What is someone called who places in the top 3 of a competition?
What is a word for getting an award in a competition or being one of the top three?
I want to ask about two sentences, and what would be the correct way to form a question:   It's very important that we eat healthy food, ...........?  Which would be the correct question tag: (isn't it) or (don't we)? It's a must that all applicants follow the rules of the competition, ..........?  Which would the correct question tag: (don't they) or (mustn't they)?   Please could you tell me the correct answers, and, more importantly, why those are correct? Why is XOR preferred over XNOR? The XOR operator seems to be a frequently used building block inside many cryptographic primitives.  As far as I can see, its most desirable properties seem be that for the XOR of two bits $a\oplus b=c$:   The information is preserved.  Either $a$ or $b$ can be recovered from $c$ and the other bit. The information is hidden.  An adversary can not learn anything about $a$ or $b$ from $c$ alone.   Neither AND nor OR has these properties but XNOR does.  It seems like any cryptographic application using XOR could be constructed equivalently with XNOR, without incurring any loss in security.  One possible reason to prefer XOR might be that XOR is equivalent to addition modulo 2, whereas XNOR would be addition plus 1 modulo 2, but this seems to be entirely mathematical, and I'm not sure if it actually results in any additional hardware complexity in practice.  Is the choice of XOR then an arbitrary one, made for historical reasons, or is there any tangible underlying motivation for using it over XNOR? Plot construction for a velocity vs position graph. Interpolation for phase-diagram construction? When wouldn't I use LASSO for model selection? Creating "Inner Buffer" using ArcGIS for Desktop? I am currently searching for how to create an "Inner Buffer" in ArcGIS for Desktop?  I have a lot of polygons (houses) in one feature class, and one big polygon (in another feature class).   I want so select all houses contained in that one big polygon within a distance to the outer border (e.g. 1000 meters).  Select Layer By Location could be one possibility to solve this but I can't figure out how. Heres a picture. im making a body, and it seems to have flash in the middle. Does infinity exist in physical systems? Sometimes people fail at asking a question by being too broad, unclear like . So I'll take a stab at what I believe to be the same question, but more concise and clearly stated:     Does infinity exist in either terms of structure, parameters, or   measurement in any physical systems?   Can infinity have any real connection with reality?  Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?  I have heard that if you model a physical system (), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong. Find the number of dissimilar terms in the expansion of $$\bigg(x+\frac{1}{x}+x^2+\frac{1}{x^2} \bigg)^{20}$$  Could someone give me some hint to proceed in this question. I used $x+1/x=t$ and wrote expansion as $(t^2+t-2)^{20}$ but even this does not convey much information. Please help. How do I interpret the Mann-Whitney U when using R's formula interface Sound of a next scene starts one moment earlier than video How to make a meta box field a requirement Raised index of partial derivative I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means $$\partial_\mu\partial^\nu \phi, \tag{$\#$}$$ where $\phi$ is a scalar field. On the one hand $\partial^\nu=g^{\lambda\nu}\partial_\lambda$ where $g_{\mu\nu}$ is the Minkowski metric, and hence we could write explicitly $$\partial_\mu\partial^\nu \phi=\sum_{\mu,\nu,\lambda}g^{\lambda\nu}\partial_\mu \partial_{\lambda}\phi=\sum_{\mu,\nu}\partial_{\mu}\partial_{\nu}\phi=\partial_\mu\partial_\nu\phi. \tag{$*$} $$ On the other hand, we may think of $\partial_\mu\partial^\nu=g(\partial_\mu,\partial^\nu)=\delta_\mu^\nu;$ so that $\partial_\mu\partial^\nu\phi=\phi?$ Maybe? I am actually not sure of what this would mean. I am really confused. Any help is appreciated. Edit: To give context of where this expression comes from: I was computing the Lagrangian $$\mathcal L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu \phi) $$ considering an infinitesimal spacetime translation $x^ \mu\to x^\mu-\alpha a^\mu$. The scalar field thus transforms like $\phi(x)\to \phi(x)+\alpha(\partial_\mu\phi(x))a^\mu.$ Plugging thins into the Lagrangian yields the term I am referring to. Edit 2: The change in placement of indices are actually my doubts. I try to elaborate. I do not have any background in using indices to talk about tensors. I am used to interpret the expressions $\partial_\mu$ as the local vector field defined in some chart (local coordinates). I think about vector fields $X$ as abstract section of the Tangent bundle, which restricted to local coordinates can be expressed as $X=X^\mu\partial_\mu$. In the context of QFT, as far as I understand, the symbol $\partial_\mu$ denotes  $(\partial_t,\nabla)$ in the local coordinates $(t,x,y,z)$. So that $\partial_\mu\phi=(\partial_t \phi,\partial_x \phi,\partial_y\phi,\partial_z\phi)$. This was supposed to be my justification on why I wrote the summation on $\mu$ and $\nu$ in $(*)$, but now I note that this only applies when $\mu$ or $\nu$ appear twice, indicating the scalar product; which leads me to the last remark. I think of $g_{\mu \nu}$ as the component of the matrix $$g=\begin{pmatrix} 1&amp;0&amp;0&amp;0\\ 0&amp;-1&amp;0&amp;0\\ 0&amp;0&amp;-1&amp;0\\ 0&amp;0&amp;0&amp;-1\\ \end{pmatrix}$$ which represents the pseudo-Riemmanian metric, which by definition acts on tangent vectors, i.e. linear combinations of the $\partial_\mu$ applied to a point. This is where my doubt comes, in which was the right way to interpret the notation; in particular what is the expresion $(\#)$ in explicit coordinates? Creating a default preamble Since I am using (almost) the same set up for my tex documents, I wonder if there is a way to create a preamble with my default packages and to recall it anytime I want to create a tex document.  If yes, how I can call this "default" preamble?  Thanks in advance The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.  So here is the puzzle:    The triangle $ABC$ is known to be isosceles, that is, $AC=BC$. The labelled angles are known to be $\alpha=\gamma=20°$, $\beta=30°$. The task is to find the angle labelled "?".  The only solution that I know of is to use the sine formula and cosine formula several times. From this one can obtain a numerical solution. Moreover this number can be algebraically shown to be correct (all sines and cosines are contained in the real subfield of the 36th cyclotomic field). So in this sense I solved the problem, but the solution is kind of a brute force attack (for example, some of the polynomials that show up in the computation have coefficients > 1000000). Since the puzzle originates from a book that deals only with elemetary geometry (and not even trigonometry if I remember correctly) there has to be a more elegant solution. Let $b_n$ be a sequence which decreases monotonically to zero, prove $\sum b_nz^n$ converges for $|z|\leq 1$ and $z\neq 1$.  So I was able to prove it converges on the open unit disk in the complex plane, but except for a few special points on the boundary, such as $-1$ and $i$, where I was able to use the alternating series test, I can't seem to prove it converges in general on the boundary minus $z=1$.  I feel I have to somehow use the fact that all the numbers on the unit circle except $z=1$ rotate around and around as you raise them to higher and higher powers, thus you get the real and imaginary parts of your series to alternate between sets of positive and negative terms, but in a very irregular way for most initial angles.  I tried to simplify the problem by breaking the series into its real and imaginary parts, and then trying to prove they each converge separately, but I still can't prove either converges:  $$\sum_{n=0}^{\infty}b_ne^{n\theta i} = \sum_{n=0}^{\infty}b_n\cos(n\theta) +i\sum_{n=0}^{\infty}b_n\sin(n\theta).$$ Do indexes work with "IN" clause If I have a query like:  Select EmployeeId  From Employee  Where EmployeeTypeId IN (1,2,3)   and I have an index on the EmployeeTypeId field, does SQL server still use that index? How to avoid using sudo when working in /var/www? What happened to the voting buttons? Starting from yesterday, the up-/downvote buttons seem to have moved from the left of a post towards above (in the middle) of the bodies of all questions and answers. Here is a screenprint of a sample (containing such buttons for both a question and an answer to it):    This happens on an iOS device (running 9.3.5), using various browsers and this in any SE site. No idea so far how it looks like in the most recent iOS version. Same for the (green) checkbox below those voting buttons (to mark an answer as accepted). Is this intended to be like this (IMO it looks ugly ... and a huge waste of available space), or is it rather some sort of CSS related issue that needs to be fixed? PS: Looking at the same page around the same time on a Win PC (using any of the common browsers) seems to work fine. How to safely upgrade from an older Ubuntu version to a newer one? How can I safely upgrade an Ubuntu installation from one release to another? What should I do before upgrading to ensure it goes smoothly, and is there any clean-up I need to do afterwards?  In particular, do I need to do anything special if I am using packages from PPAs, if I have a graphics card that needs drivers, or if I'm running a custom Linux kernel?  This is a general question , but as example we take Ubuntu 12.04 and Ubuntu 12.10 This was posted on facebook and people are saying it is incorrect, it should be: "...as you and I"   Which is correct?