What is the biological basis for tiredness after eating?
Why do the humans become sleepy after meals?
Consider the subset $B=\{(x,y):x^2+y^2\leq 1\}\ \subseteq \mathbb{R}^2$. Show that $\vert B \vert = \vert \mathbb{R}^2 \vert$.  It is easy to see tht $f:B\rightarrow\mathbb{R}^2$ defined as $f(x,y)=(x,y)$ is injective.   However, in the book I'm working out of (Book of Proof by Richard Hammack), the other injection noted is  $g:\mathbb{R}^2\rightarrow B$ defined as  $$g(x,y)=\left(\frac{x^2+y^2}{x^2+y^2+1}x,\frac{x^2+y^2}{x^2+y^2+1}y\right).$$  In the book I'm asked to verify this is indeed an injective function.  My approach was to assume $g(x,y)=g(m,n)$ and through basic algebra I was able to deduce  $x(m^2+n^2+1)(x^2+y^2)=m(x^2+y^2+1)(m^2+n^2)$  and  $y(m^2+n^2+1)(x^2+y^2)=n(x^2+y^2+1)(m^2+n^2)$  but how does this show $(x,y)=(m,n)$? Or is this the wrong approach? What are homologous chromosomes? How can I find the difference between two angles? Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra? When reading the book Measures, Integrals and Martingales written by R.L. Schilling, I saw a statement as below: $$\mathcal{A} \times \mathcal{B}:= \{A \times B: A\in \mathcal{A}, B\in \mathcal{B}\}$$ where $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras, is not $\sigma$-algebra in general.  However, I cannot construct a counterexample. Could anyone offer help here?  Kind regards Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$ About a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$  If we'll take a line $f(x)=\begin{cases} ax &amp;, x\in\mathbb R\\ 0 &amp;, x=\infty  \end{cases}$ then we'll have a bijection.   But about this function: $g(x)=\begin{cases} x &amp;, x\in\mathbb R\setminus \{0\}\\ 0 &amp;, x=\infty  \end{cases}$ ? it appears to be a bijection and it's even simpler. Word whose form is contrary to its meaning? Is there a word for a word whose form is contrary to its meaning?  For example, "quotidian" is anything but quotidian. Why is plain-hash-then-encrypt not a secure MAC? Did the Council of Elrond know that destroying the Ring would destroy Sauron? Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?  Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of? Increase close vote weight for gold tag badge holders Allowing only one download per Apple ID Remove a part of a list of names I have some files and each files have distinct name but all of them have the same "-statement" at the end like this:  flower-statement.mp3 leave-statement.mp3  ball-statement.mp3 goal-statement.mp3   So I want to remove this: -statement  Is that possible? Proving that this function is continuous on $G\times G$ The storage on my iPhone is nearly full. Most of the space is used by apps. I'd like to see which apps consume the most space on my device.  Is there a way to get a list of the apps with their corresponding used space? By "used space I mean the installation size (bonus for added user data). For example:   For an MVC architecture, should I prefer Ruby on Rails or PHP [closed] Gmail: What is a jQuery selector for incoming mail senders only? How to work with highly compressed and nested CSS in general? [closed]   Better to convert to   [closed] For an MVC architecture, should I prefer Ruby on Rails or PHP [closed] Gmail: What is a jQuery selector for incoming mail senders only? How to work with highly compressed and nested CSS in general?   I should first read the title and sometimes open the question to understand if the question is closed.  I think the [closed] mark  should be at the beginning of the title, not in the end, where I won't ever be if the title is long.  Also coloring will be a good improvement (via @Shadow Wizard). I accidentally killed the orphanage lady before I started the quest. She just pissed me off, sadly it will not acknowledge that I have finished the quest and I haven't been contacted. Is there another way to join the Dark Brotherhood? File /etc/ld.so.nohwcap is missing in Debian 7 What is the file /etc/ld.so.nohwcap used for?. In my machine (Debian 7), this file is not found. Therefore, Which package or library this file belongs to? Forcefully update to new windows The  gives the recoil force, $\mathbf{F_{rad}}$, back on a charged particle $q$ when it emits electromagnetic radiation. It is given by:  $$\mathbf{F_{rad}} = \frac{q^2}{6\pi \epsilon_0 c^3}\mathbf{\dot{a}},$$  where $\mathbf{\dot{a}}$ is the rate of change of acceleration.  If a particle has a constant acceleration, $\mathbf{\dot{a}}=0$, then there is no reaction force acting on it. Therefore the particle does not lose energy.   Does this mean that a constantly accelerating charged particle does not emit electromagnetic radiation? Why is $\sqrt{-x}*\sqrt{-x}=-x?$ Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$  Q2 - I was thinking it would be:  $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?)  Q3 - What are the formal algebra rules to use? Can I calculate this without using i such as in: $\sqrt{-x}*\sqrt{-x}=i\sqrt{x}*i\sqrt{x}=-\sqrt{x^2}=-x$. After reading this question about  and asking myself two questions about  and , I realize that these activities are for showing what you found, make the audience be attracted. If they need further explanation, they should read your paper, which has successfully defensed and proved that your work is validated. One important thing when you attempt to attract people is concentrate on the important, which means unnecessary things need to be trimmed. I think the reference is one of those. So, should we put the references in slides/poster when we already have them in paper?