We know that at point A and B there is same voltage.Then why electron moves from one point to the another one?
In an ideal circuit, How can there be a current b/w points a &amp; b, when there is no potential difference and thus no electric field between a &amp; b? If there is no current, then where does current across the resistor come from, because that means no charge is coming from the battery (?).
Why is conservation of angular momentum considered a law? We know that angular momentum is defined as the cross product of position and linear momentum. By taking the time derivative, we can deduce that the time rate of change of the angular momentum equals the net torque.  Then why is the conservation of angular momentum considered a law, when we can easily show it is conserved when no net torque is applied, by using Newton's laws?  It truly confuses me. Which tag is used in the title of a question?   It does not seem to be the most popular It does not seem to be the first in the tagging order:   This question   is prefixed with NGINX (which is actually completely fine)  And it is tagged with "php nginx fastcgi magento" in the order of popularity.  Can we choose the tag to use in the title manually? Why do we use the Lagrangian and Hamiltonian instead of other related functions? There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to first two, but we don't use them. Every where they write that using them can cause problems. But why? Fermat's Last Theorem with negative exponent Why are the Degrees of Freedom for multiple regression n - k - 1? For linear regression, why is it n - 2? I'm reading a textbook and I see this question:    So there are 200 women, and the DF is 196, implying that the equation for DF is $n - k - 1$. There are 3 variables: bp, age, and type so $k == 3$. What's the intuition behind this?  Also, why is the degrees of freedom for linear regression n - 2? What's this I hear about First Edition Unix being restored? that the source code of First Edition Unix had been restored from tapes.  The response was amazement and a request for more information.  What have people done?  Who did it?  When?  And how? Number of equivalence relations on a set What does "rigorous proof" mean? I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. NSString tokenize in Objective-C Good Afternoon all,  I am trying to typeset a symbol that I have been using to represent the integral of a regulated function. This symbol is the standard integral symbol, with the letter "R" superimposed on it.  I have tried using \mathllap, \mathclap and \mathrlap (as suggested ), however none of these have managed to give the desired effect. When I tried doing it without the limits of integration, it got close to working, however then I was unable to have the limits to the right hand side of the symbol:  \mathop{\mathrlap{\int}R}_a^b f   Any help would be greatly appreciated, thank you in advance!  Minimal Example:  \documentclass{article}  \usepackage{amsmath,mathtools}  \begin{document}  Take the regulated function $f$, we want to compare the Riemann integral, $\int_a^b f$ to the regulated integral $\mathop{\mathrlap{\int}R}_a^b f$ by taking sequences of step functions.  We will prove that $$ \mathop{\mathrlap{\int}R}_a^b f = \int_a^b f $$  \end{document} How long does it take for chat profiles to update? Following scenario:   user registers, has 1 point user visits chat, notes he cannot post because of only 1 point and needs 20 user goes back to SO, answers a question. gets enough points to chat user goes back to chat, but his profile still shows 1 point   Now my questions:   How long does it take for the profile on chat to update? Is there a way to force updating?   Update: Logging in and out did not help. What is syntax highlighting and how does it work? I noticed that sometimes my code gets highlighted in different colors when rendered.   What is syntax highlighting? How does it work? Why isn't my code being highlighted correctly? How do I report a bug or request a new language? How do I use syntax highlighting? What languages are currently available on Stack Exchange? Add self-answer delay for all users TL;DR:  I believe there are still some issues with self-answers, particularly near-immediate self-answers, which are not addressed by the current system.  This could be remedied by imposing an automated restriction on how soon self-answers will be displayed in the thread.  Allow users to submit self-answers at any time, but do not display them until certain reasonable conditions (a relatively short timeout, or the posting of a competing answer) are met.    History  This thread covers a good deal of the concerns surrounding self-answers, and some mechanisms in place to prevent "bad" self-answers from new users and other abuse of the self-answer system.    As I understand it, the following restrictions apply:   Users with &lt;100 rep cannot self-answer within 8 hours of posting a question. Self-answers cannot be accepted within 48 hours of posting. Self-answers do not provide rep from being accepted. Accepted self-answers are not pinned to the top of the thread.   Problem  This does a good job of preventing users from posting answers which should probably be comments (or not posted at all), allows superior answers to rise above accepted self-answers, and prevents some other abuses of the system.  However, there is a less tangible element of self-answers which this does not address:    Self answers are a disincentive to other users who may otherwise be willing to post more accurate, detailed material stemming from broader or deeper knowledge and/or experience than the asker has.  When someone posts an answer to their own question, it presumably means that the problem which spawned the question is solved to their satisfaction.  This is fine for the purposes of sharing information with the community, but it deters others from providing solutions to the question which may be more complete or accurate since there is not so much likelihood that the self-answerer will accept a competing answer.  Worse, in the case of near-immediate self-answers, this can remove the incentive for other users to gain reputation via up-votes because the asker has an unfair head-start.  Unless everyone is required to view new answers to questions where they have already voted on an answer, most people probably vote on a good answer and never look at the question again.  This can put (potentially) better answers at a disadvantage even when the head-start is within an hour.  Suggested Solution  Add a delay to the posting of self-answers.  The delay should be the least of these:   Until a competing answer has been posted. A reasonable amount of time to allow competing answerers a chance to post, without preventing self-answers outright.  (Suggestion: 2 hours)   To better facilitate this, and avoid completely turning people off from appropriate self-answering, I suggest the delay be handled through automation more than user limitation.  That is, the user should be allowed to submit the self-answer at any time.  But the answer will not actually be displayed to other users until one of the above conditions have been met.  This could also be applied to the existing 8-hour delay for low-rep users.  Example:  T:0 - A user finishes resolving a tough problem, just before he would have gone to SE for assistance, and decides to share his new-found knowledge with the community.  He posts a question describing his problem.  T:5 minutes - The user then posts a self-answer.  There has not yet been another answer posted to the thread.  The user is given a notice describing the restrictions in place regarding self-answers, and the answer is submitted into the database but not displayed to other users.  (Similar to how suggested edits are only displayed to the user suggesting them, until they are approved.)  T:1 hour, 15 minutes - Another user sees the unanswered question, and posts their answer.  Their answer is displayed immediately, and the self-answer is now also displayed.  T:2 hours - Had the second answer not been posted, the self-answer would automatically be displayed after this point. Blowing your own sail? Why a line is said to have infinite number of points? U.K visitor visa with very old criminal conviction - do I disclose it? I have been travelling to the U.K for 25 years at least 8-10 times a year,  receiving 5 year visas, never breaking the stay period, with no problem. I've gone through immigration with no problem till my last time in London. I was stopped for eight hours and questioned about a run with the law while I was in college. This was in the US, I was around 20, it was for a credit card use. I received a conditional discharge / probation conviction which was discharged one year later. It was stupid, I was young and in college, more than 28 years ago.  After being questioned about the issue in the US, I was allowed in. They were very polite and professional.  I was also asked if I ever changed my name: I did not change it, but one letter was added in the middle of my first and last names around 20 years ago as our names are translated from Arabic to English and before the digital biometric passports, hand written passports some times varied. It was a common thing at the time.  Now I am applying for my new 5 year visas and I intend to answer truthfully  about previous conviction (previously I answered no because my conviction was spent and I expunged my record.) I want to answer yes and explain it. Is that the right thing to do?  Should I answer yes to the question about having a different name even though it was something as simple as spelling Bryan instead of Brian or Teresa and Theresa -- not really changing it and never thought of it as a change.  I have an impeccable credit rating in the Uk and the last 28 years have never received a moving violation or ever had a run in with the law.  I made one mistake when I was very young but I am  very worried about how to go ahead. I believe honesty is the best approach, is it? Wordpress speed slow after .5 million single image post via wp-cli Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() Shortcuts in Objective-C to concatenate NSStrings Are there any shortcuts to (stringByAppendingString:) string concatenation in Objective-C, or shortcuts for working with NSString in general?  For example, I'd like to make:  NSString *myString = @"This"; NSString *test = [myString stringByAppendingString:@" is just a test"];   something more like:  string myString = "This"; string test = myString + " is just a test"; A problem about an $R$-module that is both injective and projective. Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$.   This is exercise 7.52 of Rotman's Advanced Modern Algebra. Using theorems before exercises, because $M$ is injective and $R$ is a domain, I conclude that $$\forall m\in M ,\forall r\in R\ (r\neq 0) ,\exists {m}'\in M \Rightarrow m=r{m}'$$ and also because $M$ is projective there is a surjective $\psi$ from free $R$-module $F$ with basis $\left \{ e_{i} \right \}_{i\in I}$ to $M$ and thus we can conclude that for every $m\in M$ we have $$m=\sum r_{i}\Psi (e_{i})$$ now I don't know how should I use these together.  The idea of what is happening or a suggestion or a hint will be great.