Mitosis is a complex mechanism that, through mutations and crossover, determines how a chromosome is composed. But at that point, the number of chromosomes is already determined: how can this number evolve through generations? What mechanism is involved?
I am aware of the basics of evolutionary theory, however I don't understand how mutations can add genes over time.   Am I correct in thinking that creatures within the same species who mutate to have an additional gene in their genome would normally be infertile?  Or am I misunderstanding that?  Can someone explain the process of the creation of genetic material through evolution?  Citation to a decent academic paper or book on the matter would be appreciated too.
Explicit bijection between the successor of $\alpha$ and $\alpha$ (infinite ordinal). I would like to construct an explicit bijection from $\alpha^*$ to $\alpha$, where $\alpha$ is an infinite ordinal. I have been told a possible bijection would be:  \begin{equation} f(\beta) = \begin{cases} 0 &amp; \text{if } \beta = \alpha \\ \beta^* &amp; \text{if } \beta \text{ is finite} \\ \beta &amp; \text{otherwise} \end{cases} \end{equation}  I don't understand how this is constructed, or why it is a bijection. Get last n lines of a file, similar to tail I'm writing a log file viewer for a web application and for that I want to paginate through the lines of the log file.  The items in the file are line based with the newest item at the bottom. So I need a tail() method that can read n lines from the bottom and support an offset.  This is hat I came up with: def tail(f, n, offset=0):     &quot;&quot;&quot;Reads a n lines from f with an offset of offset lines.&quot;&quot;&quot;     avg_line_length = 74     to_read = n + offset     while 1:         try:             f.seek(-(avg_line_length * to_read), 2)         except IOError:             # woops.  apparently file is smaller than what we want             # to step back, go to the beginning instead             f.seek(0)         pos = f.tell()         lines = f.read().splitlines()         if len(lines) &gt;= to_read or pos == 0:             return lines[-to_read:offset and -offset or None]         avg_line_length *= 1.3  Is this a reasonable approach?  What is the recommended way to tail log files with offsets? How to ssh into a server running on a home network, behind a router? Okay, so I have a PC in my home network, running Ubuntu on a VM with openssh server installed.    How can I remotely access it from another computer outside the home network, using the internet? What IP address do I use to connect to it? What kind of port-forwarding do I need to do? Do small title edits constitute trivial edits? Being a fairly new user something I don't seem to be able to locate is information on whether small edits to the title are considered as trivial edits. I see the usual recommendation when suggesting an edit is:     Avoid trivial, tiny one-letter edits unless absolutely necessary.   I can see the logic in that for most cases, but for example I just read a question with a title containing "Andriod" which would be trivial in one way but also affect the ability of users to locate the question using both the site and external search engines. Would it be frowned upon to suggest an edit for those cases? I was showing a GM friend of mine a build I was making that uses the Two-Handed Fighter Archetype and we are both unsure of how it interacts with the damage bonus you get from two-handed weapons (which is 1-1/2 times your STR.) It reads:     Overhand Chop (Ex): At 3rd level, when a two-handed fighter makes a single attack (with the attack action or a charge) with a two-handed weapon, he adds double his Strength bonus on damage rolls. This ability replaces armor training 1.   So, does Overhand Chop turn the 1-1/2 times multiplier from two-handed weapons into a 2 times multiplier, or do the two add up? Or, do they interact in some other way?   This assumes a Medium sized character with 20 STR and a greatsword with 3 levels. In a custom lightning component where one is using the recordEditForm tag, how can one validate fields that are filled by the user?  For example, in a quick action to create a Case record, we can have the following line inside the recordEditForm:  &lt;lightning:inputField aura:id="caseAccount" fieldName="AccountId"/&gt;   How can one enable or disable a button if certain fields are assigned by the user? Like if the account is specified, then a button on the component is enabled for the user.  Making the field required is not an option. Why does as.factor return a character when used inside apply? Reprojecting LiDAR data with libLAS, error: "latitude or longitude exceeded limits"? Let $A \in Mat_{n \times n}(F)$ a matrix which satisfies $A^n=0$ and $A^{n-1} \ne 0$ for some positive integer $n$. Let $v \in F^n$ be such that $A^{n-1}v \ne 0$.  Prove that $\{v, Av, A^2v,...,A^{n-1}v\}$ is a basis for $F^n$.     So, from definition of $A$, I know that $A^iv \ne 0$ for all $0\le i \le n-1$. I know I need to prove the set is linearly independent and span $F^n$ to prove it's a basis, but I just don't know how. What does the Web.Config file do in the views folder of a MVC project C/C++/Assembly Programatically detect if hyper-threading is active on Windows, Mac and Linux Moving app to production mode in Symfony 2 Removing Firefox in Ubuntu with all add-ons like it never existed I have been using Firefox for a long time in Ubuntu, I downloaded some add-ons but it made firefox extremely slow, even if I remove them nothing will change.  I tried to remove Firefox also, but re-installing it will somehow recover the same Firefox I had- it seems it write some info somewhere that even if you remove Firefox, it will keep them.  Moreover, I am not able to use see any flash videos or website with Firefox(chrome works fine).. Honestly I tampered with plug-in settings in Firefox that this happened. But right now I can't do anything to overwrite the setting...  My question is how I can remove Firefox in a way to remove any previous traces and re-install a new version like for the first time? I accidentally deleted a file from my laptop. I'm using Fedora. Is it possible to recover the file? I zoomed in / out of one of my UI panels and I can't figure out how to zoom back out/ in to get it normal size.   How can I reset the zoom level of UI panels? How do you get a bad transcript past Ph.D. admissions? I have a master's degree in International Studies, and a double major with Computer Science from undergrad.  My transcripts suck.  There's no other way to dress it up.  I have pretty good teaching experience, and my GRE scores are awesome, and I suspect my recommendations are as bland as everyone else's.  Basically, to an admissions committee, I suspect I'm the model of a student who is probably smart enough but didn't work hard enough.  I want to do a Ph.D. in Political Science, but the response from my applications is looking pretty grim.  Am I permanently out of the running, or is there something I can do for the next few years which will help to counterbalance my unfavorable GPA? How to determine whether the direction of rotating a 2D vector to another one is counter-clockwise or clockwise using the cross product? Vec2 a, b; // some random vector Vec2 cross = a.Cross(b); RelativeDirection(a,b,cross); // return true if the direction of                               // rotating a to b is counter-clockwise   How do I implement the function RelativeDirection()?  I got some suedo code from the textbook:  if a.Cross(b) &gt; 0 then ... else ...   but the cross product is a vector right? how do you compare it with an integer? 90s (or earlier) Japanese anime movie: two planets at war, young man is in love with princess from the other planet, and floods his own village Can I tap a creature with summoning sickness to pay "tap an untapped creature you control" costs? vim and system clipboard I can cat a file, copy and paste text with the middle button or shift-insert.  How can I copy text in vim and paste it from the system clipboard?