternating sum of products of binomial coefficients:
$$\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$$
I had the idea to use Pascal recursion to re-express $\binom n k$ so that we always have $m$ as the upper index and I have been able to come up with nice expressions for
$$ ... | Sasha | 11,069 | <p>Let $c_k = (-1)^k \binom{m}{k} \binom{n}{k}$. Then
$$
\frac{c_{k+1}}{c_k} = - \frac{\binom{m}{k+1}}{\binom{m}{k}} \cdot \frac{\binom{n}{k+1}}{\binom{n}{k}} = - \frac{k-m}{k+1} \cdot \frac{k-n}{k+1}
$$
Hence
$$
c_k = c_0 \prod_{i=0}^{k-1} (-1) \frac{i-m}{i+1} \cdot \frac{i-n}{i+1} = \frac{(-1)^k}{k!} \frac{(-m)... |
109,061 | <p>Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and while its easy to remember and its possible for a person to use this method by hand, there are many better techniques a... | Math Gems | 75,092 | <p>One important consequence of being able to explicitly solve polynomial equations is that it permits great simplifications by <strong>linearizing</strong> what would otherwise be much more complicated <strong>nonlinear</strong> phenomena. The ability to factor polynomials completely into linear factors over $\mathbb ... |
3,830,971 | <p>The function <span class="math-container">$f(x)=\cot^{-1} x$</span> is well known to be neither even nor odd because <span class="math-container">$\cot^{-1}(-x)=\pi-\cot^{-1} x$</span>. it's domain is <span class="math-container">$(-\infty, \infty)$</span> and range is <span class="math-container">$(0, \pi)$</span>.... | Martin R | 42,969 | <p>From <a href="https://mathworld.wolfram.com/InverseCotangent.html" rel="nofollow noreferrer">Inverse Cotangent</a> on Wolfram MathWorld:</p>
<blockquote>
<p>There are at least two possible conventions for defining the inverse
cotangent. This work follows the convention of Abramowitz and Stegun
(1972, p. 79) and the ... |
1,430,369 | <p>I know how to prove this by induction but the text I'm following shows another way to prove it and I guess this way is used again in the future. I'm confused by it.</p>
<p>So the expression for first n numbers is:
$$\frac{n(n+1)}{2}$$</p>
<p>And this second proof starts out like this. It says since:</p>
<p>$$(n+1... | Calvin Khor | 80,734 | <p>This is a telescoping sum. The use of $(n+1)^2 - n^2 = 2n + 1$ is a clever trick, and it is only clear why we use it once you understand the whole argument. The idea is to first find $\sum_1^n (2k+1) = 2(1+\dots+n)+(1+\dots+1)$ and use this to find $\sum_1^n k =1+\dots+n$. </p>
<p>I'm going to colour code the given... |
2,416,910 | <p>Let $A:=\{-k,\ldots,-2,-1,0,1,2,\ldots,k\}$, $k<\infty$, $k\in \mathbb{N}$.</p>
<p>Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be such that $f(-x)=-f(x)$ for all $x\in A$ and $f(x+y)=f(x)+f(y)$ for all $x,y\in A$ such that $x+y\in A$.</p>
<p>Does it follow that there is $\alpha\neq 0$ such that $f(x)=\alpha x$ for ... | J.G | 293,121 | <p>The idea is to have a random variable with extreme values that are of opposite sign from the mean, so that shifting by the mean leads to the contributions of these terms to be augmented. To do this, those extreme values must correspondingly have low weight.</p>
<p>For an explicit example, let <span class="math-cont... |
157,985 | <p>Question from a beginner. I have data containing dates and values of the format:</p>
<pre><code> data = {{{2015, 1, 1}, 2}, {{2015, 1, 2}, 3}, {{2015, 2, 1}, 4}, {{2015, 2, 2}, 5}, {{2016, 1, 1}, 6}, {{2016, 1, 2}, 7}}
</code></pre>
<p>Aim is to multiply the values of each day in a month, e.g. for January 2015,... | Alan | 19,530 | <pre><code>GroupBy[data, Part[First[#], ;; 2] &, Apply[Times, Last /@ #] &]
</code></pre>
|
157,985 | <p>Question from a beginner. I have data containing dates and values of the format:</p>
<pre><code> data = {{{2015, 1, 1}, 2}, {{2015, 1, 2}, 3}, {{2015, 2, 1}, 4}, {{2015, 2, 2}, 5}, {{2016, 1, 1}, 6}, {{2016, 1, 2}, 7}}
</code></pre>
<p>Aim is to multiply the values of each day in a month, e.g. for January 2015,... | ubpdqn | 1,997 | <p><a href="http://reference.wolfram.com/language/ref/TimeSeriesAggregate.html" rel="nofollow noreferrer"><code>TimeSeriesAggregate</code></a> can also be used, e.g.:</p>
<pre><code>#1[[1, {1, 2}]] -> #2 & @@@TimeSeriesAggregate[data, "Month", Times @@ # &]
</code></pre>
<p>yielding:</p>
<pre><code>{{2015... |
59,949 | <p>How to track <code>Initialization</code> step by step? Any tricky solution is ok. One condition, <code>x</code> should be <code>DynamicModule</code> variable.</p>
<pre><code>DynamicModule[{x},
Dynamic[x],
Initialization :> (
x = 1;
Pause@1;
x = 2;
Pause@1;
x = 3;
)]... | Kuba | 5,478 | <h3>tl;dr;</h3>
<p>Use <code>SynchronousInitialization -> False</code> and, in case of simple <code>Dynamic[x]</code>, add <code>TrackedSymbols</code>: <code>Dynamic[x, TrackedSymbols :> {x}]</code> due to the bug mentioned below.</p>
<hr />
<p>Finally I'm able to answer my question with understanding. Two years,... |
4,023,366 | <p>So the question is pretty much described in the title already. I have to show the following result. I have tried it but am failing to do so. Anyone who can please help me in understanding its proof. I have attached the picture of formulas that may prove to be helpful.</p>
<p><span class="math-container">$\exists x.P... | Graham Kemp | 135,106 | <p>Let us get you <em>started</em> with the derivation for <span class="math-container">$\exists x~.P(x)~\lor~\exists x~.Q(x)\vdash \exists x~.(P(x)\lor Q(x))$</span> using the Gentzen tree system.</p>
<p><span class="math-container">$$\dfrac{\exists x~.P(x)~\lor~\exists x~.Q(x)\\\qquad\qquad\vdots}{\exists x~.(P(x)\lo... |
412,944 | <p>Is it mathematically acceptable to use <a href="https://math.stackexchange.com/questions/403346/prove-if-n2-is-even-then-n-is-even">Prove if $n^2$ is even, then $n$ is even.</a> to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?<... | Marc van Leeuwen | 18,880 | <p>The question you linked to starts (quite naturally) with "Suppose $n$ is an integer", which clearly excludes taking $n=\sqrt 2$. Therefore it cannot be used to conclude that $\sqrt2$ is even. Obviously $\sqrt2$ being even is absurd since even numbers in particular have to be integers. I would say your "observation" ... |
412,944 | <p>Is it mathematically acceptable to use <a href="https://math.stackexchange.com/questions/403346/prove-if-n2-is-even-then-n-is-even">Prove if $n^2$ is even, then $n$ is even.</a> to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?<... | Peter LeFanu Lumsdaine | 2,439 | <p>Other answerers are answering several slightly different interpretations of your question; but taking literally what you asked,</p>
<blockquote>
<p>Is it mathematically acceptable to use “<a href="https://math.stackexchange.com/questions/403346/prove-if-n2-is-even-then-n-is-even">Prove if $n^2$ is even, then $n$ ... |
399,564 | <p>Using Mathematical induction on $k$, prove that for any integer $k\geq 1$,</p>
<p>$$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$</p>
<p>How should I proceed? The tutorial teacher attempted this question and forgot halfway through... <em>facepalm</em></p>
| Brian M. Scott | 12,042 | <p>HINT: Your induction hypothesis is that</p>
<p>$$(1-x)^{-k}=\sum_{n\ge 0}\binom{n+k-1}{k-1}x^n\;.$$</p>
<p>For the induction step take a look at this calculation:</p>
<p>$$\begin{align*}
\sum_{n\ge 0}\binom{n+k}kx^n&=\sum_{n\ge 0}\left(\binom{n+k-1}{k-1}+\binom{n+k-1}k\right)x^n\\
&=(1-x)^{-k}+\sum_{n\ge ... |
1,413,212 | <blockquote>
<p>Given $x$ the number of circles of radius $r$,
what is a good approximate size of the radius of a bigger circle which they fit in?</p>
</blockquote>
<p>To explain in actual problem terms. I want to move units in a video games which take up a certain amount of area around, I don't want those units ... | Zenohm | 224,178 | <p>Since it appears that you're doing basic calculus, I'll put my answer in terms of that.</p>
<p>The definition for the slope of a tangent line on a curve using limits is as follows</p>
<blockquote>
<p>$$
\lim_{x \to a} \frac{f(x)-f(a)}{x-a} = m
$$</p>
</blockquote>
<p>Now let's look at what they give you,</p>... |
2,517,601 | <p>I tried to find the residue of the function $$f(z) = \frac{1}{z(1-\cos(z))}$$ at the $z=0$. So I did:</p>
<p>\begin{align}
\hbox{Res}(0)&=\lim_{x \to 0}(z)(f(z))\\
\hbox{Res}(0)&= \lim_{x \to 0} \frac{1}{1-\cos(z)}
\end{align}</p>
<p>and I got that the residue to be $\infty$, and it seems to be wrong.</p>
| Nosrati | 108,128 | <p>$$\cos z=1-\dfrac{1}{2!}z^2+\dfrac{1}{4!}z^4-\dfrac{1}{6!}z^6+\cdots$$
then
\begin{align}
\dfrac{1}{z(1-\cos z)}
&= \dfrac{1}{z^3\left(\dfrac{1}{2!}-\dfrac{1}{4!}z^2+\dfrac{1}{6!}z^4+\cdots\right)} \\
&= \dfrac{1}{z^3}\left(2+\dfrac{1}{6}z^2+\dfrac{1}{120}z^4+\cdots\right) \\
&= \dfrac{2}{z^3}+\dfrac{1}{... |
176,167 | <p>$\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$
Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let $g$ be continuously differentiable. Consequently, the subdifferential $\partial f$ and the gradient $\nabla g$ are m... | Suvrit | 8,430 | <p>Although not monotone at the operator level (as suggested by C. Mooney's proof), the monotonicity of prox-residual norms is known (probably you are already aware of it). </p>
<p>Let $P_\eta^g$ denote the prox-operator for $g$ with index $\eta$, i.e.,
\begin{equation*}
P_\eta^g(y) := \text{argmin}_{x} \tfrac{1}{2\e... |
3,684,158 | <p>You are given two circles:</p>
<p>Circle G: <span class="math-container">$(x-3)^2 + y^2 = 9$</span></p>
<p>Circle H: <span class="math-container">$(x+3)^2 + y^2 = 9$</span></p>
<p>Two lines that are tangents to the circles at point <span class="math-container">$A$</span> and <span class="math-container">$B$</span... | secavara | 512,290 | <p>Not really an answer to the question, but I wanted to post this gif that shows the ellipse being formed</p>
<p><a href="https://i.stack.imgur.com/m2IbD.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/m2IbD.gif" alt="elipse"></a></p>
|
1,147,773 | <p>Do you have any explicit example of an infinite dimensional vector space, with an explicit basis ?</p>
<p>Not an Hilbert basis but a family of linearly independent vectors which spans the space -any $x$ in the space is a <strong>finite</strong> linear sum of elements of the basis.</p>
<p>In general the existence ... | Bernard | 202,857 | <p>Rings of polynomials over any field in any number of indeterminates (viewed as vector spaces over the field) have the monomials as a basis.</p>
<p>Ring of integer-valued polynomials with coefficients in $\mathbf Q\,$ have as a basis:</p>
<p>$$ \Bigl\{1, x, \frac{x(x-1)}2, \dots, \frac{x(x-1)\dotsm(x-k+1)}{k!},\dot... |
2,796,854 | <p>what happens if both the second and first derivatives at a certain point are $0$? Is it an infliction point, an extremum point, or neither? Can we say anything at all about a point in such a case? </p>
| Dan Uznanski | 167,895 | <p>You can tell what it does (assuming this function is well-behaved at the point in question; without that this question becomes basically pointless) based on the order of the lowest non-zero derivative beyond the first.</p>
<p>An <strong>inflection point</strong> has a non-zero <strong>odd</strong> derivative: $f(x)... |
289,405 | <p>Consider an elementary class $\mathcal{K}$. It is quite common in model theory that a structure $K$ in $\mathcal K$ comes with a closure operator $$\text{cl}: \mathcal{P}(K) \to \mathcal{P}(K), $$ which establishes a <a href="https://en.wikipedia.org/wiki/Pregeometry_(model_theory)" rel="nofollow noreferrer">pregeo... | Jay Kangel | 8,684 | <p>Marcel van de Vel's "Theory of Convex Structures" has material on dimension theory. Currently I do not have access to the book and its been a very long time since I read it. Here is a copy of the table of contents. The second and third from the last items may be of interest.</p>
<p>Table of Contents</p>
<p>Introdu... |
2,967,626 | <p>I have a graph that is 5280 units wide, and 5281 is the length of the arc.</p>
<p><img src="https://i.stack.imgur.com/4rFK4.png"></p>
<p>Knowing the width of this arc, and how long the arc length is, how would I calculate exactly how high the highest point of the arc is from the line at the bottom?</p>
| Blue | 409 | <p>Note that <span class="math-container">$b=z$</span> violates the strict inequality (the right-hand side becomes zero). Consequently, we have <span class="math-container">$0<b < z$</span>, which allows us to write </p>
<blockquote>
<p><span class="math-container">$$b=z \sin\theta \tag{1}$$</span></p>
</block... |
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | Jack D'Aurizio | 44,121 | <p>There is an interesting trick: you may couple the two equations by writing
$$ e^{ix}+e^{iy} = i \tag{1}$$
hence $e^{ix}$ and $e^{iy}$, that are two points on the unit circle, are simmetric with respect to the imaginary axis. By imposing that their sum has unit norm, we clearly get $\{x,y\}=\left\{\frac{\pi}{6},\frac... |
3,097,640 | <p>So I have the following problem: $a + b = c + c.
I want to prove that the equation has infinitely many relatively prime integer solutions.</p>
<p>What I did first was factor the right side to get:
(</p>
| individ | 128,505 | <p>In the equation:</p>
<p><span class="math-container">$$X^2+Y^2=Z^5+Z$$</span></p>
<p>I think this formula should be written in a more general form:</p>
<p><span class="math-container">$$Z=a^2+b^2$$</span></p>
<p><span class="math-container">$$X=a(a^2+b^2)^2+b$$</span></p>
<p><span class="math-container">$$Y=b(a... |
903,117 | <p>I am trying to evaluate
$$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$
Would a contour work? I have tried using a contour but had no success.
Thanks.</p>
<p>Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anywa... | Empy2 | 81,790 | <p>What happens if you split $\cos2x$ into $e^{2ix}$ and $e^{-2ix}$? The two pieces need different contours - one above the real line, the other below.</p>
|
651,707 | <p>Let $A=(a_{ij})\in \mathbb{M}_n(\mathbb{R})$ be defined by</p>
<p>$$
a_{ij} =
\begin{cases}
i, & \text{if } i+j=n+1 \\
0, & \text{ otherwise}
\end{cases}
$$
Compute $\det (A)$</p>
<hr>
<blockquote>
<p>After calculation I get that it may be $(-1)^{n-1}n!$. Am I right?</p>
</blockquote>
| Danny obama | 123,792 | <p>Just take $n=2$ then accordingly given conditions </p>
<p>$$A=\begin{bmatrix} 0 & 1\\ 2 & 0\\\end{bmatrix}$$
$Det(A)= -2=-2!$</p>
<p>& For $n=3$ $Det(A)= -6=-3!$</p>
<p>For $n=4$ $Det(A)=24=4!$</p>
<p>This suggests us the general formula for det. of such type of matrix is $$(-1)^{(n-1)(\frac{n}{2})}.... |
381,309 | <p>Let <span class="math-container">$T\in B(\mathcal{H} \otimes \mathcal{H})$</span> where <span class="math-container">$\mathcal{H}$</span> is a Hilbert space. We can define operators
<span class="math-container">$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$</span>
and if <span class="math-container">$\Sigma: ... | Alex Ravsky | 43,954 | <p>For any infinite group <span class="math-container">$G$</span> we can easily construct a Vitali subset <span class="math-container">$V$</span> of <span class="math-container">$G$</span>. Indeed, pick an arbitrary countable infinite subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$<... |
119,532 | <p>I have a <code>ListAnimate</code> where the frames consist of graphics made of lines and points. I want to make sure that, when the animation is paused and then resumed, all frames appear normal again, so if you stop it to rotate the plot (it's 3D), after a loop it looks normal again.</p>
<p>My code is the followin... | ubpdqn | 1,997 | <p>This is just illustrative.</p>
<pre><code>f[z_] := ReIm[z + 1/z]
Manipulate[
ParametricPlot[{f[Complex @@ v + r Exp[I t]],
ReIm[Complex @@ v + r Exp[I t]]}, {t, 0, 2 Pi},
PlotRange -> {{-5, 5}, {-5, 5}}, Epilog -> {Text[v, {-3, 3}]},
AspectRatio -> Automatic, Frame -> True,
PlotLabel ->... |
3,238,054 | <blockquote>
<p>Find parameter <span class="math-container">$a$</span> for which <span class="math-container">$$\frac{ax^2+3x-4}{a+3x-4x^2}$$</span> takes all real values for <span class="math-container">$x \in \mathbb{R}$</span></p>
</blockquote>
<p>I have equated the function to a real value, say, k
which gets me... | Fred | 380,717 | <p><span class="math-container">$ a \geq -9/16$</span> is not correct. For <span class="math-container">$ a \leq -9/16$</span> the domain of the function <span class="math-container">$f(x)=\frac{ax^2+3x-4}{a+3x-4x^2}$</span> is <span class="math-container">$ \mathbb R.$</span></p>
|
3,341,471 | <p>Let <span class="math-container">$M$</span> be a multiplication operator on <span class="math-container">$L^2(\mathbb{R})$</span> defined by <span class="math-container">$$Mf(x) = m(x) f(x)$$</span> where <span class="math-container">$m(x)$</span> is continuous and bounded. Prove that <span class="math-container">$M... | daw | 136,544 | <p>Your reasoning that <span class="math-container">$X\subset \sigma(M)$</span> is not correct. Your proof only yields
that the spectrum of <span class="math-container">$M$</span> contains values <span class="math-container">$\lambda$</span> for which <span class="math-container">$\{x: m(x)=\lambda\}$</span>
has positi... |
4,320,209 | <p><em>I am stuck on the following two questions on matrix Algebra:</em></p>
<p>Let <span class="math-container">$x$</span> be a vector. Find the following:</p>
<ul>
<li><span class="math-container">$\frac{\partial}{\partial x}||x\otimes x||^2$</span>, here ||.|| denotes Euclidean norm of a vector and <span class="math... | Steph | 993,428 | <p>A slightly different approach is to
note that the function you
want to differentiate can be written as
<span class="math-container">$$
\phi
= \| \mathbf{x} \otimes \mathbf{x} \|_2^2
= \| \mathbf{x} \mathbf{x}^T \|^2_F
$$</span></p>
<p>The differential writes
<span class="math-container">$$
d\phi
= 2 \mathbf{x} \mat... |
3,745,097 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<p>If <span class="math-container">$F$</span> is a non-empty countable subset of <span class="math-container">$\mathbb R$</span>, prove that <span class="math-container">$F$</span> is not an open set, but that <span class="math-contain... | Henno Brandsma | 4,280 | <p>Well <span class="math-container">$\Bbb Z$</span> is countable and closed, and <span class="math-container">$\Bbb Q$</span> is countable and not closed (even dense). So both are possible for countable sets. All that wass asked essentially is two examples.</p>
|
2,010,329 | <p>The function under consideration is:</p>
<p>$$y = \int_{x^2}^{\sin x}\cos(t^2)\mathrm d t$$</p>
<p>Question asks to find the derivative of the following function. I let $u=\sin(x)$ and then $\tfrac{\mathrm d u}{\mathrm d x}=\cos(x)$. Solved accordingly but only to get the answer as
$$ \cos(x)\cos(\sin^2(x))-\cos(... | Lutz Lehmann | 115,115 | <p>You have, in fundamental theorem terms, the expression $y=F(\sin x)-F(x^2)$ where $F'(t)=f(t)=\cos(t^2)$. Thus
$$
y'(x)=F'(\sin x)·\cos x-F'(x^2)·2x=f(\sin x)·\cos x-f(x^2)·2x
$$
per the chain rule.</p>
|
2,010,329 | <p>The function under consideration is:</p>
<p>$$y = \int_{x^2}^{\sin x}\cos(t^2)\mathrm d t$$</p>
<p>Question asks to find the derivative of the following function. I let $u=\sin(x)$ and then $\tfrac{\mathrm d u}{\mathrm d x}=\cos(x)$. Solved accordingly but only to get the answer as
$$ \cos(x)\cos(\sin^2(x))-\cos(... | RGS | 329,832 | <p>Let us say that $F $ is a primitive of $cos(t^2) $. Hence the integral of $y $ can be written as</p>
<p>$$y = \int_{x^2}^{sin\ x} cos(t^2) dt = F(sin\ x) - F(x^2) $$</p>
<p>Hence you can derive $y $ by deriving the rightmost expression and by making use of the fact that $F' = cos(t^2) $.</p>
<p>$$y' = (F(sin\ x)... |
220,907 | <p>If I have these three lists:</p>
<pre><code>list1={{0.01,87.,0.},{0.03,87.,0.18353},{0.1,87.,0.494987},{0.3,87.,0.899803},{1.,87.,1.08076},{3.,87.,1.10593},{10.,87.,1.04781},{10.,87.,1.02449},{10.,87.,0.964193},{30.,87.,1.0602},{30.,87.,1.04075},{30.,87.,1.05987},{100.,87.,1.14661},{100.,87.,1.00639},{100.,87.,1.09... | kglr | 125 | <pre><code>Flatten @ SortBy[{First}] @ Join[list1, list2, list3]
</code></pre>
<blockquote>
<pre><code>{0.01,87.,0.,0.01,75.,0.,0.01,55.,0.,0.03,87.,0.18353,0.03,75.,0.,0.03,55., 0.,
0.1, 87.,0.494987,0.1,75.,0.0959691,0.1,55.,0.,0.3,87.,0.899803,0.3,75.,0.37954,0.3,55., 0.,
1., 87.,1.08076,1.,75.,0.678807,1.,55.,0... |
4,180,344 | <p>I started learning from Algebra by Serge Lang. In page 5, he presented an equation<br><i></p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a commutative monoid, and <span class="math-container">$x_1,\ldots,x_n$</span> elements of <span class="math-container">$G$</span>. Let <span class="math-contai... | Shaun | 104,041 | <p>This is an easy exercise if you use induction on <span class="math-container">$n$</span>. Here <span class="math-container">$\psi$</span> represents a different order of the elements of <span class="math-container">$\{1,\dots, n\}$</span> and hence a different order of the <span class="math-container">$x_\nu$</span>... |
1,742,418 | <p>So far I have this:</p>
<p>What is the chance that at least one is a 2?</p>
<p>There is $\frac{5}{6}$ that you will not get any 2 on the four dices. From this we get $\frac{5^4}{6^4}$</p>
<p>The probability of getting at least one 2 is $1 - \frac{5^4}{6^4} $</p>
<p>Now I have no idea how to proceed from here.</p... | woogie | 330,017 | <p>You are correct for the first part. You just need to simplify your expression to a single fraction.<br>
$\frac{6^4-5^4}{6^4}=\frac{1296-625}{1296}$<br>
$=\frac{671}{1296}$ </p>
|
215,061 | <p>Is it possible to do a 2D plot taking the point coordinates from separate lists of x and y values? These lists were produced by calculations within the same notebook. For a small number of points it is easy to enter x & y values by hand but it gets tedious as the number of points grows (I've done it.). If this i... | Mr.Wizard | 121 | <p>Using your original definitions:</p>
<pre><code>fn = Function @@ {replace};
fn[2]
</code></pre>
<blockquote>
<pre><code>{(2 αu)/(2 + ku), 2 Du, (2 αv)/(2 + kv)}
</code></pre>
</blockquote>
<pre><code>fn /@ {1, 2}
</code></pre>
<blockquote>
<pre><code>{{αu/(1 + ku), Du, αv/(1 + kv)}, {(2 αu)/(2 + ku), 2 Du, (2 α... |
4,344,538 | <p><strong>Lemma.</strong> Let <span class="math-container">$X$</span> be a topological vector space over <span class="math-container">$\mathbb{K}$</span> and <span class="math-container">$v\in X$</span>. Then the following mapping is continuous <span class="math-container">$$\begin{align*}
\varphi _v:\mathbb{K}&\r... | Kritiker der Elche | 908,786 | <p>Let <span class="math-container">$A,B, C$</span> be topological spaces. A function <span class="math-container">$\phi : A \to B \times C$</span> is continuous iff the functions <span class="math-container">$p_B \circ \phi : A \to B$</span> and <span class="math-container">$p_B \circ \phi : A \to C$</span> are contin... |
2,298,855 | <p>Let $u,v\in H^1(\Omega )$ where $\Omega =B_2\backslash B_1$. How can I have continuity and coercivity of
$$a(u,v)=\int_{\Omega }\nabla u\cdot \nabla v-\int_{B_2\backslash B_1}3^{-d}uv\ ?$$</p>
<p>The best thing I get is $$|a(u,v)|\leq \|\nabla u\|_{L^2}\|\nabla v\|_{L^2}+3^{-d}\|u\|_{L^2}\|v\|_{L^2},$$
but I can't ... | gerw | 58,577 | <p>The bilinear form $a$ is not continuous. By plugging in the constant function $u = 1$, we find $a(u,u) = -3^{-d} \, |B_2 \setminus B_2| < 0$. Hence, $a$ cannot be coercive.</p>
|
2,983,519 | <p>I have to calculate the limit of this formula as <span class="math-container">$n\to \infty$</span>.</p>
<p><span class="math-container">$$a_n = \frac{1}{\sqrt{n}}\bigl(\frac{1}{\sqrt{n+1}}+\cdots+\frac{1}{\sqrt{2n}}\bigl)$$</span></p>
<p>I tried the Squeeze Theorem, but I get something like this:</p>
<p><span cla... | user | 505,767 | <p>As an alternative by Stolz-Cesaro</p>
<p><span class="math-container">$$\frac{b_n}{c_n} = \frac{\frac{1}{\sqrt{n+1}}+\cdots+\frac{1}{\sqrt{2n}}}{\sqrt n}$$</span></p>
<p><span class="math-container">$$\frac{b_{n+1}-b_n}{c_{n+1}-c_n} = \frac{\frac{1}{\sqrt{2n+2}}+\frac{1}{\sqrt{2n+1}}-\frac{1}{\sqrt{n+1}}}{\sqrt{n+... |
189,308 | <p>I have this problem: Find integration limits and compute the following integral.</p>
<p>$$\iiint_D(1-z)\,dx\,dy\,dz \\ D = \{\;(x, y, z) \in R^3\;\ |\;\; x^2 + y^2 + z^2 \le a^2, z\gt0\;\}$$</p>
<p>I can compute this as an indefinite integral but finding integration limits beats me. As indefinite integral the resu... | Sasha | 11,069 | <p>Use linearity:
$$
\int_D (1-z) \,\mathrm{d}V = \int_D 1 \,\mathrm{d}V - \int_D z \,\mathrm{d}V =\frac{2 \pi}{3} a^3 - \int_D z \,\mathrm{d}V
$$
The latter integral is best evaluated in cylindrical coordinates $(x,y,z) = (r \sin(\phi), r \cos(\phi), z)$. The measure $dV$ factors into $dz$ times the measure on the ... |
1,372,779 | <p>Given that $x$ is a positive integer, find $x$ in $(E)$.</p>
<p>$$\tag{E} j-n=x-n\cdot\left\lceil\frac{x}{n}\right\rceil$$
All $n, j, x$ are positive integers.</p>
| syusim | 138,951 | <p><strong>Hint</strong>: it may help to get rid of the ceiling function by using the division algorithm and letting $x = kn + a$ for some minimal $a$.</p>
|
827,072 | <p>How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.</p>
<p>I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is right.</p>
| Fardad Pouran | 76,758 | <p>Suppose the points are $A,B,C$. Then intersect the equations of perpendicular bisectors of $AB$ and $BC$. This is the center of the desired circle. (with your notation $(p,q)$)</p>
<p>Now calculate the distance between $(p,q)$ and $A$. Now $r$ is also found.</p>
<p><img src="https://i.stack.imgur.com/iBPC0.jpg" al... |
3,379,756 | <p>Why does the close form of the summation
<span class="math-container">$$\sum_{i=0}^{n-1} 1$$</span> equals <span class="math-container">$n$</span> instead of <span class="math-container">$n-1$</span>?</p>
| cansomeonehelpmeout | 413,677 | <p><span class="math-container">$$\sum_{i=1}^{n-1}1=n-1$$</span> When <span class="math-container">$i$</span> starts at <span class="math-container">$0$</span> you get an extra term
<span class="math-container">$$\sum_{i=0}^{n-1} 1=n$$</span></p>
|
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| Barry Cipra | 86,747 | <p>Note that</p>
<p>$$2^{2\times3^k}\lt\sum_{n=1}^k 2^{2\times3^n}\lt2^{2\times3^k}+(k-1)2^{2\times3^{k-1}}$$</p>
<p>hence</p>
<p>$$1\lt{\sum_{n=1}^k 2^{2\times3^n}\over2^{2\times3^k}}\lt1+{k-1\over2^{2(3^k-3^{k-1})}}=1+{k-1\over16^{3^{k-1}}}\lt1+{k\over2^k}$$</p>
<p>where the final inequality is extremely crude, i... |
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| zhw. | 228,045 | <p>Since the denominator $\to \infty,$ Stolz-Cesaro comes into play, and we consider</p>
<p>$$\frac{ 2^{2\cdot 3^{k+1}}}{2^{2\cdot 3^{k+1}}- 2^{2\cdot 3^{k}}} = \frac{1}{1-2^{2\cdot3^{k}-2\cdot3^{k+1}}} = \frac{1}{1-2^{-4\cdot3^{k}}} \to 1.$$</p>
<p>SC thus implies the limit is $1.$ </p>
|
1,424,297 | <p>Why is $a\overline bc + ab = ab + ac$? I think it has something to do with the rule $a + \overline a = 1$, right?</p>
| k170 | 161,538 | <p>$$\lim\limits_{n\to\infty} n^2\left(1-\cos\left(\frac{1}{n}\right)\right)$$
Let $m=\frac1n$, then
$$\lim\limits_{m\to 0^+} \frac{1-\cos m}{m^2}=\lim\limits_{m\to 0^+} \frac{2\sin^2\left(\frac{m}{2}\right)}{m^2}$$
Let $k=\frac{m}{2}$, then
$$\lim\limits_{k\to 0^+} \frac{2\sin^2 k}{4k^2}=\frac12\lim\limits_{k\to 0^+} ... |
675,857 | <p>I am trying to figure out easy understandable approach to given small number of $n$, list all possible is with proof, I read this paper but it is really beyond my level to fathom,</p>
<p>attempt for $\phi(n)=110$, </p>
<p>$$\phi(n)=110=(2^x)\cdot(3^b)\cdot(11^c)\cdot(23^d)\quad\text{ since }\quad p-1 \mid \phi(n)=... | Will Jagy | 10,400 | <p>Note that $$ \phi(n) \geq \sqrt {\frac{n}{2}}, $$ so this is a finite search</p>
<p><a href="https://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below">Is the Euler phi function bounded below?</a></p>
|
2,335,627 | <p>Is the following statement true? If so, how can I prove it?</p>
<blockquote>
<p>If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges for all $x
\in (x_0, x_1)$ then it converges absolutely for all $x \in
(-\max\{|x_0|, |x_1|\}, \max\{|x_0|, |x_1|\})$.</p>
</blockquote>
| Theo Bendit | 248,286 | <p>The trick is first showing absolute convergence. Once you show that, then for any $y_0 \in (-\max\lbrace |x_0|, |x_1| \rbrace, \max\lbrace |x_0|, |x_1| \rbrace)$, we can find a $y_1 \in (x_0, x_1)$, with $|y_1| > |y_0|$. Then, because of absolute convergence at $y_1$, we can use the comparison test:
$$\sum |a_n| ... |
3,571,637 | <p>I was reading Linear Algebra by Hoffman Kunze, and encountered this in the Theorem 8 of the Chapter Coordinates, the theorem is stated below : </p>
<p><a href="https://i.stack.imgur.com/6xFlJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6xFlJ.png" alt="coo"></a></p>
<p>What I get about is in ... | Trevor Gunn | 437,127 | <p>Representing <span class="math-container">$\alpha_j'$</span> in the ordered basis <span class="math-container">$(\alpha_1',\dots,\alpha_n')$</span> will just give you the standard basis vector <span class="math-container">$e_j$</span> because <span class="math-container">$\alpha_j' = 0\alpha_1' + \dots + 1\alpha_j' ... |
915,218 | <p>I'd like some help figuring out how to calculate $n$ points of the form $(x,\sin(x))$ for $x\in[0,2\pi)$, such that the Cartesian distance between them (the distance between each pair of points if you draw a straight line between them) is the same.</p>
<p><strong>My background:</strong> I know math up to and throug... | amon | 72,417 | <p>The trivial solution is to have all points be the same point, such that the distance is zero. The point $(0, 0)$ lies on the sine curve and therefore represents a solution.</p>
<p>For non-zero distances between the points, things are a bit more complicated. We start with:</p>
<ul>
<li>The number of points $n$ with... |
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