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 |
|---|---|---|---|---|
1,934,268 | <p>Let's say I have an arbitrary matrix $A$. </p>
<p>An eigenvector of $A$ would be a vector $\vec{x}$, such that </p>
<ol>
<li>$A\vec{x} = \lambda \vec{x}$</li>
<li>$\vec{x} \in N(A-\lambda I)$ i.e. $(A - \lambda I)\vec{x} = \vec{0}$</li>
</ol>
<p>where $\lambda$ is an eigenvalue of $\vec{x}$</p>
<hr>
<p>Now are ... | Charles F | 143,884 | <p>I'll to answer one question at a time.
First of all, your claims 1 and 2 are the same.
if $Ax =\lambda x$, then note that $Ax = \lambda I x$ and thus we can factor out the $x$ on both sides and then subtract the RHS from the LHS, getting $(A - \lambda I)x =0.$ This means that $x \in N(A - \lambda I).$
In other words... |
3,435,511 | <p>I'm looking to write down formally that a multiset of elements contains at least two elements that differs in value.
e.g.,
S1 = {1,1,1,1,1,1} and S2={1,1,1,1,0,1}
S1 has all identical elements, S2 has at least two elements that differs in value.</p>
| user619894 | 617,446 | <p>You can try an asymptotic expansion near <span class="math-container">$x\rightarrow 0$</span> :</p>
<p>First, lets rewrite the integral as<br>
<span class="math-container">$I(x)=x\int_1^\infty du \sqrt({1-{1\over u^2}})\log(1-e^{-xu})$</span>, </p>
<p>next, we split the integral into two ranges: <span class="math-... |
917,365 | <p>I hope to get the exact value of the following double series:</p>
<p>$$ \sum_{k \geq 0} \sum_{n \geq 0} \binom{2k}{ k} \frac{(-1)^n}{n! (2k+2n+1) 2^{4k+2n+1}}. $$</p>
<p>I am not sure it is possible or not.
I need your comments.</p>
| Jack D'Aurizio | 44,121 | <p>You can write your double series as:
$$\begin{eqnarray*}\color{red}{S}&=&\frac{1}{2}\int_{0}^{1}\sum_{k\geq 0}\sum_{n\geq 0}\binom{2k}{k}16^{-k}x^{2k}\frac{(-1)^n}{4^n\,n!}x^{2n}\,dx\\&=&\int_{0}^{1}\frac{e^{-x^2/4}}{\sqrt{4-x^2}}dx=\int_{0}^{1/2}\frac{dx}{e^{x^2}\sqrt{1-x^2}}=\color{red}{\int_{0}^{\... |
3,091,464 | <p>If I had an array of 7 numbers and I wanted all numbers to be equally spaced within it but needed to start at 5 and end at 35 how would I do this?</p>
<p>For instance if I were given:</p>
<ul>
<li>Numbers in array: <strong>7</strong></li>
<li>Largest number in array: <strong>35</strong></li>
<li>Smallest number in... | zhw. | 228,045 | <p>Hint: <span class="math-container">$\int_\epsilon^{1-\epsilon} f = \int_\epsilon^{1-\epsilon} \hat f.$</span></p>
|
3,091,464 | <p>If I had an array of 7 numbers and I wanted all numbers to be equally spaced within it but needed to start at 5 and end at 35 how would I do this?</p>
<p>For instance if I were given:</p>
<ul>
<li>Numbers in array: <strong>7</strong></li>
<li>Largest number in array: <strong>35</strong></li>
<li>Smallest number in... | OldGodzilla | 255,201 | <p>Indeed, <span class="math-container">$f$</span> can be extended to <span class="math-container">$\hat f$</span> on <span class="math-container">$[0,1]$</span>. Let <span class="math-container">$\epsilon > 0$</span>. Then <span class="math-container">\begin{align} \bigg| \int_{0}^1 \hat fdx - \int_{\epsilon}^{1-... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Mike Jones | 2,344 | <p>I have created some <a href="http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html" rel="nofollow">autocorrected online exercises</a> in Mathematics that you could use as supplemental material.</p>
<p>I have also started an online index to the fo... |
753,623 | <blockquote>
<p>Let $X$ be a linearly ordered set in the order topology which is connected. Show that $X$ is a linear continuum.</p>
</blockquote>
<ul>
<li>A <a href="https://en.wikipedia.org/wiki/Linear_continuum" rel="nofollow"><em>linear continuum</em></a> is a linearly ordered set $X$ with more than one element ... | user2566092 | 87,313 | <p>Suppose that it doesn't satisfy the least upper bound property for a proper subset that has an upper bound. Take such a subset set that doesn't satisfy the least upper bound property, and the complement of the subset, and show that they are both open.</p>
|
753,623 | <blockquote>
<p>Let $X$ be a linearly ordered set in the order topology which is connected. Show that $X$ is a linear continuum.</p>
</blockquote>
<ul>
<li>A <a href="https://en.wikipedia.org/wiki/Linear_continuum" rel="nofollow"><em>linear continuum</em></a> is a linearly ordered set $X$ with more than one element ... | Henno Brandsma | 4,280 | <p>If $A$ is non-empty and has an upper bound $u$ but no least upper bound, consider the sets </p>
<p>$$A_1 = \{ x \in X \mid \exists y(x) \in A: x \le y(x) \}$$
and
$$A_2 = X \setminus A_1 = \{ x \in X \mid \forall y \in A: x > y \}$$</p>
<p>As $A \subset A_1$, it is non-empty. As $u$ cannot be in $A$ (as an upp... |
35,539 | <p>I express multilinear functions in the following format. Is there any ready command to convert them to multilinear functions easily?</p>
<p><strong>Input</strong></p>
<pre><code>posTerms = {2, 4, 9, 13, 19};
negTerms = {6, 11, 26};
IntegerString[posTerms, 2]
IntegerString[negTerms, 2]
{"10", "100", "1001", "1101"... | Dr. belisarius | 193 | <p>Also</p>
<pre><code>-Differences[Tr[Times @@@ (Position[Reverse@#, 1] & /@ # /. {n_Integer} ->
Subscript[x, n])] & /@ (IntegerDigits[#, 2] & /@ {posTerms, negTerms})]
</code></pre>
<p><img src="https://i.stack.imgur.com/g3SAK.png" alt="Mathematica graphics"></p>
<p>Equi... |
164,871 | <p>i want to choose optimal decision from following problem
Imagine having been bitten by an exotic, poisonous snake. Suppose the ER
physician estimates that the probability you will die is $1/3$ unless you receive
effective treatment immediately. At the moment, she can offer you a choice of
experimental antivenins fro... | Old John | 32,441 | <p>Since 1 and 4 are both residues (for any $p\ge 5$), then to avoid having consecutive residues (with 1 and 4), we would have to have both 2 and 3 as non-residues, and then we have 2 consecutive non-residues.</p>
<p>Thus, we must have either 2 consecutive residues or 2 consecutive nonresidues.</p>
<p>i.e.: 1 and 4 a... |
1,598,343 | <p>If you form a set of orthonormal polynomials on $[0,1]$, by applying the Gram-Schmidt process from monomials $\{1, x, x^2, \dots \}$ then what is required to show that this is a basis for $L^2[0,1]$?</p>
<p>$\text{Span}\{ e_n \}_{n \in \mathbb{N}} = L^2[0,1]$?</p>
<p>How can I use this to prove the above?</p>
<p>... | Disintegrating By Parts | 112,478 | <p>An orthonormal set $\{ p_n \}$ in a Hilbert space $H$ is a complete orthonormal basis iff there is no non-zero element of $H$ that is orthogonal to every $p_n$.</p>
<p>If you know that $\{ e^{2\pi in x} \}_{n=-\infty}^{\infty}$ is a complete orthonormal basis, then you can use this to bootstrap to the normalized se... |
4,343,249 | <p>In a variant of Russian Roulette, where you put 2 bullets in 2 adjacent chambers, like that:<a href="https://i.stack.imgur.com/2pVxr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2pVxr.png" alt="This is how it looks might look like, red circles represent bullet" /></a></p>
<p>Image credit: Brill... | User5678 | 632,875 | <p>For "spin", you are correct on the probability -- pretty straightforward.</p>
<p>For no spinning, we know the first person didn't land on either bullet. So there are 4 slots left.</p>
<p>Of those slots, you will only die if the first person's spin landed <em>one before</em> the start of the two bullets. So... |
532,197 | <p>I am confused about solving $1+t=\sqrt{4+t^2}$.</p>
<p>When I solve it per hand I come to the conclusion that $t$ can be everything.</p>
<p>$$\begin{align*}
1+t =\sqrt{4+t^2}& \qquad | \cdot^2 \tag{1} \\
1+t^2 = 4+t^2 & \qquad | -t^2 \tag{2} \\
1 = 4 \tag{3}
\end{align*}$$</p>
<p>However wolfram alpha te... | user90189 | 39,490 | <p>The following suggestion should be a comment, but I think it is better to use the whole space here.</p>
<p>Define $X_{\delta}=\cup_{x\in X}B(x,\delta)$, since $X$ is bounded then $X_{\delta}$ has finitely many components; fill in details. For $X_{\delta}$ define $f_{\delta}$ as you did for $f$. Note that $f_{\delta... |
532,197 | <p>I am confused about solving $1+t=\sqrt{4+t^2}$.</p>
<p>When I solve it per hand I come to the conclusion that $t$ can be everything.</p>
<p>$$\begin{align*}
1+t =\sqrt{4+t^2}& \qquad | \cdot^2 \tag{1} \\
1+t^2 = 4+t^2 & \qquad | -t^2 \tag{2} \\
1 = 4 \tag{3}
\end{align*}$$</p>
<p>However wolfram alpha te... | user103402 | 103,402 | <p>For every compact set $X\in\mathbb R$ define $$d(X)=\liminf_{t\to 0^+}\, t^{-1}\left( m(\{x\in \mathbb R \, |\, \exists y \in X : |x-y| \leq t\})-m(X)\right)$$
(this quantity is always defined, though may be $+\infty$). Then</p>
<ol>
<li>$d(X)\ge 2$ for every nonempty $X$, because you get two intervals of length ... |
3,288,534 | <p><span class="math-container">$7^{-1} \bmod 120 = 103$</span></p>
<p>I would like to know how <span class="math-container">$7^{-1} \bmod 120$</span> results in <span class="math-container">$103$</span>.</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="http://mathworld.wolfram.com/CarmichaelFunction.html" rel="nofollow noreferrer">Carmichael Function</a>, <span class="math-container">$$\lambda(120)=4$$</span></p>
<p>As <span class="math-container">$(7,120)=1, 7^4\equiv1\pmod{120}$</span></p>
<p><span class="math-container">$$7^{-1}\equiv7^3\equiv3... |
2,461,773 | <p>I would like some clarification about the Cantor Set:</p>
<ul>
<li>What are the elements in the Cantor Set?</li>
<li>How do I write the Cantor Set in mathematical terms (i.e in a summation)? I have seen online a formula but I do not understand how they got it so would be grateful if you could explain why it is this... | fleablood | 280,126 | <p>The elements of the Cantor set are real numbers between $0,1$. The set is formed by iteratively and infinitely cutting all intervals in thirds and removing the middle third.</p>
<ol>
<li>$E_0 = [0,1]$</li>
<li>$E_1 = [0, \frac 13] \cup [\frac 23, 1]$</li>
<li>$E_2 = [0, \frac 19]\cup [\frac 29, \frac 13]\cup[\frac... |
804,414 | <p>A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$</p>
| Ekaveera Gouribhatla | 31,458 | <p>This is my Approach, Let me know if i am on right track...
$$E(X|Y=5)=\sum_{x\in \mathbb{X}}xP(X=x|Y=5)=\sum_{x \in \mathbb{X}}\frac{xP(X=x,Y=5)}{P(Y=5)}$$</p>
<p>The set $\mathbb{X}$ is $\left\{1,2,3,4,6,7,8,\cdots \infty \right\}$ So</p>
<p>$$E(X|Y=5)=\sum_{k=1,2,3,\cdots,\infty_{k\ne 5}}\frac{k\,P(X=k,Y=5)}{P(Y... |
804,414 | <p>A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$</p>
| André Nicolas | 6,312 | <p>In outline, and in most details, the posted solution is correct. </p>
<p>However, the probabilities from $7$ on are not correct, since "not $6$" should include the probability of getting a $5$. So for example the probability that $X=7$ and $Y=5$ is $(1/6)^2(2/3)^4(5/6)$. For $X=8$ and $Y=5$, we multiply by $5/6$, a... |
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | user1066 | 106 | <p>Not very efficient, but perhaps in line with what you ultimately want to do?</p>
<pre><code>positions= Position[IntegerDigits[declist,2], {___,1,1,___}];
</code></pre>
<p>With <code>Extract</code></p>
<pre><code> numbers=Extract[declist, positions];
</code></pre>
<p><strong>Output</strong></p>
<pre><code>number... |
1,351,513 | <p>Express the function F in the form $f \circ g \circ h$. </p>
<p>$$F(x)=\frac {9}{( x^2 + 7)}$$</p>
<p>I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried:</p>
<p>$$h(x) = (x+7)$$</p>
<p>$$g(x) = x$$</p>
<p>$$f(x) = \frac {9}{x^2}$$</p>
<p>But obviously that gives me $F(x) = \dfrac {9}{(x+... | Zain Patel | 161,779 | <p>What about $\displaystyle h(x) = x^2, g(x) = x+7$ and $\displaystyle f(x) = \frac{9}{x}$. So that we have $$g\circ h = x^2 +7$$ and hence $f \circ g \circ h = f\circ(g\circ h)$ which is just $$f \circ (g \circ h) = \frac{9}{x^2 + 7}$$</p>
|
1,835,295 | <blockquote>
<p>What is $\gcd(12345,54321)$?</p>
</blockquote>
<p>I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an easy way to calculate such greatest common divisors.</p>
| Kushal Bhuyan | 259,670 | <p><strong>Hint:</strong> Look at the sum of the digits of $12345$ and $54321$ , it's divisible by $3$. So $\ldots$</p>
|
386,655 | <p>Recently Ernest Davis asked me about the following computational problem: we're given as input a composite integer <span class="math-container">$n$</span>, a divisor <span class="math-container">$k$</span> of <span class="math-container">$n$</span>, and a subset <span class="math-container">$S \subset \mathbb{Z}_n$<... | Ernest Davis | 22,344 | <p>A couple of more simple observations about this plus a conjecture. First, it's symmetric in <span class="math-container">$S$</span> and the set of shifts; you can rephrase it, "Given <span class="math-container">$n$</span> and <span class="math-container">$S$</span>, find <span class="math-container">$A \subset... |
386,655 | <p>Recently Ernest Davis asked me about the following computational problem: we're given as input a composite integer <span class="math-container">$n$</span>, a divisor <span class="math-container">$k$</span> of <span class="math-container">$n$</span>, and a subset <span class="math-container">$S \subset \mathbb{Z}_n$<... | Mihalis Kolountzakis | 176,233 | <p>E.D., I think the conjecture is not true. If one tries it on the smallest non-periodic tiling (see below) it fails. The example is from N.G. de Bruijn, On the factorization of cyclic groups, Indag. Math. 15, 4, 1953.</p>
<p>The following Python program</p>
<pre><code># A + B is a tiling of Z mod 72
A = [0, 8, 16, 18... |
283,776 | <p>I have a curve that looks like this (it's cyclical):</p>
<p><a href="https://i.imgur.com/CRZ7Gbr.png" rel="nofollow noreferrer">Curve</a></p>
<p>I can get a partial fit by fitting a 3rd degree polynomial, but I have a feeling there must be a better fit (something that involves sin & cos). </p>
<p>The fitted ... | George V. Williams | 54,806 | <p>Assuming $\Delta \approx \dfrac{\hat\tau - \tau}{2}$, then we have:</p>
<p>$$ f(\Delta ) \sin \left(\frac{\pi x}{2\Delta}\right) $$</p>
<p>Where $f(\Delta)$ is the value of the function at $\Delta$.</p>
|
1,864,939 | <p>I am surprised that this question hasn't been asked on here</p>
<p><strong>I need to show that</strong> </p>
<blockquote>
<p>$$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$</p>
</blockquote>
<p>is a metric on $C[0,1]$</p>
<hr>
<p><strong>Proof:</strong></p>
<ul>
<li><p>As usual, positive semidefinit... | Community | -1 | <p>Verify that:</p>
<p>$$\|f\|_2 = \left( \int_0^1 |f(x)|^2 dx \right)^{1/2}$$</p>
<p>is a norm on $C([0,1])$ (hint for the triangle inequality: use the C-S inequality)</p>
<p>Now $d_2(x,y) = \|x - y\|_2$, hence it is a metric </p>
|
725,746 | <p>I am not sure I understand the $N - \epsilon$ method for proving the equality of a limit.</p>
<p>I have a past mid-semester exam question that has:
$$\lim \limits_{x \to 1} (x^2 - 4x) = -3$$</p>
<p>Now it seems I want to take the $-3$ over $\rightarrow$ $|x^2 - 4x + 3| \lt \epsilon$ $$\text{ }\text{ }\text{ }\t... | Jose Antonio | 84,164 | <p>You can also proof this using sequences, this is probably what you meant to say. Consider a sequence $(x_n)\to 1$ we shall show that $(f(x_n))\to -3$. The first thing to do is obtain an estimate </p>
<p>$$|x_n^2-4x_n+3|=|x_n-1||x_n-3|\le|x_n-1|(|x_n|+3)$$</p>
<p>Given $\varepsilon>0$, choose $n_0(\varepsilon/5)... |
943,535 | <p>Let $E,F$ two $\mathbb{K}$ vector spaces, $u\in\mathcal{L}(E,F)$.</p>
<p>a) Show that there exists $v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$</p>
<p>b) Can we additionally have $v\circ u\circ v=v$ ?</p>
<hr>
<p>I have asked in the past <a href="https://math.stackexchange.com/questions/836438/existence-... | Robert Israel | 8,508 | <p>I can do it for the infinite-dimensional case using the Axiom of Choice. I suspect it requires some form of this axiom.</p>
<p>Let $W$ be the null space of $u$, and extend a basis $\{e_\alpha\}_{\alpha \in A}$ of $W$ to a basis $\{e_\beta\}_{\beta \in B}$ of $E$. Then the range of $u$ is the linear span of $\{u... |
3,434,656 | <p>We are given a <span class="math-container">$3 \times 3$</span> real matrix <span class="math-container">$A$</span>, and we know it has three eigenvalues. One eigenvalue is <span class="math-container">$\lambda_1=-1$</span> with corresponding eigenvector <span class="math-container">$v_1=\left[\begin{matrix}
0 \... | Community | -1 | <p>You know that <span class="math-container">$Av_2=\lambda_2v_2$</span>. Take the complex conjugate of this equation and note that <span class="math-container">$A^*=A$</span> since <span class="math-container">$A$</span> is real. </p>
<p>Then <span class="math-container">$Av_2^*=\lambda_2^*v_2^*$</span>. Hence the ei... |
2,545,813 | <p>The question is:</p>
<p>If $R$ has an identity and $A$ is an $R$-module, then there are submodules $B$ and $C$ of $A$ such thatt $B$ is unitary $RC = 0$ and $A = B \oplus C.$
And if $A_{1}$is another $R$-module, with $A_{1} = B_{1} \oplus C_{1}$, $B_{1}$ unitary ,$RC_{1} = 0.$</p>
<p>Why if $f:A\rightarrow A_{1}$ ... | egreg | 62,967 | <p>I assume your modules are not necessarily unital. Let $M$ be a left $R$-module; define
$$
z(M)=\{x\in M:1x=0\}\qquad u(M)=\{x\in M: 1x=x\}
$$
and prove $z(M)$ and $u(M)$ are submodules of $M$, with $Rz(M)=0$.</p>
<p>Now let $x\in M$ and consider $x=(x-1x)+1x$; then
$$
1(x-1x)=1x-1x=0
\qquad
1(1x)=1x
$$
so $x-1x\in ... |
4,039,655 | <p>I'm currently having trouble evaluating the following sum to get a formula in terms of <span class="math-container">$k$</span>:
<span class="math-container">$$\sum_{i=0}^{k-1} 2^i\cdot 4(k-i-1)$$</span></p>
<p>I know that <span class="math-container">$$\sum_{i=0}^n 2^i = 2^{n+1}-1$$</span>
but since my <span class="... | Brian M. Scott | 12,042 | <p>I would first let <span class="math-container">$\ell=k-1$</span> and rewrite it as</p>
<p><span class="math-container">$$\begin{align*}
\sum_{i=0}^\ell 2^i\cdot 4(\ell-i)&=4\sum_{i=0}^\ell 2^i(\ell-i)\\
&=4\sum_{i=0}^\ell i2^{\ell-i}\\
&=4\cdot 2^\ell\sum_{i=1}^\ell\frac{i}{2^i}\\
&=2^{k+1}\sum_{i=1}... |
2,488,218 | <p>I encountered this problem while practicing for a mathematics competition. </p>
<blockquote>
<p>A cube has a diagonal length of 10. What is the surface area of the cube? <strong>No Calculators Allowed.</strong></p>
</blockquote>
<p>(Emphasis mine)</p>
<p>I'm not even sure where to start with this, so I scribble... | Octopus | 65,718 | <p>Without using square roots:</p>
<p>$$10^2 = a^2 + a^2 + a^2 = 3a^2 = 100 \\ $$
$$SA = 6a^2 = 2(3a^2) = 2×100 = 200$$</p>
|
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | anonymous | 34,527 | <p>(I will assume $b$ is not a square, since otherwise it would not be a nested radical.)</p>
<p>A nested radical can be denested if and only if there exist $u,v\in\mathbb{N}$ such that the nested radical is of the form $\sqrt{u^2+v\pm2u\sqrt{v}}$ in which case it is also equal to $|u\pm\sqrt{v}|$.</p>
<p>It's not ha... |
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | Michael Rozenberg | 190,319 | <p>There are the following identities.
<span class="math-container">$$\sqrt{a+\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}+\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$$</span> and
<span class="math-container">$$\sqrt{a-\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}-\sqrt{\frac{a-\sqrt{a^2-b}}{2}},$$</span>
where all numbers under radicals... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Brian M. Scott | 12,042 | <p>‘Is there a perfect number $n$ such that $n+1$ is a multiple of your number?’ If her number is $1$, the answer is obviously <em>yes</em>; if her number is $2$, the answer is <em>I don’t know</em>, since it’s not known whether there are any odd perfect numbers; and if her number is $3$, the answer is <em>no</em>, bec... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Daniel R | 83,553 | <p>$\frac12-it$ is a zero of the zeta function. Is $\frac{N-1}{2}\frac12+it$ also a zero of the zeta function?</p>
<p>$N = 1 \implies$ <strong>no</strong>, since $it$ is not on the critical strip.</p>
<p>$N = 2 \implies$ <strong>I don't know</strong>, since the Riemann hypothesis has not been proved. </p>
<p>$N = 3 ... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | shrinath | 98,437 | <p>Let the boy ask the girl: </p>
<blockquote>
<p>"Divide the number you have with the previous number. Is the result a fraction?"</p>
</blockquote>
<p>If the girl replies:</p>
<ul>
<li>"Yes" then the number is 3 because 3/2 is a fraction.</li>
<li>"No" then the number is 2 because 2/1 is not a fraction.</li>
<li... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | r.e.s. | 16,397 | <p>"Is <a href="https://en.wikipedia.org/wiki/Busy_beaver#The_busy_beaver_function_.CE.A3" rel="nofollow">$\Sigma$</a>(<a href="https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem#Friedman.27s_finite_form" rel="nofollow">TREE</a>(<em>number</em>)) an odd number?" </p>
<p>NB: $\Sigma(\mathrm{TREE}(1)) = \Sigma(1) = ... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Maxime | 47,720 | <p>The girl take a number in {1, 2, 3}. I say to her: "Ok, now, imagine I know your number and pick one of the other, is your number greater than mine?"</p>
<ul>
<li>If she has picked 1, she will answer "no" because 1 < 2 and 1 < 3.</li>
<li>If she has picked 2, she will answer "I don't know" because 2 > 1 but 2... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | peterwhy | 89,922 | <p>If there is no generally agreed "$>$" operator for complex number field: "Is $e^{\left(i\frac{n-1}{2}\pi\right)}>0$?"</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Behzad | 74,479 | <p>Here's my wife's solution:</p>
<p>The boy asks: "I have an equation of the form $ax^2+bx+c=0$ in my mind, in which $b^2-4ac \geq 0$. Is the number of its real roots less than your number?"</p>
<ul>
<li>If the girl's number is 3, then her answer is "yes". </li>
<li>If the girl's
number is 2, then her answer is "I d... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | kiss my armpit | 26,975 | <p>Let $n$ be the number you are thinking. And let $x$ and $y$ be positive integers I am thinking. Is there a positive integer solution $z$ for the following equation? </p>
<p>$$
x^n+y^n=z^n
$$</p>
<ul>
<li>Yes then $n=1$</li>
<li>I don't know then $n=2$</li>
<li>No then $n=3$ because of the following proven conjectu... |
774,537 | <p>I use <a href="http://en.wikipedia.org/wiki/Polish_notation" rel="nofollow noreferrer">Polish notation</a>. All systems have detachment and uniform substitution as the only primitive rules of the system.</p>
<p>A user named John told me in an <a href="https://philosophy.stackexchange.com/questions/1365/why-is-this... | Mauro ALLEGRANZA | 108,274 | <p>You may see this useful collection of <a href="http://en.wikipedia.org/wiki/List_of_logic_systems#Implicational_propositional_calculus" rel="nofollow">axiom systems</a> for <em>propositional calculus</em>.</p>
|
4,036,704 | <blockquote>
<p><span class="math-container">$$(A ∧ (A → B)) ⇒ B$$</span></p>
</blockquote>
<p>I know that this is a tautology, but apart from setting up a truth-table I dont know how I would go about a formal proof in discrete mathematics. thanks</p>
<p><a href="https://i.stack.imgur.com/j6cHN.png" rel="nofollow noref... | Patrick Stevens | 259,262 | <p>To go from another direction, you can construct a proof using the deduction theorem to transform the problem first into <span class="math-container">$\{A \wedge (A \to B)\} \vdash B$</span>. Then this is proved by the following:</p>
<ol>
<li><span class="math-container">$A \wedge (A \to B)$</span> (hypothesis).</li>... |
4,036,704 | <blockquote>
<p><span class="math-container">$$(A ∧ (A → B)) ⇒ B$$</span></p>
</blockquote>
<p>I know that this is a tautology, but apart from setting up a truth-table I dont know how I would go about a formal proof in discrete mathematics. thanks</p>
<p><a href="https://i.stack.imgur.com/j6cHN.png" rel="nofollow noref... | Floridus Floridi | 841,808 | <p>A semantic reasoning ( not sure it counts as a " formal" proof)</p>
<p>Suppose the antecedent does not imply logically the consequent.</p>
<p>It means there is ( at least) one interpretation inwhich the antecedent is true and the consequent false.</p>
<p>Hence, an interpretation in which <span class="math-... |
174,339 | <p>This one is somewhat hard to explain, but I'll try my best to.<br>
I'm trying to generate a list of numbers (containing only pi/10, 0, -pi/10). Numbers are randomly selected from these 3, where the probability of getting 0 is always 70%. But the probability of getting pi/10 and -pi/10 depend on the previous outcomes... | Whelp | 1,287 | <p>Here's a way using NestList:</p>
<pre><code>probabilities = {0.25, 0.05};
f[{result_, pList_}] := With[{x = RandomReal[]},
Which[x < 0.7, {0, pList},
(x > 0.7) && (x < 0.7 + pList[[1]]), {Pi/10, Reverse@pList},
True, {-Pi/10, Reverse@pList}]];
</code></pre>
<p>Then run it, for example 10 ti... |
15,013 | <p>I am doing a plot where I have multiple shaded regions, and I want the line that separates the two regions to be dashed with dashes being alternating colors (so the demarcation stands out from both regions).</p>
<p>For example, say I am plotting the two regions shown here</p>
<pre><code>Plot[{1, Abs[BesselJ[1, x]]... | J. M.'s persistent exhaustion | 50 | <p>Here's one possibility (incorporating Mike's enhancements):</p>
<pre><code>Plot[Abs[BesselJ[1, x]], {x, 0, 20},
Filling -> {1 -> Axis, 1 -> Top},
FillingStyle -> {Opacity[1/5, ColorData[1, 1]], Opacity[1/5, ColorData[1, 2]]},
Mesh -> Full, MeshFunctions -> {#1 &}, MeshShading -&... |
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | Hans Meierwurst | 82,683 | <p>Talking about your question is meaningless, unless you say something about the domain of your function. The expression1/x is not defined for x=0 of course.</p>
<p>A function does not have to be continous in some point, to be defined there, e.g. take the characteristic function of the rational numbers in the set of ... |
421,951 | <p>I did search for whether this question was already answered but couldn't find any.</p>
<p>Does a function have to be "continuous" at a point to be "defined" at the point?</p>
<p>For example take the simple function $f(x) = {1 \over x}$; obviously it is not continuous at $x = 0$. However it does have the $-$ and $+... | egreg | 62,967 | <p>The most common definitions of continuity agree on the fact that a function can be continuous only on points of its domain.</p>
<p>Asking whether $f(x)=1/x$ is continuous is like asking what's the preferred food of unicorns.</p>
<p>You're being misled by the phrase "point of discontinuity". Well, the truth is that... |
624,672 | <blockquote>
<p>Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove $H\rtimes_{\phi}K\simeq H\rtimes_{\psi}H$?</p>
</blockquote>
<p>My idea is to use the following</p>
<p>Theor... | Derek Holt | 2,820 | <p>Second attempt. I believe this is false when $K$ is infinite cyclic, because automorphisms of finite quotients of $K$ do not usually lift to automorphisms of $K$. For example, the two groups</p>
<p>$\langle x,y \mid x^{11}=1, y^{-1}xy=x^4 \rangle$ and
$\langle x,y \mid x^{11}=1, y^{-1}xy=x^5 \rangle$</p>
<p>(with ... |
810,514 | <p>How to compute the following series:</p>
<p>$$\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$$</p>
<p>I tried</p>
<p>$$\frac{n+1}{2^nn^2}=\frac{1}{2^nn}+\frac{1}{2^nn^2}$$</p>
<p>The idea is using Riemann zeta function</p>
<p>$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$</p>
<p>but the term $2^n$ makes complicated. I k... | Leucippus | 148,155 | <p>The series
\begin{align}
S = \sum_{n=1}^{\infty} \frac{n+1}{2^{n} \ n^{2}}
\end{align}
can be expressed as
\begin{align}
S = \sum_{n=1}^{\infty} \frac{1}{2^{n} \ n} + \sum_{n=1}^{\infty} \frac{1}{2^{n} \ n^{2}}
\end{align}
and is seen to be
\begin{align}
S = - \ln\left( 1 - \frac{1}{2} \right) + Li_{2}\left(\fra... |
2,464,726 | <p>I have quite a basic question but I don't know how to do it.
EDIT: Sorry for writing it unclearly I hope I can clarify it. </p>
<p>I want to write an if-else statement as a vector in linear algebra. However, this vector is the result of an if-else-statement of the form: </p>
<pre><code>if b>1
then a=0
else a=... | Elsa | 278,945 | <p>I would define a square matrix S with zeros everywhere except on the diagonal where we put a $1$ if the corresponding element in $b$ is greater than $1$ and $0$ otherwise.</p>
<p>The diagonal may be obtained as the vector $(b\geq 1)$, assuming you use a programming language where false is zero and true is one. </p>... |
264,130 | <p>I have been trying to find an analytical expression for the following:</p>
<p>$\frac{\partial {X^{+}}}{\partial {X}}$</p>
<p>In my case, $X$ has a constant rank.
I've found the formula for differentiating a pseudoinverse in <a href="http://i.stanford.edu/pub/cstr/reports/cs/tr/72/261/CS-TR-72-261.pdf" rel="nofollo... | Shake Baby | 1,172 | <p>Pseudo-inverse is differentiable when the underlying matrix has a constant rank. An explicit formula is then available. This is the classical reference on the subject: <a href="http://epubs.siam.org/doi/abs/10.1137/0710036" rel="nofollow noreferrer">http://epubs.siam.org/doi/abs/10.1137/0710036</a></p>
|
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | Zau | 307,565 | <p>Like <a href="http://www.spoj.com/problems/CMPLS/" rel="nofollow noreferrer">this problem</a>, using <a href="https://www.quora.com/How-do-I-solve-a-complete-the-sequence-problem-on-SPOJ" rel="nofollow noreferrer">difference of differences method</a>,</p>
<p><a href="https://i.stack.imgur.com/11IKM.png" rel="nofoll... |
3,796,937 | <p>Prove that <span class="math-container">$2^n+1$</span> is not a cube for any <span class="math-container">$n\in\mathbb{N}$</span>.</p>
<p>I managed to prove this statement but I would like to know if there any other approaches different from mine.</p>
<p>If existed <span class="math-container">$k\in\mathbb{N}$</span... | SB1729 | 466,737 | <p>Let <span class="math-container">$$2^n=m^3-1\\\implies 2^n=(m-1)(m^2+m+1)\\\implies(m-1)=2^a\text{ and }(m^2+m+1)=2^b\\\implies3m=(m^2+m+1)-(m-1)^2=2^b-2^{2a}$$</span> Now, since <span class="math-container">$m$</span> is odd, we must have <span class="math-container">$a=0$</span> or <span class="math-container">$b=... |
2,709,454 | <p>In Basket $A$ we have $2$ Blue and $1$ Red Balls and in Basket $B$ we have $3$ Blue and $3$ Red Balls. we randomly choose 2 balls from each basket (2 from $A$ and 2 from $B$). Then we put the balls from $A$ into $B$ and the balls from $B$ into $A$. Then, We chose one of the baskets randomly and pick a random ball fr... | saulspatz | 235,128 | <p>Suppose there are $n_1$ balls in bucket 1, and $n_2$ balls in bucket 2. If we pick a bucket uniformly at random, then pick a ball from that bucket uniformly at random, what is the probability that a particular ball is the one picked?</p>
<p>It depends on which bucket the ball is in. If the ball is in bucket 1, th... |
2,709,454 | <p>In Basket $A$ we have $2$ Blue and $1$ Red Balls and in Basket $B$ we have $3$ Blue and $3$ Red Balls. we randomly choose 2 balls from each basket (2 from $A$ and 2 from $B$). Then we put the balls from $A$ into $B$ and the balls from $B$ into $A$. Then, We chose one of the baskets randomly and pick a random ball fr... | antkam | 546,005 | <p>This post hopefully sheds some additional light on WHY the "coincidence" happens that the result is the same as if the two baskets are mixed together. Let $n_A, n_B$ be the no. of balls originally in A and B, and let $n_C$ be the no. of balls transferred from each basket to the other. (In the OP, $n_C=2$.) </p>
<... |
4,242,877 | <p>I was reading the book of Real Analysis by H.L.Royden and they have given the definition of Riemann sums and Riemann integral as follows:<a href="https://i.stack.imgur.com/cFfRc.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cFfRc.jpg" alt="enter image description here" /></a></p>
<p>After this d... | Paramanand Singh | 72,031 | <p>Let's focus on upper Darboux sums and upper Darboux integral. The story for lower sums and integrals is similar.</p>
<p>Let us repeat the definitions for sake of clarity. Assume <span class="math-container">$f:[a, b] \to\mathbb {R} $</span> is bounded on <span class="math-container">$[a, b] $</span> with supremum <s... |
1,410,164 | <p>I've found the following identity.</p>
<blockquote>
<p>$$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ </p>
</blockquote>
<p>I could verify it by using CAS, and calculate the integrals in term of <a href="http://mathworld.wolfram.com/ExponentialIntegral.html">exponential</a> and <a ... | Olivier Oloa | 118,798 | <p><strong>Hint.</strong> One may observe that</p>
<blockquote>
<p>$$
\frac1{1+\ln^2 x}=-\Im \frac1{i-\ln x}=-\Im \int_0^{\infty}e^{-(i-\ln x)t}dt,\quad x \in (0,1),
$$ </p>
</blockquote>
<p>gives</p>
<blockquote>
<p>$$
\begin{align}
\int_0^1 \frac{1}{1+\ln^2 x}\,dx&=-\Im \int_0^1\!\!\int_0^{\infty}e^{-(i-\l... |
633,522 | <ol>
<li><p>$ \lim_{n\to \infty} \sqrt[n]{3^n+4^n} $ . I think the limit is $4$. I did :
$ \sqrt[n]{3^n+4^n} = 4 \sqrt[n]{(\frac{3}{4}) ^n+1}$ .Am I right?</p></li>
<li><p>$ \lim_{n\to \infty} \frac{1}{1\cdot 4 } + \frac{1}{4\cdot 7} +...+\frac{1}{(3n-2)(3n+1)} $. I know that for each $k$ , this sequence is the sum of... | Mikasa | 8,581 | <p>Hint:</p>
<p>For the second one use this fact that :
$$\frac{1}{(3n+1)(3n-2)}=\frac{-1}{3(3n+1)}+\frac{1}{3(3n-2)}$$ So the sum can be reduced to this one: $$S_n=\frac{1}3\left(\frac{1}{1}+\frac{1}{(3n+1)(3n-2)}\right)$$</p>
|
907,055 | <p>I've begun a course in "Real Analysis" recently and I have this trivial exercise. Could someone check if my proof is correct?</p>
<p>Proposition: There exists Injective function $ f: A \rightarrow B \iff $ there exists function $ g: B \rightarrow A $ is surjective</p>
<p>Proof: Firstly, we prove injective function... | Barkas | 170,882 | <p>I'm sorry, but your proof seems to have some issues.</p>
<p>First of all, you should clarify, what you mean by </p>
<blockquote>
<p>Proposition: Injective function f:A→B⟺ function g:B→A is surjective </p>
</blockquote>
<p>Is $g$ supposed to be the inverse function of $f$? If so, the statement itself is not quit... |
65,731 | <p>This was <a href="http://www.quora.com/How-many-topologies-are-there-on-set-of-real-numbers">asked</a> on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other axioms. A quick search also failed... | Eric Wofsey | 86,856 | <p>Let me give a slightly simplified version of Stefan Geschke's argument. Let $X$ be an infinite set. As in his argument, the key fact we use is that there are $2^{2^{|X|}}$ ultrafilters on $X$. Now given any ultrafilter $F$ on $X$ (or actually just any filter), $F\cup\{\emptyset\}$ is a topology on $X$: the topolo... |
3,321,098 | <p>I'm wondering if there is also another easy way of solving question C.
In the book, they use <span class="math-container">$A = PDP^{-1}$</span> but the only method I know is by drawing the transformation with vectors <span class="math-container">$ e_1$</span> and <span class="math-container">$e_2$</span> and then lo... | Anti | 523,511 | <p>Gallois, thanks a lot for your reply. I think it could be thought that way. And: Thanks a lot for you hint with <span class="math-container">$t_1$</span> and <span class="math-container">$t_0$</span>. I edited my post.</p>
<p>So, how did I came up with <span class="math-container">$P_1^{t_1}$</span>? I remove <span... |
4,186,303 | <p>Given the following definitions, taken from Wikipedia:</p>
<blockquote>
<p><strong>Support of a function</strong>. When <span class="math-container">$X$</span> is a topological space and <span class="math-container">$f : X \to \mathbb C$</span> is a continuous function, the <em>support</em> of <span class="math-cont... | paul garrett | 12,291 | <p>Following up on @DavidC.Ullrich's answer:</p>
<p>First, to understand these definitions, rather than trying to parse the formalities, it is better for many people to think about <em>intentions</em> and <em>examples</em>.</p>
<p>So, seriously, the support of a distribution should be an extension (in a colloquial sens... |
8,764 | <p>Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??</p>
<p>I think the result is
Q[x]/(x) \ otimes_{Q[x]}Q(x) but on the other hand, from another way, I know it should be Q[1/x]/Q.</p>
<p>But ho... | S. Carnahan | 121 | <p>If you localize Q[x] at zero, you get the field of rational functions Q(x), so your module localizes to a vector space over this field (see <a href="http://en.wikipedia.org/wiki/Localization_of_a_module" rel="nofollow">Wikipedia</a>). Your module is finitely generated, in fact by a single element, so you get a vect... |
3,639,122 | <blockquote>
<p>Let <span class="math-container">$X:\varOmega \to \mathbb{R}$</span> be a random variable. Random variable <span class="math-container">$X$</span> is <em>degenerate</em> if
for some <span class="math-container">$c\in \mathbb{R}$</span> we have <span class="math-container">$\mathrm{P}(X=c)=1.$</span>... | Kavi Rama Murthy | 142,385 | <p>Hint: Suppose <span class="math-container">$P(X<a)=0$</span> or <span class="math-container">$1$</span> for all <span class="math-container">$a$</span>. Let <span class="math-container">$c=\sup \{a: P(X<a) =0\}$</span>. Show that <span class="math-container">$P(X<c)=0$</span> and <span class="math-container... |
3,463 | <p>I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to <em>itself</em>?" Call these maps endomaps. Permutations of elements of the set are a special case of endomaps: they are bijective ... | Olod | 1,100 | <p>One of the important properties of endomaps of an infinite set $X$ is that the expressive power of the logic that allows you to quantify over all endomaps of $X$ is the same as the expressive power of the full second order logic on $X.$ Roughly, this means that any relation on $X$ (a subset of $X^n$) can be modelled... |
103,164 | <p>I want to solve an equation which contains an infinite continued fraction $F(n)$. Then I must (obviously) truncate this continued fraction at $n=2000$.</p>
<p>The problem here is that <em>Mathematica</em> does not display the $2000$ terms of this fraction on the screen. The screen closes directly.</p>
<p>Please, h... | Michael E2 | 4,999 | <p>A common way, for a long time, of denoting a (generalized) continued fraction is to list the partial numerators and denominators, sometimes with $+$ and fraction bars like this:
$$
F = b_0+
\frac{a_1}{b_1+}\,
\frac{a_2}{b_2+}\,
\frac{a_3}{b_3+}\cdots
$$
It is also an efficient way to store a continued fraction (cf. ... |
10,992 | <p>Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685</p>
<p>I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web ca... | Raphael | 3,330 | <p>Non-answer: Export the data to CSV and use software that is suited for statistical work, e.g. R.</p>
|
3,658,545 | <p>I have a question similar to the one posed <a href="https://math.stackexchange.com/questions/2128719/how-many-ways-can-4-males-and-4-females-be-seated-in-a-row-with-no-same-sex-sitt">here.</a></p>
<p>How many ways can 4 things of type A, 4 things of type B, and 4 things of type C be arranged so no two things of the... | Dhanvi Sreenivasan | 332,720 | <p>A general solution for <span class="math-container">$n$</span> types of indistinguishable objects having <span class="math-container">$m$</span> each is</p>
<p><span class="math-container">$$f(n,m) = n!.((n-1).(n-1)!)^{m-1}$$</span></p>
<p>To obtain this, you have to arrange the objects in such a way that there ar... |
3,658,545 | <p>I have a question similar to the one posed <a href="https://math.stackexchange.com/questions/2128719/how-many-ways-can-4-males-and-4-females-be-seated-in-a-row-with-no-same-sex-sitt">here.</a></p>
<p>How many ways can 4 things of type A, 4 things of type B, and 4 things of type C be arranged so no two things of the... | antkam | 546,005 | <p>I don't see a simple way to do this, but the example is just small enough to do a case analysis by hand.</p>
<p>For now assume the four As are indistinguishable, and same for Bs and Cs. We will simply multiply by <span class="math-container">$(4!)^3$</span> at the end.</p>
<p>We will first place As and Bs, and by... |
1,709,915 | <p>Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense of Model Theory).</p>
<p>The theorem is often stated as being of fundamental importance to Universal Algebra. As fa... | Larry | 312,933 | <p>Reformulating part of Alex Kruckman's answer: Knowing that some class $\mathcal{A}$ of algebras has an equational presentation is useful because we then know that an algebra belongs to $\mathcal{A}$ if every finitely generated subalgebra does.</p>
|
2,879,041 | <blockquote>
<p>Let $f: [a,b] \to \mathbb R$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a) = 0$ and $|f'(x)|\le k|f(x)|$ for some $k$, then $f(x)$ is zero on $[a,b]$. </p>
</blockquote>
<p>I tried proving it using Legrange's Mean Value Theorem but couldn't get it.</p>
<p>$f(x)$ is differentiable ... | DanielWainfleet | 254,665 | <p>Since $f$ is continuous on $[a,b]$ and $f(a)=0,$ if $f$ is not $0$ on $[a,b]$ then $f(c)\ne 0$ for some $c\in (a,b).$</p>
<p>Suppose by contradiction that $c\in (a,b)$ and $ f(c)\ne 0$. Because $f$ is continuous and $f(a)=0$ there exists $d\in [a,c)$ such that $f(d)=0$ and $\forall x\in (d,c]\;(f(x)\ne 0).$</p>
<... |
2,894,315 | <p>Let $a;b;c>0$ such that $a+b+c=6$. Find the maximum value of $A=a^2bc+a^2+2b^2+2c^2$</p>
<hr>
<p>WLOG $b\ge c$. I see maximum value of $A=36$ at $(a;b;c)=(2;1;3)$</p>
<p>So i need to prove $A\le 36$. Or I will prove </p>
<p>$(a+b+c)^4\ge 36a^2bc+(a^2+2b^2+2c^2)(a+b+c)^2$</p>
<p>Or $(2a-b-c)(b^3+c^3+a^2b+a^2... | Servaes | 30,382 | <p>Given that $a+b+c=6$ we can substitute $a=6-b-c$ into $A$ to get
$$A=(6-b-c)^2(bc+1)+2(b^2+c^2),$$
where $b,c>0$ and $b+c<6$. Differentiating with respect to $b$ and $c$
yields
$$\frac{\partial A}{\partial b}=-2(6-b-c)(bc+1)+(6-b-c)^2c+4b,$$
$$\frac{\partial A}{\partial b}=-2(6-b-c)(bc+1)+(6-b-c)^2b+4c.$$
If t... |
3,162,547 | <p>I am trying to convert instances of nested 'for' loops into a summation expression. The current code fragment I have is:</p>
<pre><code>for i = 1 to n:
for j = 1 to n:
if (i*j >= n):
for k = 1 to n:
sum++
endif
</code></pre>
<p>Basically, the 'if' conditional is c... | Ertxiem - reinstate Monica | 649,368 | <p><strong>Hint:</strong></p>
<p>The "for" cycle in <span class="math-container">$k$</span> is very easy to turn into something simpler...</p>
<p>As for the other parts, see if you can split the problem into easier steps. For example, what happens for <span class="math-container">$i=1$</span>? And what happens for <s... |
927,566 | <p>We define measurable function from a measure space to a topological space as space which pulls back open sets to measurable sets. How can we prove measurable functions pulls back Borel set also to measurable sets.
where Borel sets are elements in sigma algebra generated by topology</p>
| drhab | 75,923 | <p>Let $f:X\rightarrow Y$ be a function and let $\mathcal{O}\subseteq\wp\left(Y\right)$. In your case $\mathcal O$ is the topology.</p>
<p>It comes to proving that $\sigma\left(f^{-1}\left(\mathcal{O}\right)\right)=f^{-1}\left(\sigma\left(\mathcal{O}\right)\right)$
where in general $\sigma\left(\mathcal{V}\right)$ den... |
2,828,636 | <p>I wanted to show that both sets are equal. My Textbook says following:</p>
<p>$\in$ means "is element of", $\land$ is the and operator, $\lnot$ is the not operator, $\notin$ means "is not element of", $\lor$ is the or operator</p>
<p>$$\def\-{\setminus}\begin{split} x \in A\-A\-B &\iff\\
(x \in A) \land (\lnot... | Alejandro Sánchez Yalí | 217,786 | <p><a href="https://i.stack.imgur.com/ue270.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ue270.png" alt="enter image description here"></a></p>
<p>You can see that the vector $\overrightarrow{OB}= (d\cos a , d\sin a )$ is orthogonal to the line. If you consider other point P(x, y) in the line, th... |
1,661,941 | <p>Evaluate </p>
<p>$\binom{n}0-2\binom{n}1 + 3\binom{n}2 +···+(−1)^n(n+1)\binom{n}n$ </p>
<p>which is the same as:</p>
<p>$\sum_{k=0}^n=(-1)^k(k+1)\binom{n}k$</p>
<p><strong>My attempt:</strong></p>
<p>Using the Binomial Theorem, we get:</p>
<p>$(1+x)^n=\binom{n}0+\binom{n}1x+\binom{n}2x^2+...+\binom{n}nx^n$</p>... | marty cohen | 13,079 | <p>More than that is true.
For any $a$ and $b$,
if $n \ge 2$,
$\sum_{k=0}^n(-1)^k(ak+b)\binom{n}k
= 0
$.</p>
<p>If $n \ge k \ge 1$,
$k\binom{n}k
=k\frac{n!}{k!(n-k)!}
=\frac{n!}{(k-1)!(n-k)!}
=n\frac{(n-1)!}{(k-1)!(n-k)!}
=n\binom{n-1}{k-1}
$.</p>
<p>Therefore,
since,
for $n \ge 1$,
$\sum_{k=0}^n(-1)^k\binom{n}k
=0$,... |
2,451,281 | <p>$$y=x+\frac{1}{x-4}$$ I tried to find range of this function as below
$$y(x-4)=x(x-4)+1\\yx-4y=x^2-4x+1\\x^2-x(4+y)+(4y+1)=0\\\Delta \geq 0 \\(4+y)^2-4(4y+1)\geq 0 \\(y-4)^2-4\geq 0\\|y-4|\geq 2\\y\geq 6 \cup y\leq2$$ this usuall way. but I am interested in finall answer... is the other idea to find function range ... | Khosrotash | 104,171 | <p>use this fact : $|a+\frac 1a|\geq 2$
$$\quad{y=x+\frac{1}{x-4}\\y-4=x+\frac{1}{x-4}-4\\y-4=(x-4)+\frac{1}{x-4}=a+\frac 1a\\|y-4|\geq 2}$$</p>
|
233,397 | <p>could anyone please clarify me the meaning of the term 'hypothesis'? </p>
<p>with relation to terms 'reasoning' and 'assumption' ?</p>
<p>Many thanks</p>
| hmakholm left over Monica | 14,366 | <p>"Hypothesis" is one of those words that have a whole slew of interrelating meanings in different contexts, but no overarching simple definition that describe all of them exactly.</p>
<p>The most primitive and original meaning of "hypothesis" appears to be simply <strong>whatever we're reasoning FROM in a logical ar... |
898,002 | <p>Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ exists iff $f'(c-)=g'(c+)$, where $f'$ and $g'$ are obtained using derivative rules?</p>
<p>This would seem reasonable ... | Joonas Ilmavirta | 166,535 | <p>No.
Consider for example $c=0$, $g(x)=0$ and $f(x)=x^2\sin(1/x)$.
(This example is from <a href="https://math.stackexchange.com/questions/896604/f-exists-but-lim-fracfx-fyx-y-does-not-exist">this other question</a>.)
The derivative $f'$ has no limit at zero, but the piecewise defined function is differentiable at ze... |
115,433 | <p>Mathematica has a lot of machinery for working with predefined probability distributions. It is not clear how to make that machinery work with a new distribution.</p>
<p>Suppose I want to define a brand new distribution</p>
<pre><code>MyDistribution[a, b, c]
</code></pre>
<p>What is the minimum I need to specify ... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "10.4.1 for Mac OS X x86 (64-bit) (April 11, 2016)" *)
dist[m_, s_] =
ProbabilityDistribution[
1/(E^((m - x)^2/(2*s^2))*(Sqrt[2*Pi]*s)), {x, -Infinity, Infinity},
Assumptions -> {s > 0, Element[m, Reals]}];
AppendTo[$ContextPath, "Statistics`Library`"];
(fi1 = FisherInformat... |
729,373 | <p>I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration.</p>
<p>Now I used an indented semicircular contour, and the parts lying on the real line and the big arc were no problem, but the small arc is being resistant, and I'm not sure what to do. Usually, on the small arc f... | DonAntonio | 31,254 | <p>Define</p>
<p>$$f(z)=\frac{e^{iz}}{z(z^2+1)}\;,\;\;C_R:=[-R,-\epsilon]\cup\gamma_\epsilon\cup[\epsilon,R]\cup\gamma_R\;\;,\;\;\text{with}$$</p>
<p>$$\gamma_a:=\{z=ae^{it}\;,\;\;a,t\in\Bbb R_+\;,\;\;0<t<\pi\}$$</p>
<p>Then the function has one simple pole within the domain defined by the above contour:</p>
... |
143,721 | <p>The exercise is about convex functions:</p>
<p>How to prove that $f(t)=\int_0^t g(s)ds$ is convex in $(a,b)$ whenever $0\in (a,b)$ and $g$ is increasing in $[a,b]$?</p>
<p>I proved that </p>
<p>$$f(x)\leq \frac{x-a'}{b'-x}f(b')+\left(1-\frac{x-a'}{b'-x}\right)f(a')$$</p>
<p>when we have </p>
<p>$$x=\left(1-\fra... | David Mitra | 18,986 | <p>You could use the fact that a function $f$ is convex on a nonempty, open interval $(a,b)$ if and only if
$$
{f(x)-f(c)\over x-c}\le {f(d)-f(x)\over d-x}
$$
whenever $a<c<x<d<b$. This follows from the fact that a chord through two points on the graph of a convex function lies on or above the graph of the ... |
143,721 | <p>The exercise is about convex functions:</p>
<p>How to prove that $f(t)=\int_0^t g(s)ds$ is convex in $(a,b)$ whenever $0\in (a,b)$ and $g$ is increasing in $[a,b]$?</p>
<p>I proved that </p>
<p>$$f(x)\leq \frac{x-a'}{b'-x}f(b')+\left(1-\frac{x-a'}{b'-x}\right)f(a')$$</p>
<p>when we have </p>
<p>$$x=\left(1-\fra... | checkmath | 25,077 | <p>I also overcome with a different approach. Assume initially $g(t)\geq 0$.</p>
<p>We will consider the auxiliary function $\int_{a'}^x g(t)dt$ to that this function satisfies the convexity conditions in $a'<x<b$, that is, </p>
<p>$$\int_{a'}^x g(t)dt\leq \frac{x-a'}{b'-a'}\left(\int_{a'}^x g(t)dt+\int_x^{b'}... |
350,211 | <p>I'm looking at the following in Jech's <em>The Axiom of Choice</em> on page 20:</p>
<blockquote>
<p><strong>2.4.1.</strong> <em>Example: The Countable Axiom of Choice implies that every infinite set has a countable subset.</em></p>
<p>Proof. Let $S$ be an infinite set. Consider all finite one-to-one finite... | user642796 | 8,348 | <p>It is likely that Jech is using the following definition: A set $X$ is <em>infinite set</em> iff it is not finite (<em>i.e.</em>, there is no natural number $n$ such that $X$ is in one-to-one correspondence with $\{ 0 , \ldots , n-1 \}$). (I cannot find a statement of this definition in <em>The Axiom of Choice</em>... |
4,455,201 | <p><em>Question:</em><br />
Let <span class="math-container">$\Omega=\{a,b,c\}$</span>. Give an example for <span class="math-container">$X, F_1, F_2$</span> in which
<span class="math-container">$E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$</span></p>
<p><em>My answer:</em><br />
I am not at all sure of my answer. If you hav... | paperskilltrees | 774,351 | <p>Take this with a grain of salt, as I am also a learner of Probability.</p>
<p>Your example is correct in the design (take <span class="math-container">$F_{1,2}$</span> such that <span class="math-container">$F_1 \not\subseteq F_2, F_2 \not\subseteq F_1$</span>), but is wrong in the implementation. In particular, you... |
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | Qiaochu Yuan | 232 | <p>The constraint $x^2 + y^2 + z^2 \le 2$ is equivalent to the constraint
$$(x + 4z)^2 + (4x - z)^2 + 17y^2 \le 34$$</p>
<p>and this gives $x + 4z \ge - \sqrt{34}$ with equality when $y = 4x - z = 0$. </p>
<p>The lesson to take away from this argument is that balls are particularly easy to deal with because they can ... |
3,567,228 | <p>I am really struggling and have spent hours proving this theorem so I would greatly appreciate some help. I think I have nearly proved the theorem except for showing that <span class="math-container">$f^{-1}(-\infty,0]$</span> is a regular sublevel set of <span class="math-container">$f$</span>. </p>
<p>My attempt ... | 萧宇轩 | 900,073 | <ol>
<li><p>We choose a cover of <em>N</em> by: <strong>a)</strong> <strong>Coordinate balls U</strong> contained in <span class="math-container">$ \text{Int}D $</span> or <span class="math-container">$ N\setminus D $</span> and denote its coordinate chart is <span class="math-container">$ \phi $</span>.
<strong>b)<... |
762,483 | <p>I understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over addition e.g. $3(2+3) = 3(2) + 3(3)$ so if an exponent is just repeated multiplication why shouldn't the same be true for expon... | ShreevatsaR | 205 | <blockquote>
<p><em>Multiplication distributes over addition [..], so if [exponentiation] is just repeated multiplication why shouldn't the same be true [..] ?</em></p>
</blockquote>
<p>Multiplication distributes over addition, and if you repeatedly multiply by the <strong><em>same</em></strong> quantity or quantiti... |
1,040,846 | <p>I want to prove, that $a_n$ is a null sequence if $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1$$</p>
<p>That means that $\forall \epsilon > 0: \exists N \in \mathbb{N}: n \ge N: |\frac{a_{n+1}}{a_n} - c| < \epsilon$</p>
<p>How can I get rid of the $a_{n+1}$ and the $c$, to show $\forall \epsilon >... | user860374 | 137,485 | <p>This statement can be proven by providing the proof of the Ratio Test:</p>
<blockquote>
<p><strong>Ration Test</strong></p>
<p>To apply the Ratio Test to a given infinite series $\displaystyle \sum_{k=1}^\infty a_k$, we evaluate the limit $$\lim_{k \to \infty}\bigg|\frac{a_{k+1}}{a_k}\bigg|=L$$
There are 3... |
2,041,499 | <blockquote>
<p>The solution to the BVP <span class="math-container">$\frac{d^2y}{dx^2}+y =\csc x$</span>, <span class="math-container">$0 < x < \frac{\pi}{2}$</span> <span class="math-container">$y(0)=0$</span>, <span class="math-container">$y(\pi/2)=0$</span> is</p>
<p><span class="math-container">$(A)$</span> ... | 2'5 9'2 | 11,123 | <p>Here is an answer without explicitly solving the equation. It's a multiple choice question, so I will make an assumption that there is exactly one correct answer. [Which turns out to be a false assumption, making this a poorly written question. See below at the end.]</p>
<p>The solution is not (A) concave. For a tw... |
1,328,738 | <p>Let $c$ be a close curve such that $c$ does not intersect itself, $c\in \mathbb{R}^2$ (in the plane), show that for all point $P$ that surrounded in $c$ there are two points $A,B$ on $c$ such that $P$ is in the middle of the interval of $A$ & $B$</p>
<p>As you can see English isn't my first language.</p>
| Mark Bennet | 2,906 | <p>So at the most basic level there is a factor $(x+a)(x-a)=x^2-a^2$ and the factorisation is $$(x^2-a^2)(x^2+bx+c)=x^4-2x^3+4x^2+6x-21$$</p>
<p>Equating powers of $x^3$ gives immediately a value for $b$. Then equating powers of $x$ gives $a^2b=-6$ so $a^2$ is known and the constant term gives $a^2c=21$ so that $c$ ca... |
1,898,225 | <p>Really basic. I have no idea what to do as I get stuck when I apply the change of base formula to $\ln(x^2)$.</p>
<p>Thank you in advance for any help.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\log a+\log b=\log(ab)$$</p>
<p>$$m\log a=\log(a^m)$$</p>
<p>$$\log\dfrac ab=\log a-\log b$$ where all the logarithm remain defined </p>
|
1,898,225 | <p>Really basic. I have no idea what to do as I get stuck when I apply the change of base formula to $\ln(x^2)$.</p>
<p>Thank you in advance for any help.</p>
| Teoc | 190,244 | <p>$$\log(2x)-\ln(x^2) $$
$$= \log (2)+\log(x)-2\ln(x)$$
$$=\log(2)+\log(x)-2\frac{\log(x)}{\log(e)}\tag{1}$$
$$=\log(2)+(\log x)(1-\frac{2}{\log(e)}) = 3$$</p>
<p>I am sure you can solve it without my help from there. </p>
<p>In regards to (1), we have $\ln(x)=\log_e(x)=\frac{\log_{10}(x)}{\log_{10}(e)}$</p>
|
4,080,680 | <p>If <span class="math-container">$\sum a_n$</span> diverges does <span class="math-container">$\sum \frac{a_n}{\ln n}$</span> necessarily diverge for <span class="math-container">$a_n>0$</span>?</p>
<p>I tried <span class="math-container">$a_n=\frac{\ln n}{n^2}$</span> to try bag an easy counterexample but it turn... | Konstantinos Gaitanas | 99,437 | <p>No this is not true.<br />
For example <span class="math-container">$\sum\frac{1}{n\cdot \ln n}$</span> diverges, but <span class="math-container">$\sum\frac{\frac{1}{n\cdot \ln n}}{\ln n}=\sum\frac{1}{n\cdot (\ln n)^2}$</span> converges by the <a href="https://en.wikipedia.org/wiki/Integral_test_for_convergence" re... |
2,076,656 | <p>As a homework, I was asked to solve this <strong>equation</strong>, $$(3x-4\lfloor x\rfloor=0),x\in \Bbb R$$ For $x\in \Bbb Z:\lfloor x\rfloor=x \implies x=0$ But for $x\not\in\Bbb Z : \lfloor x\rfloor=\frac 34x$ So now we know that $\frac 34x\in\Bbb Z$ and $x\in\Bbb R-\Bbb Z$, so maybe ? define a function such that... | user84413 | 84,413 | <p>Hint:</p>
<p>You can rewrite $3x=4\lfloor x \rfloor$ as $3\left(\lfloor x \rfloor+\{x\}\right)=4\lfloor x \rfloor$, so $\lfloor x \rfloor=3\{x\}$.</p>
<p>Since $0\le \{x\}<1,\; 0\le \lfloor x \rfloor<3$ and therefore $\lfloor x \rfloor\in \{0, 1, 2\}$</p>
|
142,879 | <p>I'm working through Dr. Pete Clark's convergence notes here: <a href="http://alpha.math.uga.edu/%7Epete/convergence.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/convergence.pdf</a></p>
<p>and I've been thinking about Exercise 3.2.2 (b),</p>
<p>The question states that for the set <span class="math-... | leslie townes | 18,076 | <p>Yes, "unconditional convergence (as defined in the link on the top of page 11) reduces to the ordinary convergence of series when the underlying set $S$ takes the place of the positive integers" is incorrect. The series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges in the usual sense, but $\sum_{n \in \mathbb{N}... |
4,520,435 | <p>(<strong>Note:</strong> I've posted <a href="https://math.stackexchange.com/a/4539959/231608">my own answer</a>, slightly redefining trirationals to be composed of reals instead of integers and addressing the problems pointed out here. Please take note of this while reading my original question.)</p>
<p><strong>Defi... | Stinking Bishop | 700,480 | <p>One formal complaint that you may need to resolve first. What if <span class="math-container">$b$</span> or <span class="math-container">$c$</span> is negative? Say, what is <span class="math-container">$(-1)^\omega$</span>?</p>
<hr />
<p>Now, to my main point:</p>
<blockquote>
<p><em>I would pay less attention to w... |
1,665,931 | <p>Let $a,c \in \mathbb{R}$ and $b \in \mathbb{C}$. We consider the equation $$a \bar{z}z+ b\bar{z} + \bar{b}z+c=0.$$ What curve it represents in the complex plane?</p>
<p>I think it a circle, but I am not able to conclude. Any help?</p>
| Joe | 107,639 | <p>Writing $z=x+iy$, $b=\alpha+i\beta$ and noticing that
$$
b\bar z+\bar b z=b\bar z+\overline{b \bar z}=2\Re(b\bar z)
$$ your equation turns as follows
\begin{align*}
&az\bar z+b\bar z+\bar b z+c=0\;\;\\
\Longleftrightarrow\;\;&a(x^2+y^2)+2\Re[(\alpha+i\beta)(x-iy)]+c=0\\
\Longleftrightarrow\;\;&a(x^2+y^2)... |
348,276 | <p>I am not an expert, thus I apologize if my question is very naive. Let <span class="math-container">$\mathsf{M}$</span> be a model category (I do not assume any functoriality on the factorization),</p>
<blockquote>
<p><strong>Q1.</strong> Is there a reference where it is proven that (all) homotopy colimits exist?</p... | John Klein | 8,032 | <p><strong>Edit:</strong> The questioner has objected to the fact that the reference I gave to Q1 assumes functoriality. </p>
<p>Here is another reference which doesn't:</p>
<p><a href="https://pages.uoregon.edu/ddugger/hocolim.pdf" rel="nofollow noreferrer">https://pages.uoregon.edu/ddugger/hocolim.pdf</a></p>
<p... |
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