Male pattern baldness is a common genetic trait. With a distinctive pattern/ density gradent (as apposed to general old age thinning as the body ages)  Hair has an advantage of protection from the sun/weather. So to lose it in a distinctive pattern (and lose that advantage) appears to have a reason*; and as it is a genetic trait, it seems like it would have been something that was selected for (or it would be more likely to be selected out). So why is it better to develop pattern baldness as someone gets older?  So what might be the biological/evolutionary advantage to developing baldness?  Traits get selected for good and bad reasons(in retrospect), however I am interested in the reason why this particular trait has been selected.  * Lets assume that there is a specific cause (causes) and it is not just a leftover evolutionary artifact.    The related question:  provides some interesting background, but doesn't specifically address the What causes in this question.
If a trait would be advantageous to an organism then why hasn't it evolved yet?  Conversely, if a trait is not advantageous or mildly disadvantageous, why does it exist?  In other words why does evolution not make the organism more "perfect"?    This is a general question that would be applicable for any kind of trait. Please keep the answers precise and scientific.   Read this meta post for more information:
Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous Suppose that $X$ is infinite and that $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous.     My attempt:  Let $|A|=n$. We will prove by induction on n. It's clear that the the theorem is trivially true for $n=0$. Assume the theorem is true for all $n=k$. For $n=k+1$, then $|A \setminus \{a\}|=k$ for some $a \in A$. Thus $X \setminus (A \setminus \{a\}) \sim X$ by inductive hypothesis, or $(X \cap \{a\}) \cup (X \setminus A) \sim X$, or $\{a\} \cup (X \setminus A) \sim X$. We have $\{a\} \cup (X \setminus A) \sim X \setminus A$ since the theorem is true for $n=1$. Hence $X \setminus A \sim \{a\} \cup (X \setminus A) \sim X$. Thus $X \setminus A \sim X$. This completes the proof.       Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help!     Update: Here I prove that the theorem is true for $n=1$.  Assume that $A = \{a\}$ and consequently $X \setminus A= X \setminus\{a\}$. It's clear that $|X \setminus A| \le |X|$. Next we prove that $|X| \le |X \setminus A|$. Since $X$ is infinite, there exists $B \subsetneq X$ such that $B \sim X$ (Here we assume Axiom of Countable Choice). Thus $|X|=|B|$. There are only two possible cases.   $a \in X \setminus B$   Then $B \subseteq X \setminus \{a\}=X \setminus A$ and consequently $|X|=|B| \le |X \setminus A|$. Thus $|X| \le |X \setminus A|$ and $|X \setminus A| \le |X|$. By Schröder–Bernstein theorem, we have $|X \setminus A| = |X|$. It follows that $X \setminus A \sim X$.   $a \in B$.   Let $b \in X \setminus B$. We define a bijection $f:X \setminus \{a\} \to X \setminus \{b\}$ by $f(x)= x$ for all $x \in X \setminus \{a,b\}$ and $f(b)=a$. Thus $X \setminus \{a\} \sim X \setminus \{b\}$. Since $b \in X \setminus B$, it follows from Case 1 that $X \setminus \{b\} \sim X$. Hence $X \setminus \{a\} \sim X \setminus \{b\} \sim X$. Thus $X \setminus \{a\} = X \setminus A \sim X$.  To sum up, $X \setminus A \sim X$ for all $|A|=1$. How to reconcile the cashless human/Federation society with human traders and merchants From a variety of sources, including , we know that the Federation (or at least the humans) don't use money. This raises the problem that humans frequently visit worlds where there is currency, and, in the case of DS9, operate a station that has currency frequently in use (for example, Federation personnel frequently attend Quark's bar). Additionally, we see human traders.   My question is, how do humans get "money" (credits, latinum, etc.)? How would a human even get the resources to purchase their own merchant ship or the resources to fill its cargo holds with merchandise? If Federation personnel can simply be given money for the asking, why haven't they poured so much money down Quark's throat that he chokes on it? Laurent series expansion No more than 10,000 results are visible in search If you make a search for something that returns more than 10,000 items in result. For example, search for  return 161,338 results on MSE. But if you select a page where 10,000+ result should be shown:        In the edge case  shows zero results:        But this is actual for any page where per_page * page_number is more than 10,000.  Though this does not happen for Stack Overflow in English. It looks like all other sites are affected.  Related bug report on ruSO.meta: How to check a particular directory is mounted on the Linux machine. For instance there is a folder named test, I want to check if it is mounted or not. If I replace my main panel, will I need to fit AFCIs? This question is about whether "circuits" can be grandfathered if the breaker or panel is changed but the circuit is otherwise not disturbed.   Code is NEC 2017.  My existing main panel has plenty of ampacity but few spaces, and thus, is totally full.  I have several potential paths for an upgrade.    One involves fitting a subpanel and moving 4 circuits (2 double-stuff breakers) there, and adding circuits there.    The other involves outright replacing the main with a very fine 60 space main panel.  No more space problems!  But what happens with AFCI and GFCI requirements? If this were a new house,  nearly all my circuits would require it, but I am grandfathered.  Will the  "new main" scenario void the grandfathering and require that nearly all circuits be upgraded to that protection?   Or can I just move over my old breakers and call it good? Is it possible to apply modifiers to multiple objects at once? What is vector division? My question is:  We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors? Stop iCloud automatic download to harddrive How do I stop iCloud from downloading files to my hard drive? I opened up a folder and it immediately started downloading all contents Extract reply addresses from all emails in an outlook folder (sent on behalf of) I've got to send a bulk email out to a bunch of people in reply to emails they sent me. I've got all the original emails stored in a single outlook folder. I want to extract all of the reply addresses from the emails in that folder so I can send an email to all of them.  The article  shows you how to export the FROM address of these emails, however a large amount of them came from a web service that sends emails 'on behalf of' the person I'm trying to mail to. So I need the reply-to address, and the export wizard doesn't do that.  Does anyone have a tip on how to do this? This is Outlook 2007 on Exchange 2010. Embedding $S_n$ into $A_{n+2}$ I am trying to prove that for all $n$, $S_n$ is isomorphic to a subgroup of $A_{n+2}$.  Say $S_n$ acts on $\{\alpha_1,...,\alpha_n\}$ and $A_{n+2}$ acts on $\{\alpha_1,...,\alpha_n,\alpha_{n+1},\alpha_{n+2}\}$.  Let $\sigma \in S_n$ and let $\phi: S_n \to A_{n+2}$ be given by $\phi(\sigma)=\sigma$ if $\sigma$ is an even permutation, and $\phi(\sigma)=\sigma(\alpha_{n+1}\alpha_{n+2})$ if $\sigma$ is an odd permutation.  This way $\phi(\sigma) \in A_{n+2}$.  Now, let $\sigma, \rho \in S_n$.  Then $\phi(\sigma \rho)=\sigma \rho (\alpha_{n+1}\alpha_{n+2})$.  Also, $\phi(\sigma)\phi(\rho)=\sigma(\alpha_{n+1}\alpha_{n+2})\rho(\alpha_{n+1}\alpha_{n+2})=\sigma\rho$.  So, $\phi(\sigma \rho)$ does not seem to equal $\phi(\sigma)\phi(\rho)$.  But the idea is that we want $\phi(S_n) \cong S_n$, and $(\alpha_{n+1}\alpha_{n+2})$ does not permute any members of $\{\alpha_1,...,\alpha_n\}$, so $\phi(\sigma \rho)$ and $\phi(\sigma)\phi(\rho)$ are equal as permutations on this (sub)set.  This feels a little subtle to me and I'm not sure if my idea is valid or not.  If it is not, can this proof still be saved?  I appreciate any thoughts on this.  Thanks. Real analysis - proof approach help Can Mac App Store installs & upgrades be automated? How can I script installing or upgrading apps from the Mac App Store? I'd prefer command line tools, but AppleScript or some other API would suffice.  Relevant puzzle pieces:   installer: note especially the -store flag which is used to simulate App Store installs for package devs. systemupdate: capable of handling updates to Apple apps, flags -i installs and -a tells it to install all available updates I would like to migrate apps and their data from one phone to another without too much faffing around, like:   rooting (very convoluted device specific process with a lot of ambiguity around it and often involving running closed source software from random vendors that you have to believe does the right thing), using app specific backup export/import facilities if at all exist (too manual, seems the only option though).   I am ok with:   using a PC to help facilitate the process, manual copying/pasting files (within reason).   Ideally, I would like to have a process that works something like this:   select apps that I would like to have in their current state on the other phone, bundle them together and store somewhere (external SD card or PC), put that bundle on the other phone and somehow install and recover their state.   A bit like  works.  So far I have tried :    - unfortunately, like many others, requires rooting,  - seemed promising but Helium has never been, nor ever will be, compatible with any device running anything less than Android 4.0 and it does not work on certain phones.  - Q: Does it backup data of apps? A: No, currently it only backup the apk files of apps. and skimmed through a number of articles including  (looks like even if I was willing to root I would have a hard time). If you are falling, can you Misty Step to the ground to avoid damage? If I fall off a cliff - say, 1000 feet - can I wait (readied action?) until I am near enough to the ground (my understanding is you fall 500 ft/round) then  safely to the ground? Or, does the momentum from the fall carry over through the teleport and I still splatter across the rocks? What is a less offensive synonym for "hypocrite"? Is there a word that describe a person who doesn't "practice what they preach"? Basically, is there a synonym for "hypocrite" that carries less pejorative connotations?   For example, let's say a friend of mine says he is a Christian. Yet he engages in behaviors considered sins by the Bible, making him a hypocrite. And I would like to tell him I think he is a hypocrite.   What would be a good word to use here, so my criticism seems constructive rather than offensive?     I tried using a , but the synonyms that came up were even worse (ie: they are more offensive than "hypocrite"). Related (but different) question: , which asks what to call someone who knows they are in the group of people they criticize. Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? Isometries of the sphere $\mathbb{S}^{n}$ Hide details of a chapter from TOC and list of tables, figures and algorithms How do I calculate $\lim_{x\rightarrow 0} x\ln x$ I was thinking about the reasons behind $0^0=1$ and I remember one of my friends studying math arguing about the continuity of the function $x^x$ in $0$. But when I write as $$x^x=e^{x\ln x}$$ I am now looking at $$\lim_{x\rightarrow 0} x\ln x$$ Graphically I can see in Mathematica that it goes to $0.$ But I can't calculate by using a Taylor expansion, because I can't expand log around $0$. How do you prove that?