Extending a class in the method signature
Java Enum definition
Product of invertibles in $\Bbb Z_n$ [Wilson's Theorem generalization] How can we compute the product of all invertibles in $\Bbb Z_n$?  In the special case $n=p$ where $p$ is a prime it is Wilson's theorem. By pairing inverses it reduces to computing the product of all $a$ such that $a^2\equiv 1\pmod{n}.\,$  How can we do that? Recent reports claim that the gravitational waves detected by LIGO match up with the signal expected from two black holes merging as predicted by general relativity.  Additionally, the masses of both black holes were estimated. Questions:  How certain are we that the detected of gravitational waves are necessarily from two black holes merging?  Since supermassive black holes are typically at the center of a host galaxy, what happened to the galaxies that contained the two merging black holes?  For example, did their host galaxies merge?    Edit After reading the answers, I understand that the signals detected by LIGO are proof of the validity of gravitational wave generational artifacts as predicted by general relativity.  When additionally gravitational waves, caused by the merger of the supermassive black holes (those which ARE indeed located in the centers of the merging galaxies) will be detected, since mergers of galaxies could be observed by the methods independent of gravitational waves such as registering the fact of the quasar appearance (). Additional questions:  Could someone offer me the historical ASTRONOMY precedent when the second degree of inference (from the mathematical model, being associated with the theory of Physics) was accepted as the discovery of the astronomical object?  Could someone give a reference to trustable scientific publication, with the subject of study being discussion of requirements satisfying claim of astronomical discovery of an astronomical object?  Have independent scientific sources (outside of members of the LIGO team) analyzed methodology and results of the LIGO signal detection, and published their conclusions with regards of what was actually discovered? See all shared items in Google Drive In Dropbox there is a place you can see all the shared items in your account (i.e., all the items you have granted others access to).  Is there a similar feature in Google Drive?  I'm not only trying to find items that are shared with other "named people", but also documents that are shared on a "anyone with link can view / edit" or "public on the internet" basis.   Franck's  seemed to work back in 2013, but Google Drive's interface has changed quite a bit since, and I'm struggling to be able to perform the same task with the new interface. I wonder if anyone knows how to do it in the new interface. Proving that without the axiom of choice there is a set with an accumulation point that isn'tthe limit of a convergeant sequence I've been reading the simple parts of Thomas Jech's book on the Axiom of Choice and came across the following proof on page 141. The proof assumes a mathematical model without the axiom of choice.     Lemma. $X$ is D-finite if and only if $X$ does not have a countable subset.       Theorem. There is a set $S$ of real numbers and a real number $a \not \in S$ such that a is in the closure of $S$ but there is no sequence $(x_n)^{\infty}_{n=0} \subset S$ such that $(x_n)^{\infty}_{n=0} \longrightarrow a$.       Proof. Let $D$ be a D-finite infinite set of reals. We claim that the set $D$ must have an accumulation point. To prove this claim suppose that all $d \in D$ are not accumulation points, then let $\{I_n:n=0,1,...\}$ be a fixed enumeration of open intervals with rational endpoints. Assign the least $n$ such that $I_n \cap D={d}$ to each $d \in D$. This makes $D$ countable and by the above lemma gives us our contradiction. So $D$ has an accumulation point $a$. Let $S=D \setminus\{a\}$ and since $S$ is D-finite, every convergent sequence in $S$ is eventually constant.   Could somebody explain this proof? Why is the last sentence true/ Why does this mean every convergent sequence is eventually constant?  I think the following theorem is important     Theorem. There is a model of ZF which has an infinite set of real numbers without a countable subset "Typos" are acknowledged as "technical" errors, ones that obviously aren't caused by the writer's poor spelling. Although less frequent than in typing, they can also occur in handwritten texts, as a result of absent-mindedness or just an accidental stroke of the pen.   Since the text wasn't actually typed, it sounds funny to call them "typos". Is there a more fitting word, one that reflects the "accidental" nature of the error? I'm having a hard time finding the solution of the following equation:  $$0 = x\cos(x)+2$$  This is part of showing that $f(x):= x^2e^{\sin(x)}$ has 3 extremas in $[-\frac{\pi}{2},\frac{\pi}{2}]$. What is an example of pairwise independent random variables which are not independent? Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$? This is a follow up question to my previous question  Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the extension of $I(V)k[x_1,\ldots,x_{m+n}]$. Now I believe I have proven this (see the proof at the bottom of my question) but when I look at Martin's answer , it is instead claimed that we  actually have  $$I(V \times W )= \sqrt{I(V) + I(W)},$$  and for $I(V) + I(W)$ to be a radical ideal we need $k$ to be algebraically closed.           My question is: What's going on here? I believe my claim is true even without the assumption that $k$ is algebraically closed.           Here's a proof user Sanchez told me of, which I have simplified:       First it is clear that we always have $I(V) + I(W) \subseteq I(V \times W)$. For the reverse inclusion consider a polynomial $f \in I(V \times W)$. Then we can always write $$f = \sum_{i=1}^n f_ig_i$$ where $f_i \in k[x_1,\ldots,x_n]$ and $g_i \in k[x_{n+1},\ldots,x_{m+n}]$. Now take any $b' \in W$. If for all $i$ we have $g_i(b') = 0$ then since $b'$ is arbitrary, $g_i \in I(W)$ for all $i$. Then $f \in I(W) \subseteq I(V ) + I(w)$ and we are done. Otherwise suppose there exists $b \in W$  and $i$ such that $g_i(b) \neq 0$. Then wlog we may suppose that  $g_1(b) \neq 0$.   Next, $\sum f_ig_i(b) = 0 $ on all of $V$ by assumption of $f \in I(V \times W)$. So $\sum f_i g_i(b) = p$ for some $p \in I(V)$. Now write $$f_1 = \frac{ p - g_2(b) f_2 + \ldots g_n(b)f_n}{g_1(b)}.$$  Substituting this for $f_1$ in $\sum f_ig_i$, followed by taking things mod $I(V)$ we get an  expression with only $n-1$ terms $\mod{I(V)}$. Continuing this process we will finally get an expression with $0$ terms $\mod{I(V)}$ so that $f \in I(V)$. This shows $I(V \times W) \subseteq I(V) + I(W)$ which completes the proof. What are the reputation requirements for privileges on sites, and how do they differ per site? Formerly Reputation requirements compared. What are the reputation requirements for privileges on the various sites in the network? Also, how do reputation requirements for various privileges compare on different Stack Exchange sites? Specifically, the following types of sites:  Designed sites e.g. Stack Overflow, Ask Ubuntu, Gaming, etc. Public beta and non-designed sites Private beta sites  ...as well as any site-specific differences. I once heard the late John Ciardi (NPR's "A Word in Your Ear") try to explain that the 1920s idiom, "copacetic" (meaning completely satisfactory), entered into the African-American vocabulary in Harlem from the days when Jews and African Americans lived there together.  He argued that copacetic has the same meaning as the Israeli idiom "kol b'seder" which literally means "all is in order."  The problem with that, said my Harlem-raised father-in-law, is that the Jews in Harlem spoke Yiddish and kol b'seder was not used in Yiddish.  The dictionary I've got is not helpful. Can someone come up with a better explanation? Why does triply periodic univariate function not exist? A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if   $$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$$   for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function.  that a triply periodic univariate function cannot exist, but why? Can't we follow the construction of elliptic functions that uses Schwarz-Christoffel formula to map upper-plane onto a triangle, and use the reflection principle to create a lattice which corresponds to three periods?     Which step fails to hold when we mimic the construction of an elliptic function to create a triply period meromorphic univariate function?   Sorry if the question is stupid. I do not have an access to the original proof that a triply periodic function cannot exist, either. So references are welcome! Direct Link to a Comment The increase in the character limit of comments has led them to become as important as the questions and answers that spawn them. On several occasions (especially on meta) I've found reason to want to link to a comment directly. I'd imagine something as simple as making the date / time stamp an anchored link which mimics the behavior of the question link. I'm trying to install Linux alongside my 64-bit Windows 8 and went into my partition manager to find this structure.    In order to create partitions for my Linux install I need to free up some space. I see that the last partition is an empty recovery partition, as you can see highlighted, it's 100% free.  I was wondering is it okay to just remove this partition seeing as it doesn't appear I'll be losing anything?  Not sure if this helps but this is on a Fujitsu AH532 running Windows 8 64-bit with UEFI. Lag compensation with networked 2D games I want to make a 2D game that is basically a physics driven sandbox / activity game. There is something I really do not understand though. From research, it seems like updates from the server should only be about every 100ms. I can see how this works for a player since they can just concurrently simulate physics and do lag compensation through interpolation.  What I do not understand is how this works for updates from other players. If clients only get notified of player positions every 100ms, I do not see how that works because a lot can happen in 100ms. The player could have changed direction twice or so in that time. I was wondering if anyone would have some insight on this issue.  Basically how does this work for shooting and stuff like that?  Thanks What is the difference between currying and partial application? I quite often see on the Internet various complaints that other peoples examples of currying are not currying, but are actually just partial application.  I've not found a decent explanation of what partial application is, or how it differs from currying. There seems to be a general confusion, with equivalent examples being described as currying in some places, and partial application in others.  Could someone provide me with a definition of both terms, and details of how they differ? Is it possible to check a bash script syntax without executing it?  Using Perl, I can run perl -c 'script name'. Is there any equivalent command for bash scripts? How to access specific raw data on disk from java Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$ Let $V$ be a real inner product space, and let $v_1,v_2, \dots ,v_k$ be an orthonormal set of vectors. How do you prove that  $$\sum_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot\|y\|?$$  When does the equality hold?  I've been trying to do this with the Bessel and Cauchy-Schwarz inequalities, but I can't make it work yet. Any help would be greatly appreciated. How to compute azimuth between QgsPointV2s? The  class has an azimuth() method but the new  class does not. What's the best way to compute the azimuth between QgsPointV2s? I couldn't find a convenient way to convert V2 to the old version, so my fallback is to construct a QgsPoint from the V2's x and y:   self.traj = QgsLineStringV2(...)     p_i = self.traj.pointN(self.i) # this returns a QgsPointV2 p_j = self.traj.pointN(self.j) a = QgsPoint(p_i.x(),p_i.y()).azimuth(QgsPoint(p_j.x(),p_j.y()))   Is there a better way? I just received a decision letter from a journal editor (after three rounds of major revision). The letter shows that the reviewers "have asked for some other minor revisions". After getting the minor revision letter, I immediately completed the revision in accordance with the feedback from the reviewers (the suggested changes just involve adding a few sentences) and immediately resubmitted the revision to ScholarOne. A week has passed since I submitted the revision, but the status in ScholarOne remains unchanged. Is this normal? Can I assume that this basically means "acceptance with minor revision"? Thanks for your insight and sharing.