I've tried a bunch of different approaches to this question (and would particularly appreciate some hints that don't give the whole thing away immediately). I've noticed that if $S$  contains some finite subsets, then for each $n \in \mathbb{N}$ there is at most one subset of size $n$ in the collection $S$ as $S$ can't contain duplicates (in order to satisfy the given condition). Certainly if $A_i$ is a finite subset contained in $S$ of size $i$, then $A_i \subset A_j$ if $i &gt; j$. I've tried considering the possibilities where there are some finite subsets in the collection $S$, and then we can also add in infinite-size subsets that are each a superset of all of the finite subsets. However it seems rather difficult to visualise what happens when you consider infinite subsets of $\mathbb{N}$ that can't be obtained by removing finitely many elements from $\mathbb{N}$. What are good approaches to take to figure out whether such a set can be uncountable - is there an element of construction here?
Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for  every pair of elements in $E$, one is a subset of the other. Clearly, $E$ could be countable by considering $E=\big\{\{1,\cdots, n\}, n\in\mathbb N\big\}$. My question is the following: could we find an example that $E$ is uncountable? Many thanks for the answer!
A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.  And what about type-II superconductors? What tips and advice do you have for photographing the Aurora Borealis? A friend has been kind enough to lend me a DSLR for a few months.  It's my first foray into DSLR photography, or indeed any kind of photography beyond simply aiming my Canon IXUS at something and clicking the button.    Over the last few weeks, I've been busy studying, practising and trying to learn as much as possible ahead of a planned trip to Norway where I hope to see, and get some photos of, the Aurora Borealis.  I currently have:   Canon Rebel Xsi (450D) Canon 50mm f/1.4 L  Canon 18-55m kit lens Aluminium full height tripod with ball head Cable release Spare battery &amp; memory cards   I'm prepared to spend the night wrapped up outdoors, in the hope that they will make an appearance. I'd like some advice on how to get the most out of this kit for the trip:   How should I go about setting up to get pics of the Aurora?  How might I achieve optimum focus? From what I've read, I'll need to use the tripod and cable release, with IS turned off, on continuous shooting mode with long exposure times. But I'd like some more specific tips in terms of which aperture and ISO settings might work best? Should I use one of the lenses above, or would it be worthwhile renting a better, wide angle lens; if so, which would you recommend? Pass commands as input to another command (su, ssh, sh, etc) There's a problem statement:     For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and      $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} \right\rfloor$$      for all $i \ge 0$.   Here, $\lfloor x \rfloor$ is the floor of $x$, the greatest integer less than or equal to $x$.   Hence, we have $\left\lfloor \sqrt{10} \right\rfloor=3$ and $\left\lfloor \sqrt{29} \right\rfloor=5$.      Show that for each positive integer $m$, the sequence $S_m$ contains the square of some integer.   I'm pretty certain that this can be proved with induction. I am just not quite sure what to induct on. Examining examples shows that $S_m$ always results in a perfect square eventually, though I'm not sure how to prove it. Access .mat file containing matlab classes in python I started reading "The Mathematics of nonlinear Programming - AL peressini" recently.  Theorem : Suppose that $f(x) , f'(x) ,f''(x)$ exist on the closed interval $[a,b]$. If $x^* ,x $ are two different points of $[a,b]$ , then there exists a point $z$ strictly between $x^*$ and $x$ such that   $$f(x)=f(x^*)+f'(x^*)(x-x^*)+\frac{f''(z)}{2}(x-x^*)^2$$  I tried to prove this theorem using The Fundamental theorem of calculus. But I can not stop it at the second derivative . Do I have to use mean value theorem ? or is there a simple proof for this ? Why do we have more male infants born? A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020) Suppose $f$ is a function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$, then we have to show that $\forall z\in \mathbb C f(z)+f(-z)=0 $.   Actually I have solved this problem and have no doubt about it but actually this question came in MADHAVA-2020. So I posted it on this site.   I am not answering the question and inviting all users to answer it.  It is a nice problem.   I will answer it within a week, but I am letting everyone try. Send an email as an attachment from iPhone How can you construct the bijection between real numbers and functions over naturals? FontAwesome font not found? Is char *envp[] as a third argument to main() portable Finding the maximum likelihood estimators for this shifted exponential PDF? Nginx: Forward all Subdomains My code doesn't work with second level tld's like domain.co.uk Here is my conf:   # add www.   if ($host ~ ^(?!www)) {    rewrite ^/(.*)$ http://www.$host/$1 permanent;   }    # remove subdomain   if ($host ~ &quot;^www\.(.*)\.(.*\.([a-z]{2,4}|[a-z]{2}\.[a-z]{2}))&quot;) {    set $host_without_sub $2;    rewrite ^/(.*)$ http://www.$host_without_sub/$1 permanent;   } How to import a module given its name as string? Allow embedded HTML5 YouTube video My text introduces multi-quibt quantum states with the example of a state that can be &quot;factored&quot; into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit $$\left|a\right\rangle = \alpha_1\left|0\right\rangle+\alpha_2\left|1\right\rangle$$ and a second qubit $$\left|b\right\rangle = \beta_1\left|0\right\rangle+\beta_2\left|1\right\rangle$$ any non-entangled joint two-qubit state of $\left|a\right\rangle$ and $\left|b\right\rangle$ will be $$\left|a\right\rangle\otimes\left|b\right\rangle = \alpha_1 \beta_1\left|00\right\rangle+\alpha_1\beta_2\left|01\right\rangle+\alpha_2\beta_1\left|10\right\rangle+\alpha_2\beta_2\left|11\right\rangle$$ but it isn't clear to me why this should be the case. It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true in a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)? Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required?  1:  (bottom of p. 18) &quot;so the state of the two qubits must be the product&quot; (emphasis added). Ubuntu on a USB stick - boot in both BIOS and UEFI modes Let $x_1,...,x_n$ be distinct integers. Prove that  $$\prod_{i&lt;j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$   I know there is a solution using determinant of a matrix, but I can't remember it now. Any help will be appreciated. How can I launch screensaver with a keyboard shortcut?