How can I formally classify a function that increases to infinity?  Intuitively I know (I don't why I know it, maybe from school) that functions like:  $$f(n) = n^2 + n + 123 + log_2n$$ $$f(n) = n^2 + n$$ $$f(n) = 12 * n^2$$  Have quadratic class  Functions like:  $$f(n) = log_2n + 2$$ $$f(n) = 5 + log_5n + 2$$  Have logarithmic class  I did this classification because it is obvious for me that $n^2$ has higher growth rate than $n$ and $n$ has higher growth rate than $log_2n$  But do we have any common and formal way to classify a function?   Or more commonly speaking, what does exactly word class means when we talk about functions?    UPDATE:  Someone suggested me that I should look at Big O notation. I don't see how it can solve my problem.  Let's say we have a function like: $$f(n) = 2^{(n + log_2n +12)} + 3^{(5n + log_2n)}$$  How Big O can help me to determine at what set (class) of functions $f(n)$ belongs to?  Moreover, we can use Big O notation only after function is been classified, not before. So I need some math framework to classify function and only then I can use Big O notation.
There is a formal definition for the Big O notation in . Up to now I have come across Big O in Numerical Analysis, Calculus and Algorithms which all are pretty distinct fields. What I am wondering is if that definition is global and is the only one that is used in every field where the Big O is involved or there are other ways that the Big O is defined in other areas  of Mathematics.
When I run the logistic regression, two independent variables have VIF values greater than 10 like 13 or so. Logistic regression is the one I will use to measure the overall change in the dependent variable with one incremental changes in every each independent variable. However, when I run the linear regression model version on my same data, the VIF values I get from all the independent variables are either slightly greater than 1 or around 5 or 6. Besides, all the coefficients are highly significant all below 0.01.  If so, is it safe to use my logistic regression model to find out the incremental effect of every each independent variable without dropping out any independent variables?   If my purpose is to find an incremental effect of every each independent variable on the overall dependent value in a logistic regression model, is multicolinearity an important problem to consider? Can I ignore it?   If my p-values are all less than 0.01 which is the same indicator as t-value, will I still need to worry about colinearity issue even though my VIF scores for two variables are around 13-14? Based on what you said, if p-value is safe enough, will this be still a problem?  I am also referring to the following website comment:                                                                     So, in sum, my ultimate goal is to use the final output from the logistic regression model generated from the independent variables and one binary dependent variable. If so, do you think I can ignore the multicolinearity problem? Is it possible to sneak up on a Force user? In Return of the Jedi, Luke and Darth Vader both sense each other's presence on the moon. In The Force Awakens, Kylo Ren   senses Han Solo on the Starkiller Base  Is it possible to sneak up on a powerful Force user? Or do you have to be   related to them  or significant to them in some way to be detected? Are there any examples of a Force user not sensing someone (in canon), or any successful attempt at disguising someone from a Force user? To clarify, I am talking about physical presence, not Force ability.   Han Solo doesn't have the Force, but Kylo sensed him anyways Can you claim a Double Sacrifice Bonus? Two very closely related questions:  If a card has two different "Sacrifice this creature for X" abilities on it, does sacrificing it give both benefits?  Likewise, if a card requires a creature to be sacrificed, but that creature has a "Sacrifice this creature for X" ability, do you get the sacrifice benefit when you sacrifice it for the other requirement? Which is correct, "you and I" or "you and me"? Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. Uninstall Kali Linux from triple boot View/edit ID3 data for MP3 files What's a quick and easy way to view and edit ID3 tags (artist, album, etc.) using C#? How do I install Java? Assume there is a public-key encryption scheme $(KeyGen, Enc, Dec)$ with perfect correctness (i.e., for all messages M and valid key-pairs (PK,SK), we have $Dec_{SK}(Enc_{PK}(M))=M$).  Will there always be a function $A$ such that for all messages M and valid public-keys PK, $A_{PK}(Enc_{PK}(M))=M$?  In other words, can a computationally unbounded adversary decrypt the message, having only the public key? How old should you be when you begin Jedi training? Anakin Skywalker was about nine years old (see script link, below) during "The Phantom Menace". Yet according to various people, he was too old (or even "far too old") to begin training as a Jedi:   QUI-GON: "I'm afraid not. Had he been born in the Republic, we would have identified him early, and he would have become Jedi, no doubt...he has the way. But it's too late for him now, he's too old." OBI-WAN: "The boy will not pass the Council's tests, Master, and you know it. He is far too old." MACE WINDU: "He is too old. There is already too much anger in him." and even ANAKIN himself: "I'm with Qui-Gon...but...they're not going to let me be a Jedi. I'm too old."   All quotes above from .  However, during "The Empire Strikes Back" Yoda gives Luke's age (18 in chapter IV according to the ) as the reason to deny him training only after first saying that Luke is:   too impatient too angry not ready is unfocused ("Never his mind on where he was") and reckless   All from .  I would have thought that, if at 9 Anakin was too old, Yoda would have seen Luke at 18+ and started off by saying, "No chance, he has".  So, what is the age to begin Jedi training? Animagi (Sirus specifically) and wands? I have 3 questions about animagus transformation and wands      Does animagus transformation require the use of a wand?   If yes, where does the wand go after the wizard transforms?   Again, if 1 is yes, how did Sirius transform into a dog in Azkaban?     Note that 2 and 3 are invalidated if the answer to 1 is no. If so, is it mentioned anywhere in the books, or can be implied by something in the books, that wand use is not required? I don't think Sirius managing to do it in Azkaban can be used as a proof here, as in Book 3 Wormtail needs a wand to change back into a rat When, if ever, should voting override answer acceptance for answer sorting? The accepted answer always appears at the top of the list, regardless of sort order. This can be misleading if the question author accepted an invalid solution and a better solution was proposed later, and eventually got more votes. I have seen somewhat obscure questions where the accepted answer has a negative score and the next answer has 10 or more up-votes. Why does a delta/wye transformer make 30 degrees phase shift ? I'v heard that a delta/delta or Wye/Wye transformers do not make any phase shift. In delta/wye or wye/delta transformers, There is a 30 degrees phase shift between the primary and secondary coils. So, Why is  there a phase shift ? and Why is it 30 degrees ?.. Why not 60 or 120 ?  I googled and I found calculations with phasor diagrams that proves the 30 degrees phase but I'm confused because of too many calculations and diagrams, I didn't get it.  Would you give me a simple answer, please ? and I prefer the physical meanings and concepts rather than equations and mathematics.  Thank you very much, How to get client's IP address using JavaScript? I need to somehow retrieve the client's IP address using JavaScript; no server side code, not even SSI.  However, I'm not against using a free 3rd party script/service. Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? When GMing a game, how much should you prepare in advance? Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? Please enable low-rep users to easily breakout up and down votes on their own questions I know once you get a certain amount of reputation, you can see the total up and down votes for questions. Well, my proposal is to always be able to see this on your own questions and answers. You can see it by going to your user page anyway, so why not make it possible to do by simply clicking on the number of votes (like you can when you have the rep)? Why does magnetic force vary in proportion to the cube of the distance instead of the square? Magnetic forces vary from gravity and electromagnetic radiation (such as light) in that gravity and radiation diminish by the square of the distance from the source, but magnetism diminishes by the cube of the distance.  Why is this? I was wondering how to make a clickable terminal shortcut for my desktop using the command easytether connect.  I don't want to have to open/close the terminal every time I wish to connect to the Internet through my phone.  I've tried updating my gnome-panel but I can only make a new directory on my desktop. What is the difference between update-grub and update-grub2? I'm making some changes on my grub file under /etc/default/grub. In some tutorials I have seen sudo update-grub and others sudo update-grub2. What is the difference?