Transformation between a dynamical system to a Hamiltonian system
When can an autonomous system be written using a Hamiltonian?
Are hand motions required to use the Force? In almost every canon (that is, visual) source, Force users typically wave a hand to invoke the Force to move or manipulate objects, people and thoughts. I get out-of-universe this is a visual cue that the Force is being used, but in-universe is it strictly necessary?  (The only exception I can think of is when Luke is training on Dagobah and is balancing rocks while standing on one hand with Yoda on his foot. He doesn't appear to be waving his hand to move the rocks.)  Is this addressed anywhere in-universe, even in Legends? Why doesn't Volumetric follow the shape of Mesh in Eevee? So I was in the middle of making a ball-shaped smoke, but the volumetric effect didn't work following the basic shape. It remains cube in Eevee Blender 2.8; while in Cycle it works, but in Cycle it takes thousands of years for Render. maybe that's an idiot question. Given a finite field $k$ and some irreducible polynomial $f(x) \in k[x]$, then $k_f \cong k[x]/(f(x))$? I know that it's true if $k$ is the prime field and I think that the statement is not true for general finite fields, however I could not find any counter example.  Thanks in advance. How to find out where an error occurs? Add high-resolution-graphics and / or .svg-icons In fact, I would like to ask more general question -- &quot;what does write permission for a directory allow you to do exactly?&quot; -- but let's approach it with a concrete example. It is a long question, if you are in a hurry read the bold -- it should cover the main part. Different sources (, , ) say something similar to the following on the directory permissions:  r, read -- reading of the directory's content (filenames inside) w, write -- changing the directory's attributes (e.g. modification time) and creating/renaming/removing entries inside x, search -- accessing files inside, you have access to the inode of the file, hence you can reach it's actual content  My problem is with the description of w. What directory's attributes does it give you access to? I cannot create/rename/remove a file inside a directory with only write permission: I make a directory (tdir/) and a file inside (afile), chmod -x-r tdir/, mv tdir/afile tdir/af, rm tdir/afile, touch tdir/newfile -- all fail with permission denial, unless I set x permission to the directory as well. And x alone doesn't give you the permission to create/rename/remove files inside the directory. In order to do that you need both x and w. But touch tdir does change the modification time of the directory with w only. I would rephrase the sources above this way for the compliance with the issue: a directory's r allows you to see the filenames inside, but no access to the actual file (to the inode); x gives you access to the inodes of the files (which means you can see their permissions and, according to it, have access to the contents), but you still cannot change anything in the directory; directory is actually some sort of a file and to change something in it you need the w permission. Thus, when you are changing something in the directory you need w permission. If your change requires inodes of the files in the directory -- you need x as well. It explains why you cannot remove a file inside a directory with w only: when removing a file you need to reduce the link count of the inode by 1 -- you need to know the inode -- thus you need x for the directory. But why do you need x for creating (you could ask the system to create a file without exposing the inode?) and renaming/moving the file (when you move a file you don't change it in any way, you only change the records inside the directories and their inode counts?)? Maybe it is just an implementation thing? I.e. indeed you don't need the inode for renaming/creating files -- you need only filenames and w permission; but inode and filename constitute one record in the directory; thus changing the filenames = changing the records = kind of accessing the inodes. And also what attributes do directories have besides modification time, permissions and files records? What else in the directory can you change with w only? CBC-MAC Padding weaknesses I'm trying to wrap my head around how different padding methods affect the strength of CBC-MAC's. Suppose a message is split into different blocks, m1, m2, m3 etc. and the last block is padded with 0's to fill it if needed.  If the message is intercepted, why can an attacker easily create another message with the same MAC tag for the same key as the message that was intercepted?  If another message is intercepted that does not contain any padding (so it completely fills the last block), how does this make it more difficult for the attacker to create a tag that matches? Does the message need to meet any conditions for the attack to work?  If this padding method is not secure, what is a better method to secure against these attacks? Would it be more secure if the padding was just random? Not sure how that would work. I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) 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? How do I check whether a file exists or not, without using the  statement? Why is $ 561 = 3*11*17 $ the smallest Carmichael number? A composite integer n is a Carmichael number if the only Fermat   witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not   coprime with $n$. The smallest example of such a number is $561 = 3 ·  11 · 17$   I think I understand the definition, but I don't quite understand why is the smallest Carmichael number is the above mentioned number. Please clarify. Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function? Let $a &lt; b$ and $f:[a, b] \rightarrow \mathbb R$ be a Riemann integral function. Suppose that $f(x) \geq 0$ for all $x \in [a, b]$. I want to show whether $\int_a^bf(x) = 0$ implies $f(x) = 0$ or not, but I'm not sure where to start. I don't know much about Riemann integral functions in general. I only know that $$\underline \int_a^b f= \overline \int_a^b f.$$ Also, I know the fact that if $f$ is continuous, then $f(x) = 0$ for all $x\in [a, b]$. Can anyone provide any hints for me, because I'm really confused about this whole thing. Add Point to Plot3D Stack Overflow is a site of passers-by They says that Stack Overflow is a community-driven web-site. But, as a matter of fact, it is not.  There is no community at all. It's more like an imageboard, as a number of participants makes them pretty much like anonymous to each other despite of registration.   Stack Overflow is a site of passers-by.  Passers-by who want to solve their silly typo problem and passers-by who eager to share their vast knowledge in spotting silly typos. And - oh - sometimes to perform a civic duty and vote to close a question they have no idea what to answer to and thus calling it "not a real question".  So, as a funny consequence, whatever question can get   answered upvoted downvoted closed deleted   depends not on its quality but only on the fluctuation happened to "community" at the given moment, at a pretty random rate.  It is not only quite discouraging. It makes the very idea of collecting and improving knowledge quite impossible. Another layman blackhole question, pulling one end of a string out from behind the event horizon No long explanation is needed,  What would happen if I were to allow one end of a rope to fall past the event horizon of a black hole while I held the other end?  Would I be able to pull it out? Would the rope feel extremely (infinitely?) heavy? How to visualize the gradient as a one-form? I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I usually visualize gradients as vector fields while I visualize one-forms with level sets. How to visualize the gradient as a one-form? Why are extra dimensions necessary? Some theories have more than 4 dimensions of spacetime. But we only observe 4 spacetime dimensions in the real world, cf. e.g.  Phys.SE post.   Why are the theories (e.g. string theory) that require more dimensions taken seriously by scientists?  Is there any proof that these extra dimensions exist? Is there a simple layman's explanation for the need [or strong hint] for extra dimensions? Are global variables bad? Can I still use a usb flash drive after spilling water on it? So a few months ago I accidentally spilled a little water on a few of my usb flash drives. I really want to test and see if they work, but before I do, I just want to make sure that there's no risk of damaging my computer. I'm pretty sure it's fine, but I just got a new laptop and would hate to ruin it over something stupid. $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$ How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.  What i thought of is to consider $$(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$$ and if we suppose that $n$ is odd then we have, $$(a-b)^{n} \equiv a^{n} -b^{n} \ (\text{mod} \ n)$$ and since $n \mid (a^{n} - b^{n})$ we have $$(a-b)^{n} \equiv 0 \ (\text{mod} \ n) $$  I think i am far away from the conclusion of the problem, but this is what i could work on regarding the problem.