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 |
|---|---|---|---|---|
481,952 | <p>Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?</p>
| Davide Giraudo | 9,849 | <p>Each subset of a metric space is a union of bounded sets (the singletons) but not every subset is bounded. </p>
|
2,143,510 | <p>NOTE: There are some other similar questions, but I got a negative answer to this question from my proof. Please find out the errors in my reasoning. </p>
<p>$\mathbf {Claim:}$ Is every point of every open set $E \subset R^2$ a limit point of E? Answer the same question for closed sets in $E \subset R^2$</p>
<p>Fr... | fleablood | 280,126 | <p>Every k is A $\iff $ there is no k that is not A.</p>
<p>So every point of the empty set is a pink alligator that eats square circles... because the empty set does not have any points that are not pink alligators that eat square circles... because the empty has no points at all so none of them can avoid being a pi... |
1,649,194 | <p>Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections
$$\sigma_+\colon S^n-S\to\mathbb{R}^n $$
and
$$\sigma_-\colon S^n-N\to\mathbb{R}^n$$
Both of these maps are homeomorphisms and they for... | user43883 | 114,453 | <p>$\def\ip {\, \lrcorner \, }$
I get a different result than the previous answer: $\sigma^-$ preserves orientation iff $n$ is even:</p>
<p>Consider $\Bbb S^{n-1}$ embedded in $\Bbb R^n$.The orientation form on $\Bbb R^n$ is $dx_1 \wedge \ldots \wedge dx_n$.</p>
<p>If $v(x)$ is a non-vanishing vector field on $\Bbb S... |
1,649,194 | <p>Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections
$$\sigma_+\colon S^n-S\to\mathbb{R}^n $$
and
$$\sigma_-\colon S^n-N\to\mathbb{R}^n$$
Both of these maps are homeomorphisms and they for... | Narasimham | 95,860 | <p><em>Inversions</em> in the plane and likewise stereographic projection from $\mathbb R^2 $ (having an inversion component) reverse sense in projections of a closed loop of the surface. </p>
|
3,403,255 | <p>I am trying to follow wikipedia's page about matrix rotation and having a hard time understanding where the formula comes from.</p>
<p><a href="https://en.wikipedia.org/wiki/Rotation_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Rotation_matrix</a> Wiki page about it.</p>
<p>what i have so far:</... | John Alexiou | 3,301 | <p>Consider a rotated rectangle with sides <span class="math-container">$a$</span> and <span class="math-container">$b$</span></p>
<p><a href="https://i.stack.imgur.com/o7w5e.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/o7w5e.png" alt="sketch"></a></p>
<p>Do the trigonometry to find the <em>x</e... |
2,634,701 | <p>Let $ f: {{\mathbb{R^n}} \rightarrow {{\mathbb{R}} }}$ be continuous and let $a$ and $b$ be points in $ {{\mathbb{R} }} $
Let the function $g: {\mathbb{R}} \rightarrow {\mathbb{R}}$ be defined as:
$$ g(t) = f(ta+(1-t)b) $$
Show that $g$ is continuous .</p>
<p>If I define a function $ h(t)=ta+(1-t)b$, then I have ... | ChoMedit | 527,854 | <p>Just use $\varepsilon-\delta$ argument to solve the problem.
Choose arbitrary point $t_0$ in $\mathbb{R}$.
$$
\forall \varepsilon>0 \exists \delta>0 : |t-t_0|<\delta \Rightarrow |h(t)-h(t_0)|<\varepsilon
$$</p>
<p>We could derive the equality, $|h(t)-h(t_0)|=|(t-t_0)(a-b)|=|t-t_0||a-b|$. Note that it is... |
2,084,624 | <p>The question is : </p>
<p>Is $\sum_{k=1}^\infty \frac{(-3)^k(k!)}{k^k}$ convergent? </p>
<p>Note : I can't find the limit of its main term. I know the answer must be related to some test about convergence of series ... I don't know which one and i can't find the limit.</p>
| Jan Eerland | 226,665 | <p>Using the ratio test:</p>
<p>$$\lim_{\text{k}\to\infty}\left|\frac{\left(\frac{\left(-\text{n}\right)^{\text{k}+1}\cdot\left(\left(\text{k}+1\right)!\right)}{\left(\text{k}+1\right)^{\text{k}+1}}\right)}{\left(\frac{\left(-\text{n}\right)^\text{k}\cdot\left(\text{k}!\right)}{\text{k}^\text{k}}\right)}\right|=\lim_{... |
31,767 | <p>I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight. </p>
<p>Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to construct a bijective mapping $f: T_k^n \to \{1, 2, \ldots, \binom{n}{k}\}$ such that computing each $f(x_1^n)$... | H A Helfgott | 398 | <p>To answer a remark above: yes, I think it would be a very good idea to get together group of interested people to build a commentary. What non-interested people (or people who haven't read and will not read Vinogradov) can do is suggest what current technical tools would be most appropriate for such a collaborative ... |
1,117,592 | <blockquote>
<p>Let <span class="math-container">$k$</span> be a finite field and <span class="math-container">$V$</span> a finite-dimensional vector space over <span class="math-container">$k$</span>.</p>
<p>Let <span class="math-container">$d$</span> be the dimension of <span class="math-container">$V$</span> and <sp... | xxxxxxxxx | 252,194 | <p>It is useful in this case to consider the dual. In the dual space, your collection of hyperplanes covering every vector corresponds to a set of <span class="math-container">$(q+1)$</span> <span class="math-container">$1$</span>-dimensional subspaces such that each hyperplane contains at least one of these <span cla... |
3,397,548 | <p>For a sequence <span class="math-container">$\{x_n\}_{n=1}^{\infty}$</span>, define <span class="math-container">$$\Delta x_n:=x_{n+1}-x_n,~\Delta^2 x_n:=\Delta x_{n+1}-\Delta x_n,~(n=1,2,\ldots)$$</span> which are named <strong>1-order</strong> and <strong>2-order difference</strong>, respectively. </p>
<p>The pro... | robjohn | 13,854 | <p>Since <span class="math-container">$x_n$</span> is bounded, choose <span class="math-container">$M$</span> so that <span class="math-container">$|x_n|\le M$</span>.</p>
<p>Suppose that <span class="math-container">$\lim\limits_{n\to\infty}\Delta x_n\ne0$</span>. Then <span class="math-container">$\exists\epsilon\gt... |
3,811,753 | <p>Show that the equation:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<p>Has an integration factor that depends on <span class="math-container">$x$</span> And solve it that way.</p>
<hr />
<p>Already we got to:</p>
<p><span class="math-container">$$
y’ + \frac{xy^3}{3x^2y^2} = \fra... | Claude Leibovici | 82,404 | <p>Make life easier letting
<span class="math-container">$$y=\frac z {\sqrt[3]x}\implies 3 x z'(x)=\frac{2}{z(x)^2}$$</span> which is simpler</p>
|
3,413,364 | <blockquote>
<p>Consider the set of points <span class="math-container">$$O = \{ x \in P \mid \alpha^* = C^T x \}$$</span> where <span class="math-container">$P \subseteq \mathbb R^n$</span> is a closed convex set, <span class="math-container">$C \in \mathbb R^n$</span> and <span class="math-container">$\alpha^* = \m... | celtschk | 34,930 | <p>First, the function <span class="math-container">$f(x)=C^T x$</span> is a finite-dimensional linear function, and therefore continuous.</p>
<p>Also, in <span class="math-container">$\mathbb R^n$</span> single-element subsets are always closed; <span class="math-container">$\{\alpha^*\}$</span> is such a set.</p>
<... |
367,364 | <p>How to calculate the following integral:</p>
<p>$\int^{R}_{0}[2 \cos^{-1}(\frac{r}{2R}) -\sin(2 \cos^{-1}(\frac{r}{2R}) ) ] dr$.</p>
<p>This is a part of a complex formula.</p>
| vonbrand | 43,946 | <p>You have $a_{n + 1} = 2 a_n - (-1)^n$, and thus:
$$
\frac{A(z) - a_0}{z} = 2 A(z) - \frac{1}{1 + z}
$$</p>
|
367,364 | <p>How to calculate the following integral:</p>
<p>$\int^{R}_{0}[2 \cos^{-1}(\frac{r}{2R}) -\sin(2 \cos^{-1}(\frac{r}{2R}) ) ] dr$.</p>
<p>This is a part of a complex formula.</p>
| Matt L. | 70,664 | <p>You can also try to see the pattern:</p>
<p>$$a_1 = 2a_0 - 1\\
a_2 = 2\cdot 2 a_0 - 2 + 1\\
a_3 = 2\cdot 2\cdot 2a_0 - 2\cdot 2 + 2 - 1\\\vdots\\
a_n = 2^na_0 + (-1)^n\sum_{k=0}^{n-1}(-2)^k$$
The sum can of course be written in closed form.</p>
|
892,114 | <p>i have three number
1 2 3 which will always be in this order {123}, i want to find out number of cases can be made,
like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" "b", i.e, each one will become different entity,like 2a,2b,3a,3b,1a,
with only exception i.e. 1 will have only one stat... | amWhy | 9,003 | <p>$$(x-2)^2 = (x - 2)(x-2) = x^2 - 2x -2x + (-2)(-2) = x^2 - 4x + 4$$</p>
<p>This is called <em>expanding</em> $(x-2)^2$. We factor $x^2 - 4x + 4$ when we write it as the product of its factors, in this case $(x-2)(x-2) = (x-2)^2$.</p>
<p>Now,
$$(x-2)^2-12=(x^2-4x+4)-12=x^2-4x-8$$
You can find the zeros of the quad... |
187,975 | <p>Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.</p>
<p>I would like to use the foll... | Henrique de Oliveira | 18,474 | <p>There's a stronger version of that (basic) theorem due to Lyapunov. It is stronger because it concerns vectors of measures, and not only a single measure. It states that given a non-atomic vector measure (a collection of $n$ measures $\mu_1,\ldots, \mu_n$ where each measure is non-atomic) always has an image which i... |
3,294,123 | <p>Define <span class="math-container">$f(x)=x^{-1}(\log x)^{-2}$</span> if <span class="math-container">$0<x<\frac{1}{2}$</span>, <span class="math-container">$f(x)=0$</span> on the rest of <span class="math-container">$R$</span>. Then <span class="math-container">$f \in L^1(R)$</span>. Show that
<span class="m... | zhw. | 228,045 | <p>A simpler way to show <span class="math-container">$Mf\notin L^1$</span> is to note there is a constant <span class="math-container">$c>0$</span> such that <span class="math-container">$|f|>c$</span> on a set of positive measure <span class="math-container">$E\subset (0,1/2).$</span> Then for <span class="math... |
1,621,363 | <p>Integrate: $$\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x.$$</p>
<p>I tried the substitution method ($\sin(x) = t$) and ended up getting $\int \frac{t}{9+32t-32t^3}\,\text{d}t$. Don't know how to proceed further. </p>
<p>Also tried adding and substracting $\cos(x)$ in the numerator which led me to get $$\sin(2x) =... | Jack Tiger Lam | 186,030 | <p>To attack this integral, we will need to make use of the following facts:</p>
<p>$$(\sin{x} + \cos{x})^2 = 1+\sin{2x}$$</p>
<p>$$(\sin{x} - \cos{x})^2 = 1-\sin{2x}$$</p>
<p>$$\text{d}(\sin{x}+\cos{x}) = (\cos{x}-\sin{x})\text{d}x$$</p>
<p>$$\text{d}(\sin{x}-\cos{x}) = (\cos{x}+\sin{x})\text{d}x$$</p>
<p>Now, co... |
2,887,440 | <p>We were asked in our Calculus class to prove that,</p>
<blockquote>
<p>$f(x+y) - f(x) = \frac {\sec^2(x) \tan(y)} {1 - \tan(x) \tan(y)}$ given that $f(x) = \tan(x)$</p>
</blockquote>
<p>I have gotten so far as:</p>
<p>$$f(x+y) - f(x)$$</p>
<p>$$\tan(x+y) - \tan(x)$$</p>
<p>$$\frac{\tan(x)+\tan(y)}{1-\tan(x)\t... | Henrik supports the community | 193,386 | <p>It's simply substitution of the argument to $f$.</p>
<p>On the other hand: $f(x+y)=f(x)+f(y)$ that you claim to have found several sources for, is <em>not</em> generally true.</p>
|
233,618 | <p>I want to be able to take a polynomial and take the 1st 5 derivatives, then add at least one root of each derivative to a list using a loop. However, each attempt I try only ends up outputting the roots of the 5th derivative, not the rest. So far I have:</p>
<pre><code>rootderivs[n_]:=(
p[x_]:= x^8-3x^5+x-1;
rootli... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
p[x_] := x^8 - 3 x^5 + x - 1
</code></pre>
<p>For the real roots</p>
<pre><code>Table[NSolve[D[p[x], {x, n}] == 0, x, Reals] // Union, {n, 5}] //
Grid[#, Frame -> All] &
</code></pre>
<p><a href="https://i.stack.imgur.com/5I0iB.png" rel="nofollow noreferrer"><img src="ht... |
596,374 | <p>I solved this , but I am not sure if I did in the right way.</p>
<p>$$2^{2x + 1} - 2^{x + 2} + 8 = 0$$</p>
<p>$$2^{x + 2} - 2^{2x + 2} = 8$$</p>
<p>$$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$</p>
<p>$$x + 2- 2x - 2 = 3$$</p>
<p>solving for $x$:</p>
<p>$$x = -2$$</p>
<p>any feedback would be appreciated.<... | Suraj M S | 85,213 | <p>let $2^{x+2}=t$ then $2^{2x+1}=\frac{1}{8}t^2$</p>
<p>substituting in the equation ,you get
$$\frac{1}{8}t^2-t+8=0$$
which is a quadratic equation and can be easily solved.</p>
|
4,297,051 | <p>Let <span class="math-container">$G$</span> be a non-abelian group of order <span class="math-container">$p^3$</span>, <span class="math-container">$p$</span> prime. Show that <span class="math-container">$Z(G)$</span> is a group of order <span class="math-container">$p$</span>. Deduce that <span class="math-contain... | Marcos | 962,125 | <p>Let us use that <span class="math-container">$G/Z(G)$</span> is cyclic iff <span class="math-container">$G$</span> is abelian. Using this, let us prove the result:</p>
<p>We know that <span class="math-container">$| Z(G)|\in\{p,p^2,p^3\}$</span> (<span class="math-container">$Z(G)\neq 0$</span> since <span class="ma... |
4,297,051 | <p>Let <span class="math-container">$G$</span> be a non-abelian group of order <span class="math-container">$p^3$</span>, <span class="math-container">$p$</span> prime. Show that <span class="math-container">$Z(G)$</span> is a group of order <span class="math-container">$p$</span>. Deduce that <span class="math-contain... | Arturo Magidin | 742 | <p>While one can prove this using the oft-quoted result that if <span class="math-container">$G/Z(G)$</span> is cyclic then <span class="math-container">$G$</span> is abelian, this fact is not necessary to prove the result.</p>
<p>Assume, for the sake of contradiction, that the order of <span class="math-container">$Z(... |
778,605 | <blockquote>
<p>Let $f,g,h : D \to \mathbb{R}$ be functions, $D \subset \mathbb{R}$. Let c be an accumulation point of $D$. Suppose that $$f(x) \le g(x) \le h(x)$$
for all $x \in D$ with $x \neq c$ and suppose
$$\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L \in \mathbb{R}$$
Prove that $\lim_{x \to c}g(x) = L... | drhab | 75,923 | <p>To be calculated is integral:</p>
<p>$\frac{1}{4}\int_{0}^{2}\int_{0}^{2}1_A\left(a,b\right)dadb$ where
$1_A\left(a,b\right)=1$ if $a>\frac{1}{4}\wedge\left|a-b\right|>\frac{1}{4}$
and $1_A\left(a,b\right)=0$ otherwise. </p>
<p>Here $a$ corresponds with Alice and $b$ with Bob.</p>
<p>It comes to determining... |
3,176,629 | <p>In the evening, pizza was ordered nine people sat around a round table, 50 slices of pizza were served to these nine people. Prove that there were two people sitting next to each other who ate at least 12 pizza slices.</p>
<p>I used the pigeon hole principle to determine 50/9 = 5.5 => 6</p>
<p>Therefore, at least ... | awkward | 76,172 | <p>Assume the contrary, i.e., every pair of adjacent persons ate no more than <span class="math-container">$11$</span> slices. </p>
<p>There are <span class="math-container">$9$</span> pairs of adjacent persons, where we count persons 1 and 9 as adjacent. So if we sum up the slices eaten by each pair (persons 1 and ... |
1,138,789 | <p><a href="http://en.wikipedia.org/wiki/Free_object" rel="nofollow noreferrer">Wikipedia</a> defines free objects as follows:</p>
<blockquote>
<p>Let <span class="math-container">$(\mathcal{C},F)$</span> be a concrete category (i.e. <span class="math-container">$F : \mathcal{C} \to {\rm \bf{Set}}$</span> is a faithful... | Will Jagy | 10,400 | <p>Lucian, I think you can do this yourself (for primes) with some hints and some formalism, from Leonard Eugene Dickson, <em>Introduction to the Theory of Numbers</em>. If a prime $q$ is represented as $q = x^2 + xy+ 15 y^2,$ then $3 x^2 + xy + 5 y^2$ does not represent it or its square or cube. If a prime $p$ is repr... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Mark Meckes | 129 | <p>I'd like to expand on points that Thomas made. Learning math is like learning a language, and a certain amount of memorization (note: not necessarily drilling!) is a necessary component of that. To use a language, you need to have immediate mental access to the basic vocabulary and grammar. (In math, that means n... |
2,257,365 | <p>In a semicircle of diameter $CD$ there's a chord $AB$ of length 7, and it's parallel to the diameter. There's also a small semicircle that is tangent to $AB$ and its diameter is a segment in $CD$ . Find the area of the semicircle without the small semicircle.</p>
<p>I'm pretty curious about this problem, i've tried... | Abhinav Dhawan | 346,937 | <p>You can calculate tan $ \angle AMD$ which equals 2. Also tan $ \angle BMC $ is 2 .</p>
<p>You can take AMD=BMC=$\theta$ and then can easily compute tan ( 180 - $\theta$) = tan $\alpha$ using identity tan (a-b)</p>
<p>Hope it helps and it's easier as you asked . Your solution have certain problems as mentioned in c... |
2,926,270 | <p>The base step is pretty obvious: <span class="math-container">$1 \geq \frac{2}{3}$</span>.</p>
<p>Then we assume that <span class="math-container">$P(k)$</span> is true for some <span class="math-container">$k \in \mathbb{Z}^{+}$</span> and try to prove <span class="math-container">$P(k+1)$</span>. So I have</p>
<... | For the love of maths | 510,854 | <p>For <span class="math-container">$P(k+1)$</span>:<br>
<span class="math-container">$\sqrt{1}+\sqrt{2}+...+\sqrt{k} + \sqrt{k+1} \geq \frac{2}{3}(k+1)\sqrt{k+1}$</span><br>
<span class="math-container">$\frac 23k\sqrt k+\sqrt {k+1}\geq \frac 23k\sqrt k+\frac 23\sqrt {k+1}$</span><br>
<span class="math-container">$\fr... |
2,926,270 | <p>The base step is pretty obvious: <span class="math-container">$1 \geq \frac{2}{3}$</span>.</p>
<p>Then we assume that <span class="math-container">$P(k)$</span> is true for some <span class="math-container">$k \in \mathbb{Z}^{+}$</span> and try to prove <span class="math-container">$P(k+1)$</span>. So I have</p>
<... | user581912 | 581,912 | <p>For the induction step, you want to show that:
$$
\frac{2k\sqrt{k} + 3\sqrt{k+1}}{3} \geq \frac{2(k+1)\sqrt{k+1}}{3} \\
2k\sqrt{k} + 3\sqrt{k+1} \geq 2k\sqrt{k+1} + 2\sqrt{k+1}\\
$$
Working backwards:
$$
2k\sqrt{k} + \sqrt{k+1} \geq 2k\sqrt{k+1} \\
4k^2 \times k \geq (4k^2 - 4k+1)(k+1) = 4k^3 - 4k^2 + k + 4k^2 - 4k+... |
541,644 | <p>I want to know why $p \leftrightarrow q$ is equivalent to $(p \wedge q) \vee (\neg p \wedge \neg q)$? Without using the truth table.</p>
<p>Thanks all</p>
| drhab | 75,923 | <p>If $p$ is true then $p\iff q$ tells us that $q$ is true as well. Also if $q$ is true then $p\iff q$ tells us that $p$ is true as well. So it cannot be that exactly one of them is true. They are both true or both not true.</p>
|
3,313,697 | <p>To calculate the <span class="math-container">$n$</span>-period payment <span class="math-container">$A$</span> on a loan of size <span class="math-container">$P$</span> at an interest rate of <span class="math-container">$r$</span>, the formula is:</p>
<p><span class="math-container">$A=\dfrac{Pr(1+r)^n}{(1+r)^n-1... | Micah | 30,836 | <p>Let's be more explicit about what's happening. If you borrow <span class="math-container">$\$10000$</span> at <span class="math-container">$5\%$</span> over a <span class="math-container">$10$</span>-year term, you must pay</p>
<p><span class="math-container">$$
A=\frac{\$10000(0.05)(1+0.05)^{10}}{(1+0.05)^{10}-1}\... |
2,654,538 | <p>If $2\tan^2x - 5\sec x = 1$ has exactly $7$ distinct solutions for $x\in[0,\frac{n\pi}{2}]$, $n\in N$, then the greatest value of $n$ is?</p>
<p>My attempt:</p>
<p>Solving the above quadratic equation, we get $\cos x = \frac{1}{3}$</p>
<p>The general solution of the equation is given by $\cos x = 2n\pi \pm \cos^{... | Mostafa Ayaz | 518,023 | <p>By replacing $\tan^2 x=\sec^2x-1$ we have$$4\sec^2x-5\sec x=5$$which yields to $$\sec x=\dfrac{5\pm\sqrt{25+80}}{8}=\dfrac{5\pm\sqrt{105}}{8}$$where only $\sec x=\dfrac{5+\sqrt{105}}{8}$ is acceptable. Also the equation $\sec x=p>1$ has exactly two roots in $[0,2\pi)$ so for having 7 distinct roots we must be in ... |
2,205,950 | <p>If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f'(a) \neq 0$ for all $a \in \mathbb{R}$, show that $f$ is one-to-one for all $a\in \mathbb{R}$.</p>
<h2>My attempt</h2>
<p>We know that $f(a)$ is not a constant because $f'(a)\neq 0$.Define $f$ by $f(a)=bx$. $f'(a)=x\neq 0$</p>
<p>If $f(x)=f(v)$ then $$bx=bv$$<... | Jonas Meyer | 1,424 | <p>"$f(a)$ is not a constant" means what? If $a$ is a number, then $f(a)$ is a number. You mean $f$ is not constant. And you can't say $f(a)=bx$, and it isn't clear what that means; what are $b$ and $x$? And this wouldn't imply $f'(a) = x$. </p>
<p>If you are going to prove that the hypothesis implies the conclusi... |
37,252 | <p>Let $V$ be a vector space with a finite Dimension above $\mathbb{C}$ or $\mathbb{R}$.</p>
<p>How does one prove that if $\langle\cdot,\cdot\rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are two Inner products</p>
<p>and for every $v\in V$ $\langle v,v\rangle_{1}$ = $\langle v,v\rangle_{2}$ so $\langle\cdot,\... | Calle | 5,966 | <p>You can use the <a href="http://en.wikipedia.org/wiki/Polarization_identity" rel="noreferrer">polarization identity</a>.</p>
<p>$\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$ induces the norms $\| \cdot \|_1$ and $\| \cdot \|_2$ respectively, i.e.:</p>
<p>$$\begin{align}
\| v \|_1 = \sqrt{\... |
1,753,620 | <p>How do I find the matrix exponential $e^{tA}$ with </p>
<p>$$A = \left(\begin{matrix} 2 & 8 \\ 0 & 2\end{matrix}\right)$$</p>
<p>The eigenvalue is 2 with multiplicity 2, but it yields only 1 eigenvector {${1, 0}$}, so the matrix isn't diagonalizable. I'm confused what to do. One option is to convert it int... | Olivier Oloa | 118,798 | <p>The given series <strong>converges uniformly</strong> on $(-\infty,\infty)$ to $f$, since each term is a continuous function on $(-\infty,\infty)$ thus $f$ is a <strong>continuous</strong> function on $(-\infty,\infty)$. The uniform convergence of the series may be obtained from the normal convergence:
$$
\left|\sum... |
3,862,408 | <p>This is the second example of 1. in <a href="http://www-personal.umich.edu/%7Ebhattb/teaching/mat679w17/lectures.pdf" rel="nofollow noreferrer">Ex. 2.0.3 </a> of Bhatt's notes in perfectoid space.</p>
<p>We define <span class="math-container">$R^{perf}:= \varprojlim ( \cdots R \xrightarrow{\phi} R)$</span> where <sp... | Eric Wofsey | 86,856 | <p>Suppose <span class="math-container">$g\in N_G(H)$</span>, <span class="math-container">$h\in H$</span>, and <span class="math-container">$x\in X^H$</span>. Then since <span class="math-container">$g$</span> normalizes <span class="math-container">$H$</span>, <span class="math-container">$hg=gh'$</span> for some <s... |
681,543 | <p>So I have a function </p>
<p>$$r= ( x^2 + y^2)^{1/2}$$</p>
<p>and I want to show that </p>
<p>$$\operatorname{grad} f(r) = f'(r)(\operatorname{grad} r).$$</p>
<p>I don't really know where to begin do you say that $f(r) = (f \circ r)(x,y)$ and then use the definition of gradient to work it out. Please give a rel... | Mark Fantini | 88,052 | <p>To compute the gradient of $f(r)$ you need to compute the partial derivatives. How to do it?</p>
<p>Use the chain rule: $f(r)$ stands for $f[(x^2+y^2)^{1/2}]$ since $r=(x^2+y^2)^{1/2}$, therefore</p>
<p>$$
\begin{align}
\frac{\partial}{\partial x} f(r) & = \frac{\partial}{\partial x} f[(x^2+y^2)^{1/2}] \\
&am... |
1,375,085 | <p>It is the first time I met such a question:</p>
<blockquote>
<p>Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?</p>
</blockquote>
<p>Intuitively I think $f(n)$ would gradually become larger as $n$ gets larger, but I find it hard to produce an argument.
Is there any trick to use for... | ccorn | 75,794 | <p>Compare $f(n)$ with $G(n)=256^{256^n}>g(n)$. You arrive at
$G(n) = 2^{2^{8n+3}}$, so you just need to compare $2^n$ vs. $8n+3$.</p>
|
3,043,846 | <p>I want to rewrite a question not so well written on this site and clarified by Mr. Lahtonen (thank you again).</p>
<p>So here the question:</p>
<blockquote>
<p>Let the extention <span class="math-container">$GF(p^m) \supset GF(p)$</span> that contains roots of
<span class="math-container">$p(x)=x^{p^{m}}-1$</s... | C Monsour | 552,399 | <p>You can even use non-isomorphic finite groups of the same order. The smallest example is <span class="math-container">$Aut(C_4\times C_2)\cong Aut(D_8)\cong D_8$</span>.</p>
|
2,699,621 | <p>To show $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}$ is it enough to tell that the taylor series expansion of $\sqrt{1+x}$ around $0$ has more positive terms?</p>
| DeepSea | 101,504 | <p>If $x > 4$, then $LHS < 1 < \sqrt{x+1} = RHS$ and inequality holds. Assume $0 < x \le 4\implies LHS > 0$, square both sides and simplify:$$1+\dfrac{x^2}{4}+\dfrac{x^4}{64}+x- \dfrac{x^2}{4}- \dfrac{x^3}{8}< 1+x\iff \dfrac{x^4}{64}< \dfrac{x^3}{8}\iff\dfrac{x^3}{8}\left(\dfrac{x}{8}-1\right)< ... |
2,699,621 | <p>To show $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}$ is it enough to tell that the taylor series expansion of $\sqrt{1+x}$ around $0$ has more positive terms?</p>
| K B Dave | 534,616 | <p>From the <a href="https://en.wikipedia.org/wiki/Taylor%27s_theorem#Explicit_formulas_for_the_remainder" rel="nofollow noreferrer">integral form of the remainder</a> for Taylor's theorem,
$$\begin{split}(1+x)^{1/2}&=1+\tfrac{x}{2} -\tfrac{x^2}{8} +\tfrac{x^3}{16} \int_0^1 (1+tx)^{-5/2}3(1-t)^2\mathrm{d}t\\
&&... |
2,669,277 | <p>In his textbook <em>Calculus</em>, Spivak presents integration by parts as follows: </p>
<p>If $f'$ and $g'$ are continuous then
\begin{align*}
\int fg'&=fg-\int f'g\\
\int f(x)g'(x)\,dx&=f(x)g(x)-\int f'(x)g(x)\,dx\\
\int_a^b f(x)g'(x)\,dx&=f(x)g(x)\bigg|_a^b-\int_a^b f'(x)g(x)\,dx\\
\end{align*}
I und... | RRL | 148,510 | <blockquote>
<p>Spivak's statement for integration-by-parts (in the context of Riemann
integration) holds when $fg’$ and $gf’$, individually, are
Riemann integrable, and it is enough just that $f’$ and $g'$ be Riemann
integrable when $f$ and $g$ are continuous.</p>
</blockquote>
<p>The "counterexample" in the ... |
895,759 | <p>Is there some sorts of Krull's theorem (that every ring has maximal ideal) for rings that do not have multiplicative identity (unit)? So I know that non-unital rings do not satisfy Krull's theorem, but for some types of non-unital rings, theorem does get satisfied. So what is it?</p>
<p>Edit: Wikipedia seems to men... | rschwieb | 29,335 | <p>As you noted, there is no version of Krull's theorem that asserts the existence of maximal ideals in all rngs. The best you can hope for is truth in certain subclasses of rngs. One obvious example is if you ask for the maximal condition on right/left or two-sided ideals.</p>
<p>Here's the result that Wikipedia is r... |
50,227 | <p>The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in $(x,y,z)$ form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in $(x,y)$ form.</p>
<p>My guess would be it'd assign an arbitrary up vector and then doing some... | marty cohen | 13,079 | <p>My take on this is that you want to find a mapping of the form
<span class="math-container">$(x, y, z) \mapsto (ax+by+cz, dx+ey+fz) $</span>
so the resulting triangle in the plane is the same shape and size as the original triangle.
Let <span class="math-container">$L_{ij}$</span> be the distance between points <spa... |
2,174,061 | <p>in $\Delta ABC$ if the $AD\perp BC$,$D\in BC$,and such $$|BC|=2|AD|$$
show that
$$\dfrac{|AB|}{|AC|}\le\sqrt{2}+1$$
<a href="https://i.stack.imgur.com/SXDvI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SXDvI.png" alt="enter image description here"></a></p>
<p>since
$$\cot{B}+\cot{C}=\dfrac{BD}... | tehjh | 101,039 | <p>This is more of an algebra problem in fact, and I misled myself into thinking of Stewart's theorem at first. In fact if we let the length of $AD$ be 1 and $BC$ be 2 it becomes equivalent to trying to maximise an expression like $\frac{x^2+1}{(2-x)^2+1}$ where $x$ varies between $0$ and $2$ and since the numerator ... |
4,170,940 | <blockquote>
<p><a href="https://www.isical.ac.in/%7Eadmission/IsiAdmission2017/PreviousQuestion/BStat-BMath-UGA-2016.pdf" rel="nofollow noreferrer">Question 36</a>: Finding graph corresponding to <span class="math-container">$\int_0^{\sqrt{x} } e^{ -\frac{u^2}{x} } du$</span>
<a href="https://i.stack.imgur.com/KIVRA.p... | saulspatz | 235,128 | <p>You haven't done the differentiation correctly, but that's not the best way to do the problem, anyway.</p>
<p>Write <span class="math-container">$$e^{-u^2/x}=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\left(\frac{u^2}{x}\right)^n$$</span></p>
<p>Term-by-term integration gives <span class="math-container">$$\sum_{n=0}^\infty\... |
2,302,067 | <p>I'm trying to prove that if ${\kappa}$ is an infinite cardinal, then there are $2^{\kappa}$ bijective functions from ${\kappa}$ to ${\kappa}$. I would greatly appreciate any tips. Thank you. </p>
| Asinomás | 33,907 | <p>I would do it differently.</p>
<p>The number of bijections is clearly not more than $\kappa^\kappa=2^\kappa$.</p>
<p>Let $S$ be the set of subsets of $\kappa$ that leave more than $1$ element in the complement. Clearly $S$ has the same cardinality as $2^\kappa$.</p>
<p>To show that there are at least $2^\kappa$ b... |
2,250,638 | <p>I feel I am doing the problem correctly however my answers are not following the solution.</p>
<p>My attempt: </p>
<p>$y^{2}+2y+12x-23=0$</p>
<p>$(y^{2}+2y+1) +12x = 23-1$</p>
<p>$(y+1)^{2}+12x=22$ </p>
<p>$\dfrac{(y+1)^{2}}{22}+\dfrac{6x}{11}=1$</p>
<p>Note $a > b$</p>
<p>$a^{2}=22$</p>
<p>$b^{2}=11$</p>... | Mick | 42,351 | <p>This is not a solution. The following figure is just my guess on the meaning of the question.</p>
<p><a href="https://i.stack.imgur.com/Pwh6T.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Pwh6T.png" alt="enter image description here"></a></p>
|
281,714 | <p>The absolute maximum value of $f\left(x\right) = x^3-3x^2+12$ on closed interval $\left[-2,4\right]$ occurs at $x = $ </p>
<p>Confused what does <em>absolute maximum value</em> means. </p>
<p>Does it mean </p>
<ol>
<li>The largest of the large values? $\max \{f\left(x\right)\mid x\in [-2,4]\}$</li>
<li>The larges... | Rustyn | 53,783 | <p>It means your first assumption. </p>
<p>Setting the derivative equal to $0$, we obtain:<br></p>
<p>$3x^2 - 6x = 0 \Rightarrow$<br>
$x(3x - 6)=0 \Rightarrow$ <br></p>
<p>$x=0,$ or $x=2$</p>
<p>$f(2) = 8$, $f(0) = 12$</p>
<p>Now we test end points,</p>
<p>$f(4) = 64 - 48 + 12 = 28$ <br>
$f(-2)= -8 -12 + 12 = -8... |
281,714 | <p>The absolute maximum value of $f\left(x\right) = x^3-3x^2+12$ on closed interval $\left[-2,4\right]$ occurs at $x = $ </p>
<p>Confused what does <em>absolute maximum value</em> means. </p>
<p>Does it mean </p>
<ol>
<li>The largest of the large values? $\max \{f\left(x\right)\mid x\in [-2,4]\}$</li>
<li>The larges... | Martin Argerami | 22,857 | <p>I would guess it is your first option. A usual terminology in calculus is about absolute and relative (or local) maxima and minima. </p>
<p>The absolute maximum would be then $\max\{f(x):\ x\in[-2,4]\}$. </p>
<p>The phrase "absolute maximum <em>value</em>" probably has to do with the fact that when looking at extr... |
1,581,545 | <p>I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them.</p>
<p>I am coding this proof in Coq, and i'm doing it over <em>natural numbers</em>. I aim to prove the uniqueness of prime factorization (So I cannot use this lemma!). Howev... | robjohn | 13,854 | <p>Claim 2 below should answer the question.</p>
<p>Since the only <a href="https://en.wikipedia.org/wiki/Unit_%28ring_theory%29" rel="nofollow">unit</a> in $\mathbb{N}$ is $1$, we have</p>
<p>$p$ is <a href="https://en.wikipedia.org/wiki/Prime_element" rel="nofollow">prime</a> iff $p\mid ab\implies p\mid a\lor p\mid... |
4,317,945 | <p>A function <span class="math-container">$h : A → \mathbb{R}$</span> is Lipschitz continuous if <span class="math-container">$\exists K$</span> s.t.</p>
<p><span class="math-container">$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$</span></p>
<p>Suppose that <span class="math-container">$I = [a, b]$</span... | heropup | 118,193 | <p>It's really difficult to read your writing; for me to try to detect where you might have made a calculation error would probably take longer than to just write my own step-by-step solution.</p>
<p>Let <span class="math-container">$f(x) = x^2$</span>, <span class="math-container">$g(x) = (x+1)^n$</span>, and <span cl... |
163,465 | <p>I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.</p>
<p>1) <strong>Does this quantity $f(X,t)$ have a name</strong>? As the title question suggests, it is a tail of the integral that is cut out when computing t... | eisit | 48,584 | <p>I think results from extreme value theory will be helpful here. The standard condition to put on $p(x)$ in these kinds of situations is that $p$ is regularly varying in $x$, and that there exists a constant $\gamma > 0$ such that
$$ \lim_{x \rightarrow \infty} \frac{1 - F(tx)}{1 - F(x)} = t^{-1/\gamma} \text{ for... |
189,689 | <pre><code>CountryData[ "UnitedStates", {"Population", 2014} ]
</code></pre>
<blockquote>
<p>322 422 965 people</p>
</blockquote>
<pre><code>CountryData[ "UnitedStates", {"Population", 2015} ]
</code></pre>
<blockquote>
<p>Missing[ "NotAvailable" ]</p>
</blockquote>
<p>How can I update it?
I am new to Mathemati... | Edmund | 19,542 | <p>You should report to WRI. As a workaround you may use the entity functions instead; it is a bit verbose.</p>
<p>First you can get a list of qualifier values of an <code>EntityProperty</code> by</p>
<pre><code>EntityValue[EntityProperty["Country", "Population"], "QualifierValues"]
</code></pre>
<blockquote>
<pre><... |
1,369,482 | <p>I'm have problem proving: Law for Scalar Multiplication :</p>
<p>Vector spaces possess a collection of specific characteristics and properties. Use the definitions in the attached “Definitions” to complete this task.</p>
<p>Define the elements belonging to $\mathbb{R}^2$ as $\{(a, b) | a, b \in \mathbb{ R}\}$. Com... | Patrick Da Silva | 10,704 | <p>If you're asking if vector spaces are closed under multiplication by a scalar, then yes, it is true. If you're asking why, it's because it's written in the definition of a vector space that it must be true ; there is nothing to prove here. It's true because we assume it is when we speak of a vector space.</p>
<p>ED... |
1,369,482 | <p>I'm have problem proving: Law for Scalar Multiplication :</p>
<p>Vector spaces possess a collection of specific characteristics and properties. Use the definitions in the attached “Definitions” to complete this task.</p>
<p>Define the elements belonging to $\mathbb{R}^2$ as $\{(a, b) | a, b \in \mathbb{ R}\}$. Com... | Rescy_ | 248,257 | <p>If $V$ is a vector space over the field $\mathbf F$, then it must satisfy two properties, namely closure under addition and closure under multiplication.</p>
<p>For closure under multiplication, we demand that if $u \in V$, $a \in \mathbf F$, then $a \mathbf F \in V$. Note that the 'multiplication' needs to be defi... |
3,854,446 | <p>I am reading a textbook on representation theory which says the following.</p>
<p><span class="math-container">$G$</span> is a finite group with irreducible representation <span class="math-container">$\rho:G\to GL(V)$</span> over field <span class="math-container">$k$</span> (possibly algebraically closed, there's ... | runway44 | 681,431 | <p>Suppose <span class="math-container">$T=aI$</span> is a scalar multiple of the identity map.</p>
<p>Taking traces yields <span class="math-container">$\mathrm{tr}(T)=(\dim V)a=(\phi,\chi)=0$</span>, so <span class="math-container">$a=0$</span>.</p>
<p>(I assume <span class="math-container">$\dim V$</span> is inverti... |
2,021,557 | <p>I'm not really sure how to do this, I guessed it had something to do with Vector Functions but overall couldn't find a way to do it. Can you please help?</p>
<p>The equations are:</p>
<p>$$f(x,y) = x^2 + y^2
\ g(x,y) = xy + 10 $$</p>
<p>and I need a Vectorial equation.
Thank you in advance!</p>
| mvw | 86,776 | <p>$f$ is a paraboloid (red colour), $g$ is a hyperbolic paraboloid (green colour):</p>
<p><a href="https://i.stack.imgur.com/JAc4mm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JAc4mm.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/J0c8Rm.png" rel="nofollow norefe... |
1,103,624 | <p>Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem.</p>
<p>Using only the natural numbers 1 through 9 without repeating any of them (natural because there cannot occur any rationals anywhere in this process, i.e. th... | Fizz | 173,347 | <p>I do have a suggestion for an algorithm that would reduce the cases to try. If I understand correctly that there isn't an operator precedence, but all operations are simply done left to right, then last thing you do is a multiplication. But $100 = 2 \times 2 \times 5 \times 5$. So the last box can only be 1, 2, 4 or... |
1,103,624 | <p>Ashamed to admit that I cannot aid my friend's niece with her second grade homework problem. So much for that college education, eh? Here's the problem.</p>
<p>Using only the natural numbers 1 through 9 without repeating any of them (natural because there cannot occur any rationals anywhere in this process, i.e. th... | epi163sqrt | 132,007 | <p>As it is a second grade homework problem, I don't think the problem is stated with the intention to find <em>all</em> solutions.</p>
<p>Note, the problem does <em>not ask</em> to find all solutions. To me it seems more of being a stimulus to <em>play</em> with numbers and fundamental operations in order to increase... |
2,781,801 | <p>When asked to evaluate $g$ at the point specified above we would get $\dfrac{1}{e} \cdot \log_e(\frac{1}{\sqrt e})$ and that evaluates to some -0.18393... but the correct answer is -1/2e. How does it get simplified to that?</p>
| max_zorn | 506,961 | <p>You did well. The "correct" answer is just written sloppily. It should be
$$-1/(2e)$$
which evaluates to what you found. </p>
|
3,306,571 | <p>I know that the function <span class="math-container">$f(x) = \frac{x}{x}$</span> is not differentiable at <span class="math-container">$x = 0$</span>, but according to the definition of differentiable functions:</p>
<blockquote>
<p>A differentiable function of one real variable is a function whose derivative exi... | mlchristians | 681,917 | <p>To see why the function <span class="math-container">$f(x) = \frac{x}{x}$</span> is differentiable everywhere except at <span class="math-container">$0$</span>, and has derivative equal to <span class="math-container">$0$</span> where it is differentiable, consider the following: </p>
<p>The graph of <span class="m... |
3,699,105 | <p>If <span class="math-container">$T$</span> is normal operator and <span class="math-container">$T^3=T^2$</span>,then show that <span class="math-container">$T$</span> is idempotent .</p>
<ol>
<li><span class="math-container">$TT*=T*T$</span> </li>
<li><span class="math-container">$T^3=T^2$</span></li>
<li>We are to... | fleablood | 280,126 | <p>For a counter example just remove an essential item from R but keep it in S. And maybe (but not necessarily) vice versa</p>
<p>For example if <span class="math-container">$T= R\cup S = \mathbb Z \times \mathbb Z$</span> be the equivalence relationship on <span class="math-container">$\mathbb Z$</span> that <em>ever... |
2,567,332 | <p>A Greek urn contains a red, blue, yellow, and orange ball. A ball is drawn from the urn at random and then replaced. If one does this $4$ times, what is the probability that all $4$ colors were selected?</p>
<p>I approached this questions by doing $(1/4)^4$ because there's always a $1/4$ chance of selected a specif... | Remy | 325,426 | <p>The first ball can be any of the four with probability $\frac{4}{4}$</p>
<p>The second ball must be any of the other three with probability $\frac{3}{4}$</p>
<p>The third ball must be any of the other two with probability $\frac{2}{4}$</p>
<p>The fourth ball must be the ball that hasn't been selected yet with pro... |
2,567,332 | <p>A Greek urn contains a red, blue, yellow, and orange ball. A ball is drawn from the urn at random and then replaced. If one does this $4$ times, what is the probability that all $4$ colors were selected?</p>
<p>I approached this questions by doing $(1/4)^4$ because there's always a $1/4$ chance of selected a specif... | Benji Altman | 398,014 | <p>We could do this by counting the number of ways to draw four balls and the number of ways to draw four balls without getting any duplicates.
There are $4!$ ways to not get a duplicate as every drawing can be thought of as an ordering and if we don't allow duplicates then we have a permutation. There are $4^4$ diffe... |
1,102,885 | <p>I have exams in Machine Learning coming up and I need help answering this question.</p>
<blockquote>
<p>There are a million identical fish in a lake, one of which has
swallowed the One True Ring. You must get it back! After months of
effort, you catch another random fish and pass your metal detector
over it... | ChocolateBar | 161,284 | <p>You have $2$ cases in which the detector will beep:</p>
<p>$1:$ You have found the fish with the ring and the detector beeps. Probability: </p>
<p>$$\frac{1}{10^6} \cdot (1-\frac{1}{10^9}) = 9.99999999 \cdot 10^{-7}$$</p>
<p>$2:$ You have not found the fish with the ring but the detector beeps anyway. Probability... |
1,566,471 | <p>Hi can someone please help?</p>
<p>I need to evaluate this indefinite integral:</p>
<p>$$\int \frac{(\ln x)^5}x dx$$</p>
<p>I know I need to use substitution, so if I let <em>u= x</em> but I can't figure out the antiderivative for the top portion.</p>
<p>Thank you!</p>
| Hagen von Eitzen | 39,174 | <p>Actually, that is the <em>definition</em> of independent. Then again, $P(B\mid A)$ is not $=1$ ("if $A$ happens, $B$ must happen"), it is not even defined.</p>
|
874,300 | <p>I'm having trouble grasping how to set these types of problems. There are a lot of related questions but it's difficult to abstract a general procedure on finding constants that give the given function bounding constraints to make it big-theta(general function). </p>
<p>so $\frac{x^4 +7x^3+5}{4x+1}$ is $ \Theta ... | David | 119,775 | <p>Here is a nice simple method.</p>
<p>If $x>1$ then
$$\frac{x^4 +7x^3+5}{4x+1}<\frac{x^4+7x^4+5x^4}{4x}=\frac{13}{4}x^3$$
and
$$\frac{x^4 +7x^3+5}{4x+1}>\frac{x^4}{4x+x}=\frac{1}{5}x^3\ .$$
That is, we have shown that if $x>1$ then
$$\frac{1}{5}x^3<f(x)<\frac{13}{4}x^3\ .$$</p>
|
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | Aryabhata | 1,102 | <p>Elementary method: use AM $\ge$ GM!</p>
<p>$n-2$ copies of $1$, two copies of $\frac{1}{\sqrt{2}}$ gives</p>
<p>$$ \frac{n-2 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{n} \ge \left(\frac{1}{2}\right)^{1/n}$$</p>
<p>$$\frac{n-c}{n} \ge \left(\frac{1}{2}\right)^{1/n}$$</p>
<p>for some $c \gt 0$ ($ c= 2 - \sqrt{2}$... |
91,590 | <p>So I'm reviewing old homeworks for an upcoming comp sci test and I came across this question:</p>
<p>Say whether the following statement is True, False or Unknown: </p>
<blockquote>
<p>The problem of checking whether a given Boolean formula has exactly
one satisfying assignment, is NP-complete</p>
</blockquote... | Kyle Jones | 21,376 | <p>For the problem to be NP-complete it has to be in NP, which means it has to have a polynomial time verifier for all "yes" answers. If the exactly-1-satisfying-assignment question is in NP, that means there must be "yes" answers and verifiers for the following two questions:</p>
<p>Does this formula have a satisfyi... |
37,325 | <p>Suppose Bob has 50% chance to stand at each of two points $p_1, p_2$ on a unit circle. If one tries to naïvely answer the question "what is Bob's average position on the circle", an ambiguity shows up: the answer can be either the midpoint along the shortest arc between $p_{1,2}$ or along the longest. </p>
<p>More ... | Alon Amit | 308 | <p>If you wish to have a specific, unambiguous "average position" for any distribution then, in particular, you expect to have one for the uniform distribution. This implies that whatever your definition is, it must give preference to some point on the circle.</p>
<p>Granting that, you can parametrize the circle by th... |
37,325 | <p>Suppose Bob has 50% chance to stand at each of two points $p_1, p_2$ on a unit circle. If one tries to naïvely answer the question "what is Bob's average position on the circle", an ambiguity shows up: the answer can be either the midpoint along the shortest arc between $p_{1,2}$ or along the longest. </p>
<p>More ... | joriki | 6,622 | <p>It depends on what you mean by "systematic". You can arbitrarily place a cut somewhere and parametrize the circle using angles $\phi\in[0,2\pi]$. The moments of these angles are unambiguous, but also arbitrary. There can't be a non-arbitrary way to do this because of the symmetry of the circle. If you have two point... |
3,760,594 | <p>Is there a proper notation to <em>compose</em> sets and produce a set of sets? (<em>I am referring to this as compose due to ignorance of a proper manner to call it</em>)</p>
<p>To illustrate what I want, let me <em>suppose</em> that <span class="math-container">$\otimes$</span> does the job, so that</p>
<p><span cl... | Tuvasbien | 702,179 | <p>If <span class="math-container">$\nabla f=\nabla g$</span>, then <span class="math-container">$\frac{\partial f}{\partial x_k}=\frac{\partial g}{\partial x_k}$</span> for all <span class="math-container">$k\in\{1,\ldots,n\}$</span>. Thus there exists <span class="math-container">$c_k(x_1,\ldots,x_{k-1},x_{k+1},\ldot... |
804,882 | <p>If both $L:V\rightarrow W$ and $M:W\rightarrow U$ are linear transformations that are invertible, how can you prove that the composition $(M\circ L):V\rightarrow U$ is also invertible.</p>
| Kevin Sheng | 150,297 | <p>Hint: Suppose $L$ and $M$ are two invertible linear transformations. This means that the standard matrix $[L]$ and $[M]$ are invertible. The composition of two linear transformations is just the product of their standard matrices. Go from there.</p>
|
1,560,209 | <p>Prove that $f(x):\mathbb{R}\to\mathbb{R}$ , $x \mapsto x^3$ is injective.</p>
<hr>
<p>I want to prove this claim is true. </p>
<p>Here is my outline so far:</p>
<hr>
<p>We want to show that $f(a)=f(b)$ implies that $a=b$, for all $a,b \in \mathbb{R}$</p>
<p>We have $f(a)=a^3$, and $f(b)=b^3$</p>
<p>So, if $f... | Alekos Robotis | 252,284 | <p>Here is an alternative suggestion for a proof. First, show that $f$ is strictly increasing. Consider $x\in\mathbb{R}$, and define some $\epsilon>0$.
$$ f(x+\epsilon)=(x+\epsilon)^3=x^3+3x^2\epsilon+3x\epsilon^2+\epsilon^3>x^3=f(x),\forall x\in\mathbb{R}. $$
Now, suppose this function were not injective, that i... |
160,518 | <p>In Mathematics, we know the following is true:</p>
<p>$$\int \frac{1}{x} \space dx = \ln(x)$$</p>
<p>Not only that, this rule works for constants added to x:
$$\int \frac{1}{x + 1}\space dx = \ln(x + 1) + C{3}$$
$$\int \frac{1}{x + 3}\space dx = \ln(x + 3) + C$$
$$\int \frac{1}{x - 37}\space dx = \ln(x - 37) + C$$... | Nicholas Kirchner | 33,957 | <p>It boils down to $u$-substitution (which if you haven't covered yet, you soon will). You know that
$$ \int \frac{1}{x} dx = \ln x + C $$
(I'll not bother with absolute values here. You should remember them on problems that you do for class, but they aren't the focus of this question.)</p>
<p>Now, to handle
$$ \in... |
160,518 | <p>In Mathematics, we know the following is true:</p>
<p>$$\int \frac{1}{x} \space dx = \ln(x)$$</p>
<p>Not only that, this rule works for constants added to x:
$$\int \frac{1}{x + 1}\space dx = \ln(x + 1) + C{3}$$
$$\int \frac{1}{x + 3}\space dx = \ln(x + 3) + C$$
$$\int \frac{1}{x - 37}\space dx = \ln(x - 37) + C$$... | Joe | 24,942 | <p>Generally speaking, "using $\ln (x)$" as a rule or technique is unheard of. When one speaks of techniques, they usually include integration by substitution, integration by parts, trig substitutions, partial fractions, etc. With introductory calculus in mind, $\ln |x|$ is <strong>defined</strong> as $\int \frac{1}{x... |
63,015 | <p>Why does assigning a DownValue using <code>Apply</code>, e.g.,</p>
<pre><code>Remove[a]
index={3,4};
(a @@ index) = 5;
a @@ index
(*Set::write: Tag Apply in a @@ {3, 4} is Protected. >>*)
(*a[3,4]*)
</code></pre>
<p>not work, while an assignment such as</p>
<pre><code>Remove[a]
a[Sequence @@ index] = 5
a @@... | Chris Degnen | 363 | <p>Evaluation happens too late, but you can fix it by forcing evaluation first:</p>
<pre><code>Remove[a]
index = {3, 4};
Evaluate[a @@ index] = 5;
a @@ index
</code></pre>
<blockquote>
<p>5</p>
</blockquote>
<p>This can only be done on the initial definition though. After that</p>
<pre><code>a[Sequence @@ index... |
3,529,359 | <p>Let <span class="math-container">$\Omega$</span> be a bounded and smooth domain and let <span class="math-container">$J:H^1(\Omega) \times H^1_0(\Omega) \to \mathbb{R}$</span> be defined by</p>
<p><span class="math-container">$$J(u,v) = \int_\Omega f(u)|\nabla v|^2$$</span>
where <span class="math-container">$f\col... | daw | 136,544 | <p>Here is the proof of @JohannesHahn in a more elementary way.
The idea is to show that
<span class="math-container">$$
(u,v) \mapsto \sqrt{ f(u)}\nabla v
$$</span>
is weakly sequentially continuous from <span class="math-container">$H^1(\Omega)\times H^1(\Omega)$</span> to <span class="math-container">$L^2(\Omega)$</... |
2,603,239 | <p>(The Cauchy principal value of)
$$
\int_0^{\infty}\frac{\tan x}{x}\mathrm dx
$$</p>
<p>I tried to cut this integral into $$\sum_{k=0}^{\infty}\int_{k\pi}^{(k+1)\pi}\frac{\tan x}{x}\mathrm dx$$
And then
$$\sum_{k=0}^{\infty}\lim_{\epsilon \to 0}\int_{k\pi}^{(k+1/2)\pi-\epsilon}\frac{\tan x}{x}\mathrm dx+\int_{(k+1/2... | robjohn | 13,854 | <p>This can also be handled using contour integration. Since there are no singularities inside $\gamma$,
$$
\int_\gamma\frac{\tan(z)}z\,\mathrm{d}z=0
$$
where $\gamma$ is the contour</p>
<p><a href="https://i.stack.imgur.com/4HUqG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4HUqG.png" alt="enter... |
1,151,653 | <p>How can I express the following as a function sequence? Namely, how can I properly express <span class="math-container">$f_n(x)$</span>?</p>
<p>Here are the following function graphs:</p>
<p><img src="https://i.stack.imgur.com/2GFYj.png" alt="enter image description here" /></p>
<p>Text only (color-coded with image)... | AvZ | 171,387 | <p>A unit circle will have the equation $$x^2 + y^2=1$$<br>
To find the intersection points of this with the parabola $y=x^2$ we substitute the value of $y$.<br>
We can write this as<br>
$$x^2 + x^4=1$$<br>
This is a quadratic equation in terms of $x^2$.
We can now write $$x^2=\frac{-1+\sqrt{5}}{2}$$
Note that a real... |
3,956,292 | <p>Consider the Euclidean ball <span class="math-container">$B^n(x,r)$</span> in <span class="math-container">$\mathbb{R}^n$</span> given by:
<span class="math-container">$$B^n(x,r) = \{z\in\mathbb{R}^n : ||z-x||_2 \leq r\}$$</span> with centre <span class="math-container">$x\in\mathbb{R}^n$</span> and radius <span cla... | Andreas Blass | 48,510 | <p>To simplify notation, I'll assume, without loss of generality, that the center of the ball is at the origin.
Suppose, toward a contradiction, that you have a counterexample in some high dimension. Say your set contains <span class="math-container">$p$</span> and <span class="math-container">$q$</span> but not some p... |
3,973,006 | <p>The question is fully contained in the title.</p>
<p>I tried to prove maximality (if that happens, <span class="math-container">$I$</span> is prime as well) in <span class="math-container">$\mathbb Z[X]$</span>, but I am not able to figure a strategy out for that purpouse. Obviously, if <span class="math-container">... | Bernard | 202,857 | <p><strong>Hint</strong>: <span class="math-container">$I$</span> contains <span class="math-container">$\;7(X^3+2X^2+1)-X^2(7X+14)=7$</span>, hence
<span class="math-container">$$\mathbf Z[X]/I\simeq \mathbf Z/7 \mathbf Z[X]/(I/7 \mathbf Z[X])= \mathbf Z/7 \mathbf Z[X]/(X^3+\bar 2X^2+\bar 1).$$</span>
Can you show tha... |
64,544 | <blockquote>
<p>Please let me know what is the standard notation for group action.</p>
</blockquote>
<p>I saw the following three notations for group action.
(All the images obtained as <code>G\acts X</code> for different deinitions of <code>\acts</code>.) </p>
<p>(1) <img src="https://lh5.googleusercontent.com/_7... | Theo Johnson-Freyd | 78 | <p>It is always a shame, of course, when none of the many LaTeX packages has precisely the symbol that you might use on the chalkboard. In writing, you should try to use words when possible, or at least supplement your symbols with words. Someone just reading the words should be able to pick up the nuances of your no... |
64,544 | <blockquote>
<p>Please let me know what is the standard notation for group action.</p>
</blockquote>
<p>I saw the following three notations for group action.
(All the images obtained as <code>G\acts X</code> for different deinitions of <code>\acts</code>.) </p>
<p>(1) <img src="https://lh5.googleusercontent.com/_7... | Pete L. Clark | 1,149 | <p>As it happens, I can still remember being confused the first few times I saw this notation: not to put too fine a point on it, but there is something syntactically new going on there beyond the usual function / arrow notation.</p>
<p>In my opinion this is not notation at all but rather <strong>shorthand</strong>. ... |
1,695,261 | <p>Is it true that for every $ε > 0$, there is $δ > 0$, such that $0 < |x−2| < δ ⇒ |(x^2 −x)−2| < ε$?</p>
<p>Now I know that $|(x^2 −x)−2|$ is same as $|(x-2)(x+1)|$, but I am not sure how to link that with the first bit of info given. In general epsilon-delta proofs confuse me. </p>
<p>So I start by s... | Clarinetist | 81,560 | <p>I always like to refer people to my answer <a href="https://math.stackexchange.com/questions/418961/epsilon-delta-proof-that-lim-limits-x-to-1-frac1x-1/418991#418991">here</a> when it comes to simple polymonial $\delta$-$\epsilon$ proofs. Read this link so that you understand my methodology here.</p>
<p><strong>Scr... |
2,755,143 | <p>Find Number of integers satisfying $$\left[\frac{x}{100}\left[\frac{x}{100}\right]\right]=5$$ where $[.]$ is Floor function.</p>
<p>I assumed $$x=100q+r$$ where $0 \le r \le 99$</p>
<p>Then we have </p>
<p>$$\left[\left(q+\frac{r}{100}\right)q\right]=5$$ $\implies$</p>
<p>$$q^2+\left[\frac{rq}{100}\right]=5$$</... | Jason | 130,776 | <p>A less-than-satisfying, but simple and fast computational approach -- this double-flooring operation is increasing in $x$. When $x=200$, you get 4. When $x=300$, to get 9. This just leaves 100 integers to check with a simple program. Here is one such program in Mathematica:</p>
<blockquote>
<p>Tally@Table[Floor[x... |
3,519,515 | <p>Here, I wonder what is a good way to use the epsilon delta definition or converging sequences to show that the set S containing quotients on [0,1] have/does not have volume 0, (i.e. whether there exist a <strong>finite</strong> number of intervals which union contain all of S such that the <strong>sum</strong> of le... | Henno Brandsma | 4,280 | <p>Suppose <span class="math-container">$\lambda(\Bbb Q \cap [0,1]) < 1$</span>, where by <span class="math-container">$\lambda$</span> I mean the volume of the set (as (sort of) defined in the comments to the question.</p>
<p>This would mean that we can cover <span class="math-container">$\Bbb Q \cap [0,1]$</span>... |
58,525 | <p>I am trying to make surface plots of squashed spheres. The spheres are defined by a list of points. For simplicity, consider the round sphere:</p>
<pre><code>pts = Flatten[
Table[{Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ],
Cos[θ]}, {θ, 0, π, π/14}, {ϕ, 0,
2 π, 2 π/14}], 1];
</code></pre>
<p>One way to plot thi... | Dr. belisarius | 193 | <p>Just format your list such that <code>SphericalPlot3D[]</code> can handle it:</p>
<pre><code>pts = Flatten[Table[{Theta, Phi, Cos[Theta]}, {Theta, 0, Pi, Pi/14}, {Phi, 0, 2 Pi, 2 Pi/14}], 1];
f = Interpolation[pts];
SphericalPlot3D[f[Theta, Phi], {Theta, 0, Pi}, {Phi, 0, 2 Pi}, ColorFunction -> "Rainbow",
... |
1,981,360 | <blockquote>
<p>Given function $f:\mathbb{R}_0^+ \to \mathbb{R},~f(x) = x^2 + 4x + 4$ prove that it is injective.</p>
</blockquote>
<p>Using definition of injectivity $(\forall x_1, x_2 \in \mathbb{R}_0^+)(x_1 \neq x_2 \implies f(x_1) \neq f(x_2))$ I'm doing the following:</p>
<p>$$x_1^2 + 4x_1 + 4 = x_2^2 + 4x_2 +... | hamam_Abdallah | 369,188 | <p>$\exists (x_1,x_2) \in [0,+\infty)^2 : x_1^2+4x_1+4=x_2^2+4x_2+4 \; \implies$</p>
<p>$(x_1-x_2)(x_1+x_2+4)=0 \implies$</p>
<p>$x_1=x_2 $ since $x_1+x_2+4\geq4$.</p>
<p>thus $f$ is injective.</p>
|
191,548 | <p>Say I have a list:</p>
<pre><code>{{Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]},
{Line[{{-Sqrt[5/8 + Sqrt[5]/8],1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, ... | Sjoerd Smit | 43,522 | <p>You can use <code>Cases</code> for this:</p>
<pre><code>lines = {{Line[{{-Sqrt[5/8 - Sqrt[5]/8],
1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,
1}}]}, {Line[{{-Sqrt[5/8 + Sqrt[5]/8],
1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8],
1/4 (-1 - Sq... |
191,548 | <p>Say I have a list:</p>
<pre><code>{{Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]},
{Line[{{-Sqrt[5/8 + Sqrt[5]/8],1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, ... | gwr | 764 | <h2>Answer to what you want</h2>
<pre><code>lines = {
{ Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]
, Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]
}
, { Line[{{-Sqrt[5/8 + Sqrt[5]/8], 1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}]
, Line[{{Sqrt[5/8 ... |
2,256,973 | <p>I'm writing a computer algorithm to do binomial expansion in C#. You can view the code <a href="https://gist.github.com/jamesqo/01015428601641347e436129c1ae0079#file-multinomial-cs-L29-L39" rel="nofollow noreferrer">here</a>; I am using the following identity to do the computation:</p>
<p>$$
\dbinom n k = \frac n k... | yberman | 92,108 | <p>You might be better off memoizing the traditional formula? If you have a 32-bit roof there isn't much that can take that long.</p>
<pre><code> #include <stdio.h>
#define N 50
int cache[N][N];
int f(int n, int k) {
if (k == 0 || k == n) {
return 1;
}
... |
2,781,017 | <p>I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if</p>
<p>$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$ </p>
<p>holds true for $a_i$, $b_i \in (0,1)$.</p>
| Ingix | 393,096 | <p>$a_1=0.1, a_2=0.2, b_1=0.3, b_2=0.4$ lead to the incorrect statement</p>
<p>$$\frac13 + \frac24 \le \frac37$$</p>
<p>In reality, the opposite inequality is true. You can see that if you rename $a,b$ to $x,y$ and rewrite it as</p>
<p>${\sum y_i} \sum \frac{x_i}{y_i} \geq \sum x_i$. This is the first inequality you... |
2,948,045 | <p>In Eric Gourgoulhon's "Special Relativity in General Frames", it is claimed that the two dimensional sphere is not an affine space. Where an affine space of dimension <em>n</em> on <span class="math-container">$\mathbb R$</span> is defined to be a non-empty set E such that there exists a vector space V of dimension ... | J. Darné | 611,408 | <p>The thing is, you might want to get some topology in the picture. In fact, if you do not, you can choose any bijection between the sphere and a <span class="math-container">$\mathbb R$</span>-vector space, and you end up with a structure of vector space on your "sphere" (by transporting the structure). My point is, ... |
312,878 | <p>Why is $\mathbb{Z} [\sqrt{24}] \ne \mathbb{Z} [\sqrt{6}]$, while $\mathbb{Q} (\sqrt{24}) = \mathbb{Q} (\sqrt{6})$ ?</p>
<p>(Just guessing, is there some implicit division operation taking $2 = \sqrt{4}$ out from under the $\sqrt{}$ which you can't do in the ring?)</p>
<p>Thanks. (I feel like I should apologize for... | Zev Chonoles | 264 | <p>No need to apologize; and your instinct is correct. Note that $\sqrt{6}\notin\mathbb{Z}[\sqrt{24}]$; indeed,
$$\mathbb{Z}[\sqrt{24}]=\{a+b\sqrt{24}\mid a,b\in\mathbb{Z}\}=\{a+2c\sqrt{6}\mid a,c\in \mathbb{Z}\}.$$
Thus, $\mathbb{Z}[\sqrt{24}]$ consists of the elements of $\mathbb{Z}[\sqrt{6}]$ for which the number of... |
1,061,311 | <p>Suppose $\sum_{n=0}^\infty a_n$ and $\sum_{m=0}^\infty b_m$ converge absolutely. I have to show that $$\left(\sum_{n=0}^\infty a_n\right) \cdot \left(\sum_{m = 0}^\infty b_m\right) = \sum_{m, n}^\infty a_nb_m.$$ But I do not understand what the sum on the right-hand side means (i.e. what limit this represents). Coul... | Etienne | 80,469 | <p>You are perfectly right in non-understanding what this "double sum" means.</p>
<p>Here is one possible interpretation of what you have to prove. This relies on the notion of <em>summability</em> for a family of real numbers. Let $(c_i)_{i\in I}$ be a family of real numbers indexed by some set $I$. Then, $(c_i)_{i\i... |
1,821,437 | <p>I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. <em>If it's any indication of difficulty, the exercise is only Part 3-A of the sheet, graded for 10%</em></p>
<blockquote>
<p>Solve the equation $z^5=-32$ and draw its s... | JasonM | 343,478 | <p>The solutions should just be $2e^{\frac{2\pi ki}{10}}$, for odd integers $0 \leq k <10$. They are the divisions of the circle of radius 2 centered at the origin, divided into 5 pieces, rotated about the origin $\frac{2 \pi}{10}$ in the complex plane. </p>
<hr>
<p>Here's more details:</p>
<p>$z^5=-32 \implies... |
98,088 | <p>I can't understand a sentence in a textbook: if $x$ is a transitive set, then $\bigcup x^+=x$? Could someone help me to understand?</p>
<p><strong>added:</strong> $x^+=x\cup\{x\}$</p>
| Michael Greinecker | 21,674 | <p>The set $x$ is <em>transitive</em> if $z\in y \in x$ implies $z\in x$. Moreover, $x^+=x\cup\{x\}$. So let $x$ be a transitive set.</p>
<p>We have $x\subseteq\bigcup x^+$ since $x\in\{x\}$. Now let $z\in \bigcup x^+$. Then $z\in\bigcup x\cup x$. If $z\in\bigcup x$, then there exists $y\in x$ with $z\in y$. But since... |
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