Given that we have a recursive function defined as $\begin{matrix} f_{0}=0 \\  f_{n}=f_{n-1}+1 \end{matrix}$ We can come up with a related non-recursive closed form function defined as $g(x)=x$ Where for every value of $n$ $f_{n}=g(n)$ Do all recursive functions defined with $\begin{matrix} f_{0}=\text{an integer} \\  f_{n}=\text{an algebraic function(1)} \end{matrix}$ (1): where said function involves only algebraic operations, like, addition, subtraction, multiplication, and division, as well as fractional or rational exponents have such a related non-recursive closed form function $g(x)$? I'm not looking for how to find $g(x)$ if it exists, I`m just wondering if one always exists.
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?
How to change css property using javascript Show function is continuous at origin $$ \begin{cases}xy^2\sin\left(\frac 1 y\right), \text {if $y$ $\neq$ 0} \\ 0, \text{if $y$ = 0} \end{cases}$$  I'm not sure how to work out this limit as $x$ goes to infinity. If it were just the y part, I know that $y^2$ goes to 0 and $\sin$ is a limited function, so it has to go to 0, but I don't know what to do with the $x$ term. wpa_supplicant store password as hash (WPA-EAP with phase2="auth=PAP") Is there a way to store my password in /etc/wpa_supplicant/wpa_supplicant.conf as some hash instead of plaintext?  By "password" I refer here to the password used for phase2 authentification. I do not refer to the Preshared Key (PSK) which could be hashed using wpa_passphrase.  For phase2 MSCHAPv2 or MSCHAP authentification I could store the password as MD4 hash using nt_password_hash (see example wpa_supplicant.conf line 659).  Is there any way to store my PAP password as hash?  Or: Is there a way to store the password in some sort of external storage? The example wpa_supplicant.conf suggests the use of such an external storage (using ext:????) but I could not find any documentation about it.  I am aware that storing the password as hash does not increase wifi security. But as the password MUST be the same as the password for other services (account management, subscriptions, payments ...) I don't want it to be stored as plaintext. optimality of the Carnot cycle Reference Books on Cryptography What is the best way to initialize a private, static data member in C++?  I tried this in my header file, but it gives me weird linker errors:  class foo {     private:         static int i; };  int foo::i = 0;   I'm guessing this is because I can't initialize a private member from outside the class.  So what's the best way to do this? Why isn't Obi-Wan's ghost young? One of the changes made to the Special Editions of Star Wars: Episode VI: Return of the Jedi was at the end of the film.  Luke sees 3 ghosts during the celebration on the forest moon of Endor, Yoda, Anakin, and Obi-Wan.  In the Special Editions, the actor who played Anakin earlier in the film, Sebastian Shaw, is replaced with Hayden Christiansen who played young Anakin in Episodes 2 and 3.  My question is, why is Anakin the only one de-aged?  Shouldn't Obi-Wan also be his younger self, along with Yoda? I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' How do Muggle Wars affect the magical world? This came up to me while I was reading the question about Magical being using firearms in battle to raise their power (which I think I have a good answer for, but it's too long to fit in this question... and it would also be inappropriate).  Anyways, I was reading it and then the following thought came into my mind: If wizards decided to use guns on their battles (the ones agains Voldemort for instance), that would create that opening for much bigger battles than we see in the books (after all even a muggle can fire a gun, so lots of guns + the Imperio curse, and presto manifesto (is that racist? talking about wizards and all) you have yourself an army). Those kinds of battles, however, would most definitely not go unnoticed by the muggles - even though they wouldn't be able to link it to magic.  Well, that thought brought up my one and final question, which I bring forth to you guys. Is there any mention on how the Wizards and Witches viewed the two world wars (which had taken part by the time Harry got to Hogwarts, and which he had probably heard about in muggle school), and what was the Wizards' part in it? What about the cold war? Did the magical world even recognise the existence of weapons such as nukes, which could render all life on earth extinct?  EDIT  DVK has brought up an  regarding the connections between the Second World War and the wizarding world (more specifically, the wizard Grindelwald). Although it doesn't fully answer my question, it does help by showing that there was indeed some form of connection between the two world at the time. Why is /tmp a symlink to /private/tmp on Mac OS X? In other words, why isn't /tmp just a regular directory, like on Linux or BSD? I understand how it works and I don't mind it, I'm just interested in the (historical?) reasoning behind it. I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the same function but with different domain how is this possible?  Or in other words why don't we simplify $f(x) = \frac{x-1}{x-1}$ to $f(x) = 1$ before plotting the points. Is it just defined this way or is there a particular reason ?  Note: my book says the domain of $f(x) = 1$ is $\mathbb{R}$ and the domain of  $f(x) = \frac{x-1}{x-1}$ is $\mathbb{R}$ except $1$. As the question says, I need to edit my /proc/acpi/wakeup file, but when I open it with any text editor, it shows blank file. cat command, on the other hand, displays correct file contents in terminal. When I navigate to that file in file manager, it says it's size is 0 bytes. I'm using Xubuntu 15.04. I'm thoroughly confused. Is there a way to send a file using POST from a Python script? I want to say:  Please, check out our handbook/knowledge base.  ...but it can be interpreted in two ways:  Please, check out our (handbook/knowledge) base. — as if there is a &quot;handbook base&quot; and a knowledge base. Please, check out our handbook/(knowledge base). — the meaning I'm after.  I thought that maybe I should put spaces around the slash to better separate the two choices:  Please, check out our handbook / knowledge base.  ...but I'm not sure that does it. I checked out  on the topic and it didn't answer my question.  Is it possible to convey the right meaning with a slash, or should I just avoid it altogether? How do I install KDE on Ubuntu server? What class of Devil Fruit does the Gomu Gomu no Mi come under? Considering the three types of Devil Fruits that has been explained are 1. Paramecia Type - where the user can get super human abilities and change their body parts into weapons, etc. 2. Zoan type - where the use can turn into a particular animal and inherit its abilities 3. Logia Type - where the user's body inherits a particular element and the body becomes like air which cannot be touched or damaged by non Haki users.  So what exactly is the type of Gomu Gomu no Mi? It cannot be Zoan or Logia. It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type! So is it some special type of Devil Fruit? What's the 1 inch square knob sticking out of the wall? Algorithm faster than Newton's for calculating $\sqrt{2}$ Can Xcode 6 and Xcode 5 coexist on the same computer? Change the default editor when sudo visudo When doing sudo visudo, the default editor is nano. I want to change it to vi or vim.     I already made vim the default editor, and use it as a substitute for gedit to open php, .txt, .c and .h files, by placing a vim.desktop file in ~/.local/share/applications/ and by correctly editing either  ~/.local/share/applications/mimeapps.list or /etc/gnome/defaults.list. However apparently this does not apply to nano.    Any clue ?