Why can't voltage in a capacitor change instantaneously?
Why is the voltage across a capacitor and the current across an inductor continuous?
Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$ A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$ 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? Pluralization rule for "five-year-old children", "20 pound note", "10 mile run" Why are year, pound and mile in the singular form in the phrases below?   five-year-old children  20 pound note 10 mile run   Is that because they're acting as adjectives, which are always invariable in English?  Is it incorrect to say...   five-years-old children? 20 pounds note? 10 miles run? Using matplotlib in pygame 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). Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie.  length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal to $|b-a|$.  It seems intuitive, but I don't have any idea how can I show this in the formal way. Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation Switching to HTTPS - redirect question In Europe usually you don't need to go through immigration to change planes (unless you are entering/exiting Schengen). Is it the case in the US as well? May I catch a connecting flight on one of the US airports if I don't have US visa? Is there any way you could revive Guest Sessions in a future LTS distro?  Guest sessions are good security - they give me just enough power to do what I need to do on a daily basis without risking my system. In short, Guest sessions save me from myself.  I had upgraded from 16.04 to 18.04, gotten a problem I couldn't fix and was all to happy to wipe and rebuild my hard drive and return to 16.04.  I can't be the only one who feels this way. I want to determine whether a certain n-dimensional, simply-connected vector field is conservative. I can very easily calculate its Jacobian in numerical form.  In 3 dimensions, the problem is easy: the Jacobian is a 3x3 matrix, and the curl can be obtained from its elements:  [jacobian[2,1] - jacobian[1,2],  jacobian[0,2] - jacobian[2,0],  jacobian[1,0] - jacobian[0,1]]  A curl of zero would mean that the vector field is conservative.  But what if my Jacobian has 100 elements? The curl is only defined for 3 dimensions, but is there a formula for some high-dimensional generalization? In Random forest algorithm, Breiman (author) constructs similarity matrix as follows:   Send all learning examples down each tree in the forest If two examples land in the same leaf increment corresponding element in similarity matrix by 1 Normalize the matrix with number of trees   He says:     The proximities between cases n and k form a matrix {prox(n,k)}. From their definition, it is easy to show that this matrix is symmetric, positive definite and bounded above by 1, with the diagonal elements equal to 1. It follows that the values 1-prox(n,k) are squared distances in a Euclidean space of dimension not greater than the number of cases.    In his implementation, he uses sqrt(1-prox), where prox is a similarity matrix, to convert it to distance matrix. I guess it has something to do with the "sqaured distances in a Euclidean space"-quoted above.  Can somebody shine a little light on why it follows that 1-prox are squared distances in a Euclidean space and why he uses squared root to get distance matrix? Ubuntu 12.04 software upgrade does not show 12.10 upgrade Why can't I tame my ocelot? I'm trying to make a cat house for my zoo on Minecraft. I have some untamed ocelots and I also want a few tamed ocelots. Fish only puts them in breeding mode. What exactly is "base ISO" and how do I find what is base ISO on my camera? If I understood correctly, base ISO is in full stop steps from the lowest possible ISO setting on my camera. For example if the lowest setting on my camera is ISO 100, than the following table would be base ISO:    100, 200, 400, 800, 1600, 3200, 6400   While steps in between like   125, 160, 250, 320, 500, 640, 1000, 1300, 2000, 2500, 4000, 5000   would be third-stop "pulled" values, which result more noise than perhaps higher base ISO values (for example 2500 is worse than 3200).   Did I get this right or missed by a mile, what did I miss? Do non-base ISO values generate more noise than next higher base-value ISO stop? How do I find what are my base-value ISO stops (is it the lowest ISO value multiplied by 2n)? I'm assuming that the answer to the question in the title is a resounding yes. Since Baryonic matter and dark matter interact via gravitational forces.  If this is the case how is information not lost if, as is presumed, dark matter does not interact via the electromagnetic force? Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact? Non superscript citations for figures Is it possible to reference figures like so   taken from \emph{ref. \cite{reference}}   with non superscript numbers, while leaving the rest of the references in the bulk of the document untouched? This is using BibTeX and the . How do I quickly rename a MySQL database (change schema name)? The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than .