I was thinking about this today. And although I've made some trials to answer it, none of them seem to fit. So, how do we define interior and exterior of a geometric figure? I guess that perhaps there might be some connection with the idea of open/closed sets in topology. But I'm not sure.
A question occurred to me when browsing SE this evening, just curious.  What specifies the inside of a 3D construct? If I have a hollow sphere, what's to say that the world isn't the inside, and the center of the sphere isn't the outside. Picture the Asylum in the Hitchhiker's Guide to the Galaxy series, if you have read that.  Additionally, if the inside of a 3D object is defined as what is smaller than the outside, what happens when there is a a sphere with a volume equivalent to 51% of the universe?
Highlighted source code inline with Latex/Minted? Can you help me with my capacity planning? What is the difference between Θ(n) and O(n)? c stdout print without new line? only display df lines that have more fs usage than 80% root@SERVER ~$ df Filesystem  512-blocks  Free    %Used   Iused   %Iused  Mounted on /dev/YXCV   655360      365632  45% 6322    13% / /dev/ASDF   3801088     670648  83% 41759   32% /usr /dev/ASR    1048576     500496  53% 5555    9%  /var   How can I pipe the df command's output to only display lines that have more usage than 80%?  e.g.: it would only display:  /dev/ASDF   3801088     670648  83% 41759   32% /usr So, according to Einstein's theory of general relativity, gravity is not a force instead it is a consequence of objects with mass deforming spacetime, right? And so, according to him, there is no difference between standing on a platform accelerating upwards at 9.81 m/s^2 and standing on Earth, right? But, there is a difference. At least, I think it's a valid difference. If you drop two apples from a high altitude on Earth, not only will they accelerate downwards at 9.81m/s^2 but they will also slightly move towards each other because the Earth is a sphere and gravity acts radially inwards pulling the two apples closer to each other. But if you drop the apples from the same height above the accelerating platform, the apples will only appear to accelerate downwards at 9.81 m/s^2 but will not move towards each other like they would on Earth. Why is this true? Is this even a valid difference? If it isn't, why isn't it? Neural networks output probability estimates? Suppose my training data contains ~100 variables, and each example is tagged as "success" or "failure".  I understand how a neural network can be used to try and predict success vs failure based on the variables.  However I am interested in the neural network outputting the posterior probability rather than success or failure. In fact, I evaluate the efficacy of the NN based on how accurate the probabilities are (eg. AUROC over the entire dataset) rather than % of cases with correct prediction.  Are NN's the right tool for the job here and if so how do you structure the NN to output this?  (NOTE: I'm a ML newbie!) Why does time change in Ubuntu after installing Windows Today I was installing Windows 7 on the same computer I have Ubuntu. For some reason, after installing Windows 7 and going back to Ubuntu the time went from the correct hour at that moment, 12:30PM to 7:30AM. This happened after installing Windows 7. For what I understand the BIOS has a time set, Ubuntu has the same time that is shown in the BIOS when installing it. At least up to this point is what I have seen. Now after installing Windows 7 (And believe me this is only for testing the performance of Windows 7 against Ubuntu in which case Windows 7 is falling behind a lot on the same hardware) the time changes in Ubuntu to the wrong one that is set in Windows 7.  I actually do not change the time in Windows 7 since this is one of the checks I am doing to see if the operating system can correctly grab the time. Windows 7 fails. Ubuntu does not BUT Windows 7 actually changes the time for Ubuntu if I boot with it afterwards.  So my question is, why/how does time change in Ubuntu after installing Windows.  NOTE - I have set Ubuntu time to "manual" instead of grabbing it from the Internet just to test out this. Existence in ZF of a set with countable power set Is it consistent with ZF for there to exist a set $S$ such that the power set $P(S)$ is countable? If so, what is the weakest form of the axiom of choice needed to prove that no such set exists? Asking distant cousins for DNA? In my family we have a DNA contradiction – two cousins (at about the 4th cousin level) took a Y-DNA test, but did not match and had two completely different haplogroups.   Since then, I have traced living cousins with the same surname – the only ones left living that I am aware of – and I would love to pay to have them tested to see if we can figure out where the non-paternity event (NPE) or other event happened (i.e. this new cousin could help us determine which line the NPE occurred in depending on which cousin he matches).   Does anyone have a sample letter, or know the best way to go about asking a distant cousin for DNA?   I have their address and am planning to write a letter, but I can't imagine receiving such a letter, so I am not quite sure how to ask without scaring them off. Reputation and inbox notification shows up again after refresh Is it a good concept to use multiple inheritance or can I do other things instead? Public-key cryptographic systems are often implemented on top of an existing problem. For example, RSA is built on top of integers and the difficulty of factoring. ECDSA/ECIES are built on top of elliptic curves and the difficulty of the discrete logarithm.  My question is, what are, mathematically speaking, the exact primitives needed to implement public-key based signature and encryption cryptography in a generic way? How to get the length of longest string in an array Proof maximum number of equidistant points for a given dimension The question is quite straight forward, on $\mathbb{R}^2$ the maximum number of equidistant distinct points that can be layed (using the euclidean distance) is 3, forming thus an equilateral triangle. On $\mathbb{R}^3$ the maximum number of points is 4, forming thus a tetrahedon. My question is quite simple, is the maximum number of points that can be layed in $\mathbb{R}^n$ equal to $n+1$? How to filter a list by hour Solve this integral for free WiFi How do I install Adobe Flash player? The equation $X^{n} + Y^{n} = Z^{n}$ , where $ n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are intergers. Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$ How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$  for any two integers $a$ and $b$?  Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors between them resulting in them becoming coprime. However, how would you prove this rigorously and mathematically?