This might seem a bit weird, but why can only women produce milk? Wouldn't it be more beneficial to our species if all humans could produce it? I can't think of any reason only one gender would need it, unless it would somehow be harmful to men.  I know that men can secrete milk in some cases, but why not in general?
In course of evolution (possibly), mammary glands became vestigial in male mammals, but became fully developed in females. Is there any plausible explanation for this characteristic?
If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. How do I write a correct micro-benchmark in Java? How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: Downloading all photos from iCloud Photo Library without Photos app What is the variance of the weighted mixture of two gaussians? There is a lemma that says if a group $G$ has no proper nontrivial subgroups, then $G$ is cyclic. And here is the proof of the lemma:     Suppose $G$ has no proper nontrivial subgroups. Take an element $a$ in $G$ for which $a$ is not  equal to $e$. Consider the cyclic subgroup $\langle a \rangle$. This subgroup contains at least $e$ and $a$, so it is not trivial. But $G$ has no proper subgroups, so it must be that $\langle a \rangle = G$. Thus $G$ is cyclic, by definition of a cyclic group.   But here i do not understand the following: Why must $\langle a \rangle$ be a subgroup of $G$? For every single element $a$ in $G$, if $\langle a \rangle$ is a subgroup of $G$, then every group should have at least as many subgroups as the number of its elements. I would appreciate any help. Thanks I met an interesting issue about C#. I have code like below.  List&lt;Func&lt;int&gt;&gt; actions = new List&lt;Func&lt;int&gt;&gt;();  int variable = 0; while (variable &lt; 5) {     actions.Add(() =&gt; variable * 2);     ++ variable; }  foreach (var act in actions) {     Console.WriteLine(act.Invoke()); }   I expect it to output 0, 2, 4, 6, 8. However, it actually outputs five 10s.  It seems that it is due to all actions referring to one captured variable. As a result, when they get invoked, they all have same output.  Is there a way to work round this limit to have each action instance have its own captured variable? Why does Li+ ion move to cathode in Li-ion battery? Mis-aligned shapefiles in QGIS? In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? At school I learnt the  to remember the resulting direction of different phenomena, such as geometrical cross products, mechanical torque, or the direction a screw will move when turning it. This directions are however decided by humans, and we could have easily defined the directions in the opposite way.  However the right hand rule also predicts phenomena which is not arbitrarily set by humans, but is intrinsic to the nature. If you have a coil in which you put some current, and the nature creates a magnetic field in a direction perpendicular to it, the direction depends on the right hand rule.  My question is, why this direction and not the opposite? Nature tends to be symmetric and not have apparent preferences for two equal options, so in a so why is one direction preferred over the other one? Why doesn't a "left hand" rule apply? Simplify $\sum (x_i- \mu)^2$ How can I simplify this relation? $$\sum_{i = 1}^m(X_i-\mu)^2=\sum_{i = 1}^m(X_i-\bar{X}_m)^2+m(\bar{X}_m-\mu)^2$$ I know how to set up: $$\sum_{i = 1}^m(X_i-\mu)^2=\sum_{i = 1}^m[(X_i-\bar{X}_m)+(\bar{X}_m-\mu)]^2$$ but when I open the bracket, how do I handle the mixed term? $$2\sum_{i = 1}^m(X_i-\bar{X}_m)(\bar{X}_m-\mu)$$  Thank you! Salesforce to Xero integration using Named Credentials by OAuth 2.0 product of normal and lognormal variates If x and y are uncorrelated normal variates, x*exp(y) will have a symmetric unimodal distribution with positive excess kurtosis. Has this distribution been named and studied? I followed the official step-by-step "Visual Studio Code and Unity" guide, but it seems I messed up something, because Intellisense doesn't see any Unity related stuff.  Like the MonoBehaviour class and its members (like .transform), or the Unity callbacks, like Start or Update. Using Label Expression with Text Formatting Tags and Ampersand How to find the URLs accessed by iOS Apps? What would happen if you used a dead person's hair in Polyjuice Potion? Polyjuice Potion is changing based on someone's current physical state. It shows tattoos (or lack of) and such from DH when Ron changed to Harry.     "I knew Ginny was lying about that tattoo,” said Ron, looking down at his bare chest.    What would happen if someone used hair of a dead person, taken from their body after death, and drank the potion? Would they die as well? Is the tabu package obsolete? Examples of topologies in which all open sets are regular? An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia . Furthermore, the regular open subsets of a space (any space) form a complete Boolean algebra.  I'm coming to this from logic and algebra, with not much background in topology. I can't figure out which topologies have "interesting" collections of regular open sets. For example, in the trivial topology and the discrete topology, every open set is regular, if I'm not mistaken. Those are not "interesting" topologies. I assume there are other topologies in which the space X and the null set are the only regular open sets. If so, those aren't "interesting" either, at least not with regard to their regular open ets.  I believe what I'm looking for is topologies in which every open set is regular, other than the ones I've just described.  Thanks for any help. How do I count the number of occurrences of a char in a String? I have the string   a.b.c.d   I want to count the occurrences of '.' in an idiomatic way, preferably a one-liner.  (Previously I had expressed this constraint as "without a loop", in case you're wondering why everyone's trying to answer without using a loop). I want to change my thesis topic (to the extent that the ethos is still in tact) but my advisor does not agree. What can I do? I am doing a phd where my aim is to solve a problem A which is very interesting. My supervisor has got a lot of connections in a technology B and he claims to be an expert in B. I came to this group because of these accomplishments. But soon I realized that B is not really a great technology for A.   So in my phd proposal I proposed a method which encompasses B but is more general (say R). Now I realize that I do not want to use B at all because my preliminary results are not encouraging. In fact, I am inclined to use E as a base technique which does not have any impact on R.  But my supervisor, who is interested in only publishing in B  and frowns upon everything else (it was a nightmare to convince him to let me work on R) is dead against it. He says that he is an expert in B and his vision of the group is not inclined with me trying other things.  There are other professors in my university who have worked in E and could collaborate but my supervisor won't collaborate with people from outside his kingdom. I also know professors from outside my university who could help me if needed. I could knock the doors of school committee in order to  help me but that would create a lot of problems as my supervisor wields a lot of power and influence everywhere.    Another query is, is it ethical to collaborate with people outside the university (of course after letting my supervisor know) for my PhD work. My Masters supervisor would be happy to help me out.  Edit: After looking at comments, I am thinking that after all, it is MY PhD. Yes, I have spent 1.5 years, but 1.5 years is left. Why should I spend any moment of my life on something I find to be not exciting, particularly less exciting because it is forced on me. And in those 1.5 years I learnt to write well and also learnt 'A' which is important. Its not total waste. My supervisor has his agenda, and why should I give up mine to make him happy? His vision is fine, but he does not have the right to force his vision on me.