Finite Definition of Span
The definition of span
a cold vs flu / the flu Have you got a cold? Have you got flu? Have you got the flu?  Why can't we say a flu or the cold in the previous examples? Here I offer a "proof" of why the power set of the natural numbers is countable by creating a supposed bijection. I know that my proof is wrong, but where? If anyone could point it out, that'd be awesome.  Consider the power set of the natural numbers. We can form a bijection $f:\mathbb{N}\to\mathcal{P}(\mathbb{N})$ as follows.  For any $f(x)$ convert $x$ to its binary representation, which is a string of $0$s and $1$s. If the $i$-th digit is $1$, then include $i$ in the output set $S=f(x)$. Otherwise do not include $i$. Note that $i$ is $0$-indexed and starts from the right-most digit.  So in this case $f(5)=\{1,3\}$. Supposedly we can construct any set in $\mathcal{P}(\mathbb{N})$ with an appropriate choice of $x$.  Where does this proof blow up? The old advice used to be to double-quote any expression involving a $VARIABLE, at least if one wanted it to be interpreted by the shell as one single item, otherwise, any spaces in the content of $VARIABLE would throw off the shell.  I understand, however, that in more recent versions of shells, double-quoting is no longer always needed (at least for the purpose described above).  For instance, in bash:  % FOO='bar baz' % [ $FOO = 'bar baz' ] &amp;&amp; echo OK bash: [: too many arguments % [[ $FOO = 'bar baz' ]] &amp;&amp; echo OK OK % touch 'bar baz' % ls $FOO ls: cannot access bar: No such file or directory ls: cannot access baz: No such file or directory   In zsh, on the other hand, the same three commands succeed.  Therefore, based on this experiment, it seems that, in bash, one can omit the double quotes inside [[ ... ]], but not inside [ ... ] nor in command-line arguments, whereas, in zsh, the double quotes may be omitted in all these cases.  But inferring general rules from anecdotal examples like the above is a chancy proposition.  It would be nice to see a summary of when double-quoting is necessary.  I'm primarily interested in zsh, bash, and /bin/sh. What issues should be considered when overriding equals and hashCode in Java? What issues / pitfalls must be considered when overriding equals and hashCode? Indefinite Integral with "sin" and "cos": $\int\frac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $ From what I know, it is unknown whether $x_n=x_{n-1}^2 + 1$ has a closed form. Is there a recurrence relation which is known to have no closed form with a proof of inexistence?  Assuming a closed form is a non recursive description using the elementary operations of addition multiplication and power, or, assuming any other good definition of "closed form".  Edit  There are similar questions out there, but the answers are a little bit going around the question.   So is there one with proof or is it unknown? And if it depends on the definition of a closed form, then what are the (or some) options? How strong can a closed form definition be to still have a recursive relation that can be proved to not being able to have its form?  Update  Because it is a big question, and I don't want to open a new small one, I think a good representative small question about this subject is:  Is there a recurrence relation with a domain of the natural numbers, described by elementary functions, which has no closed form defined as:  A closed form of a recurrence relation with a domain of natural numbers is a function $f$ that can be described as a composition of a constant number of elementary functions without reccursion. i.e, $f(n) = f_1(f_2( \dots (f_k(n))) \dots) $ where $f_i$ is elementary for all $1 \leq i \leq k$   I think this definition is the most intuitive and basic.  For example, the relation $x_n = x_{n-1}+1$ has a closed form in this sense: $$f(n) = f_1(f_2(f_3(f_4(n))))$$  Where $f_1$ is division, $f_2(n)=(n,2)$, $f_3$ is multiplication and $f_4(n) = (n,n+1)$   Is there a recurrence relation which has no closed form in this sense? I have a High poly mesh and I have decided to make the low poly by merging as it is easiest for me. Would it be possible to change the W Hotkey , so that it merges to the center automatically? It is time consuming pressing W > Merge > Center. How to get horizontal black line when coloring cells? Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$ Let $a_1, a_2, a_3, \dots , a_n$ be positive real numbers whose sum is $1$. Prove that $$\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \ldots +\frac{a_n^2}{a_n+a_1} \geq \frac12\,.$$  I thought maybe the Cauchy and QM inequalities would be helpful. But I can't see how to apply it. Another thought (might be unhelpful) is that the sum of the denominators on the left hand side is $2$ (the denominator on the right hand side). I would really appreciate any hints. Inertial forces and centre of mass Is inertial force always attached to the centre of mass of the object? Why? What rules cause this to happen?  See the  below of the leaned bike in a corner.    The force $m\frac{v^2}{r}$ pointing to the right is the centrifugal force (inertial force - because it's a rotating reference frame, therefore a non-inertial reference frame). It is attached to the centre of gravity of the bike.  I'm thinking that inertial force is attached to the centre of mass because of the fact that it should simply avoid creating a torque on the object. That's intuitive, but there must be a general rule that allows us to conclude that. What is Ruby's double-colon `::`? Is it possible to get the ID of a Contact which is part of a Person Account?  I have a custom object which has a custom field referencing the Person Account, though it seems to be referencing the Contact directly. I have created a button which I am using to pass the ID into a Visualforce Page.  With the Insert Field drop-down on the create/edit button page I can put in {!Account.Id} or {!Contact.Id} but neither of these work. The Account ID is placed in the URL but is not the correct ID and the Contact ID is blank, which I assume is because the button is on a Person Account page and not a Contact Page. Does Google penalize daily updated  tags in sitemaps if the data is not daily updated? I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a Riemannian Manifold are very similar except for the detail that makes them Riemannian (having a inner product on the tangent space of every element which varies smoothly with respect to this element of the R. Manifold).  So my question is: is there any example of a Smooth Manifold that is NOT a Riemannian Manifold?  Thanks. How can I render bokeh circles?  Nothing I do seems to work.   I tried using an empty as the focus point for a cube, with some point lamps in the background with about 2000 strength, but it doesn't work. It just makes everything blurry.  I also raised the blades to 4/5/6/12, and it doesn't make a difference.  I'm using Blender 2.74. Large snap packages because of dependencies grep a file, but show several surrounding lines? I would like to grep for a string, but also show the preceding five lines and the following five lines as well as the matched line. How would I be able to do this? Is there a way to see the history of questions protected by me or another user? Having a record of all actions taken on a site, as is provided by the &quot;activity&quot; and &quot;votes&quot; tabs of our profile, comes in handy very often. Perhaps the most common case is being able to find a question I've interacted with in the past, to see if anything has changed about it. This applies to questions a user has protected as much as anything else. Users are theoretically encouraged to unprotect questions when it's no longer necessary, as emphasized , but in practice they often do not. In many cases it's hard to blame them: unless they specifically marked it as a favourite, the site doesn't keep any record of the question to allow them to find it again. We should help users more easily review their own protections in the future.    For personal review, the votes tab is adequate, but there may also be value in displaying this publicly in the activity tab. There may be users with a history of repeatedly protecting questions over-eagerly. It would be useful to be able to review other users' protection histories to see if this is the case. Even if this is not available to regular users, it may be helpful to moderators, unless their user activity views already allow them to filter down to see this. I have a DB Server running CentOS 6.4 and MySQL 5.5 from RPMs (not source).  I have dozens of databases but one of them is being hit quite hard with large selects which locks out other updates which in turn locks out other selects (at least I think this is what's happening!)  To alleviate this I would like to replicate one of the databases onto the same server so that the writes can write to the "master" and reads can be done on the "slave". I don't have the luxury of additional hardware.  I have found a lot of guides on the internet describing how to replicate to a different server, and one () describing how to replicate to the same server but it doesn't seem to be geared towards retrofitting this functionality to an existing database with 10s of GB of data in it already. It also talks about compiling MySQL from source.  Does anyone know how I can retrospectively convert my MySQL 5.5 config to replicate one of my databases on the same server?  My existing DB runs on port 3306 so I feel it should be as easy as firing up another instance of MySQL on port 3307 and making some changes to my.cnf but not sure what changes.  Thanks in anticipation. How to build a difference distribution table?