What is the Difference between Lorentz Invariant and Lorentz Covariant? Like my title, I sometimes see that my books says something is Lorentz invariant or Lorentz covariant. What's the difference between these two transformation properties? Or are they just the same thing? If they are different, is there a term called "Lorentz contravariant" like tensors?
Definitions and usage of Covariant, Form-invariant & Invariant? Just wondering about the definitions and usage of these three terms.  To my understanding so far, "covariant" and "form-invariant" are used when referring to physical laws, and these words are synonyms?  "Invariant" on the other hand refers to physical quantities?  Would you ever use "invariant" when talking about a law? I ask as I'm slightly confused over a sentence in my undergrad modern physics textbook:     "In general, Newton's laws must be replaced by Einstein's relativistic laws...which hold for all speeds and are invariant, as are all physical laws, under the Lorentz transformations." [emphasis added]      ~ Serway, Moses &amp; Moyer. Modern Physics, 3rd ed.   Did they just use the wrong word?
Efficient computation of $\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$ I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression?  $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$  I've noticed $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor = \left\lfloor\frac{1}{2}\sum_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor} \left\lfloor\frac{n}{k}\right\rfloor\right\rfloor + \sum_{k=1,\ odd(k)}^n \left\lfloor \frac{n}{k}\right\rfloor$$  But I can't see an easy way to let the second term enter in a recursion. I look for similar questions to this one but my problem is different. I want to join this community:    so I click in     but I got this error:    but then, I found that the community is already in list of communities:    However, if I click on the community, I'm an anonymous member     and even I have the option to join again     but if I click on it, the problem starts again :( what can I do? How did "When" become the customary answer to "say when"? When a waiter at a restaurant comes by with pepper or Parmesan cheese, he says, "say when" and starts putting it on your food.  Many people will say "OK" or "that's enough," but it seems that the customary answer is "when".  How did this become the customary answer?  Did everyone decide to make the exact same joke, until it became commonplace.  I thought it was funny when I was a kid, but even now as an adult, I still say it.  For evidence that this is indeed commonplace, see .  The answer "when" is used and is external to the joke of the strip. But even so, I'm curious as to just how commonplace this really is.  I see it a lot in the US.  Is it used elsewhere in the world?  (by the way, I'm new to asking questions on this site, and I have no idea what tags to put on this question) "SELECT *" why is it an antipattern On multiple questions here and on stack overflow i saw people saying in the comments and answers that select * from table is almost always an antipattern without any explaination why. Althrough I can sort of deduce why it is an antipattern. I may be looking over a detail that someone else with better understanding of the issue noticed.  So here is my question, why people say that select * is an antipattern. In .NET why is String.Empty read only instead of a constant? I'm just wondering if anyone knows what the reasoning was behind that decision. Accessing private member variables from prototype-defined functions How would I give a simple example of a function $f$ differentiable in a deleted neighborhood of $x_0$ such that $\lim_{x\to x_0}f^\prime(x)$ does not exist? I cannot seem to think of an example.  A delete neighborhood is an open interval about $x_0$ which does not contain $x_0$. So, $(x_0-\delta,x_0+\delta)-\{x_0\}$ for some $\delta&gt;0$.  How would something be differentiable in a deleted neighborhood if at the point of the derivative, the limit does not exist. Presumably, the derivative ends up looking something like $\lim_{x\to x_0} \dfrac{1}{x}$, if it does not exist. Is there a way to convert number words to Integers? How to fake a package version installed? Being an early adopter and using 11.04 daily builds I sometimes meet a situation when some packages require earlier versions of other packages than those I have installed. Usually newer (but refused) versions of needed packages provide all the functionality of their older versions.  If I use dpkg -i --force-depends-version to install dependent packages everything works fine, I've never seen a problem.  Instead of manually downloading and installing every package, can I just make the system pretend there is a specific package version installed when there is a newer? Or is there a --force-depends-version key equivalent for apt-get? I have a random sample of Bernoulli random variables $X_1 ... X_N$, where $X_i$ are i.i.d. r.v. and $P(X_i = 1) = p$, and $p$ is an unknown parameter.  Obviously, one can find an estimate for $p$: $\hat{p}:=(X_1+\dots+X_N)/N$.  My question is how can I build a confidence interval for $p$? Why does Microsoft Word 2010 show an error for the following sentence?     1.&nbsp;Where should this car be parked?   Word 2010 also suggests changing the sentence to     2.&nbsp;Where this car should be parked?   Is the sentence (1) wrong? If yes, why? What about sentence (2)? Show that $\mu (E \setminus A)>0$ if $A$ is a nonmeasurable subset of $X$ and $A \subseteq E$, $E$ is measurable. This question was asked , but the question wasn't asking for an answer, just a clarification on the proof by contradiction they already had, so I'm reposting it to ask for help arriving at the answer.  Because the asker above said they proved it by contradiction, that is what I'm attempting. But I'm having issues wrapping my brain around the concept of set measurability. So let's give it a good college try with what I can understand.  For clarity, $\mu$ is an outer measure.  A is nonmeasurable, so $\exists B \subseteq X$ such that $\mu (B) \ne \mu (B \bigcap A) + (B \bigcap A^c)$.  E is measurable, so $\mu (B) = \mu (B \bigcap E) + (B \bigcap E^c)$.  This essentially means $\mu (B) \ge \mu (B \bigcap E) + (B \bigcap E^c)$ (Since monotonicity is alredy implied by the definition of outer measure).  So we can attempt a proof (or the start of one) by contradiction.  Assume $\mu (E \setminus A)=0$. So $E \setminus A$ is a null set.  So $\mu (B) \ge \mu (B \bigcap E) + (B \bigcap E^c)$. I assume I'm supposed to show a contradiction using that A is nonmeasurable. I'm unsure how.  Another way I think could work would be to use: $\mu (E \setminus A)= \mu (E \bigcap A^c) = 0$ and arrive at a contradiction given that A is nonmeasurable, but again, I'm unsure how.  I just need some direction from here. Thank you in advance for any advice. It happens sometimes to receive or see a very good answer that arrives very late and that would deserve more upvotes. What can be done to give a boost to them? I would happily spend some of my reputation points to be able to give them such boost. This is not merely a question of reputation, but the idea is to let the answer &quot;bubble up&quot; faster so that it can be seen easily by other people to look into question. I hope I explained my question so that is understandable. Most of you got the point and there are some interesting proposals like the one that proposes to put randomly an answer at top. But I do have not enough experience to say if there is one that could solve the issue or not. I would be curious to know what Jeff thinks about it, if it feels there is something that can be improved or not. ModelBuilder not adding output feature class to Table of Contents? I'm trying to create a tool in ModelBuilder that will take a new point from a user, create a feature class for this point and display it on a map.  What I've done is:   I've created a feature class for a point-like feature in ArcCatalog. I've prepared a really simple model in ModelBuilder, that starts with a feature set variable (which takes the feature class as a schema and symbology file). The variable is marked as a parameter. Next I'm using the Copy Features tool that takes the variable as input and saves it as a feature class in my scratch workspace. The output feature class is marked as a parameter with the 'add to display' option.   After the module is run I get a shapefile in my scratch workspace. I can add it manually to TOC and it contains the selected point. However no matter what I do, I can't display it in TOC automatically after the model completes.  There is a known issue described in several places that doesn't allow to display newly created layers in TOC when model finishes with the 'add to display' option. However there's a quick workaround, which is to mark the output layer as parameter. In my case it doesn't work.  I'm using ArcGIS for Desktop 10.1 on Windows 7 x86_64. The difference between a poor or meh question and a stellar question can often simply be someone understanding it and providing it a great answer. I can't begin to count the number of times I've justified re-opening a question as a moderator by saying:     Look at the answer it got, though. This isn't something we want deleted, this is something we want fixed, because it's obviously valuable. Someone with domain knowledge can easily edit that question based on the answer it received.   This doesn't only apply to borderline-poor or poor questions – sometimes a good edit means the difference between 100 and 10,000 people finding something through typing things into Google. Good titles are hard.   We think we've come up with a way to provide incentive for good, opportunistic edits to happen more often, ideally without questions needing to go through closure or collecting a centimeter of dust before they do. What if we simply rewarded folks that answered and edited questions with a special, relatively difficult to earn badge?  Let's say that you answer something, and:   You edit the question 12 hours before or after answering it. This allows you to edit now, answer later &mdash; or answer now and edit later, when you have the time. Your edit isn't rolled back, or outright rejected if it was a suggested edit The question is not closed for any reason, even simply being a duplicate Your answer has a score of 1 or higher   &hellip; then you've done something that we probably want to recognize. You understood something, you provided the knowledge that you have, and then you provided an edit to make sure that more people in need of this knowledge can find it, while raising the overall quality of the site.  Here's the proposed tiers for the badge (within the above stated criteria):   Explainer (Bronze): Answered &amp; edited 10  questions Refiner (Silver): Answered &amp; edited 50 questions Illuminator (Gold): Answered &amp; edited 500 questions   We think this is going to be pretty hard to game. You can't go leaving streaks of terse snark all over the place on questions you know are going to be closed anyway, and junk edits aren't going to fly.  We're talking about rewarding folks that take a tiny bit more time out of their day just to write a more descriptive title so that folks can find the awesome answer that they wrote. Just doing that alone can make a big difference. Sure, it's a tad hard to get at the gold level, but it's supposed to be. If you routinely edit questions that you answer and have done so for some time, you'd probably get it the day we roll it out.  While this is technically something that we came up with on the quality initiative ( | ) &mdash; this badge could be earned on all of our sites.  What do you think? What did we miss? What could be better about the idea? how to disable system apps without root? How to disable system apps without root?  My phone model is vivo x5 max+ and android 4.4.4  I have tried Debloater but it failed to connect (I am sure I've allowed USB debugging). Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$ I'm trying to solve this limit     $$\lim_{x\to 0} \frac{\sin x-x}{x^3}$$   Solving using L'hopital rule, we have:  $$\lim_{x\to 0} \frac{\sin x-x}{x^3}= \lim_{x\to 0} \frac{\cos x-1}{3x^2}=\lim_{x\to 0} \frac{-\sin x}{6x}=\lim_{x\to 0} \frac{-\cos x}{6}=-\frac{1}{6}.$$  Am I right?  I'm trying to solve this using change of variables, I need help.  Thanks  EDIT  I didn't understand the answer and the commentaries, I'm looking for an answer using change of variables. 17.10 Super key to open applications I've upgraded my laptop from 17.04 to 17.10 and the super key no longer brings up the applications menu. Does anyone have any idea how I can change this?   Shortcuts menu doesn't let me change Super + A to just Super.  Edit: I'm running the standard desktop version of Ubuntu 17.10 I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff. Should a function have only one return statement?