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How much time have the Straw Hats spent with each other? Conjectures that have been disproved with extremely large counterexamples? I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.  I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is odd.  The conjecture states that if this operation is repeated, all numbers will eventually wind up at $1$ (or rather, in an infinite loop of $1-4-2-1-4-2-1$).  I fired up Python and ran a quick test on this for all numbers up to $5.76 \times 10^{18}$ (using the powers of cloud computing and dynamic programming magic).  Which is millions of millions of millions. And all of them eventually ended up at $1$.  Surely I am close to testing every natural number? How many natural numbers could there be?  Surely not much more than millions of millions of millions. (I kid.)  I explained this to my friend, who told me, "Why would numbers suddenly get different at a certain point?  Wouldn't they all be expected to behave the same?"  To which I said, "No, you are wrong!  In fact, I am sure there are many conjectures which have been disproved by counterexamples that are extremely large!"  And he said, "It is my conjecture that there are none! (and if any, they are rare)".  Please help me, smart math people.  Can you provide a counterexample to his conjecture?  Perhaps, more convincingly, several?  I've only managed to find one! (Polya's conjecture). One, out of the many thousands (I presume) of conjectures. It's also one that is hard to explain the finer points to the layman. Are there any more famous or accessible examples? Add an API key to QGIS for QOSM layer What is the mass density distribution of an electron? I've a set of regions within a city. I'd like to split each region into five subregions and then get the centroid of each one of them.  I know how to make a vector grid, but the problem I have is that the generated points are not taking into account the regions, they're only related to the general map.  I have this:    And I want to split each region in five equal size areas. Something like this:    I'd like to build a visualization similar to this one: . In order to accomplish that, I need to split each region, because the original ones are pretty big. Evaluate $\int \frac{1}{(x^2+1)^2}dx$ $$\int \frac{1}{(x^2+1)^2}\mathrm dx$$ Look simple, but I stuck a little bit. What method is better in this situation? Please, help. I have a question related to modeling short time-series. It is not a question , but how. What method would you recommend for modeling (very) short time-series (say of length $T \leq 20$)? By "best" I mean here the most robust one, that is the least prone to errors due the fact of limited numbers of observations. With short series single observations could influence the forecast, so the method should provide a cautious estimate of errors and possible variability connected to the forecast. I am generally interested in univariate time-series but it would be also interesting to know about other methods. How disk space is used on Android device? I have critical low storage level and can't understand what takes much of it and how that's possible that the installed applications can use 7 GB? I only use a few social media clients without any games or soft like Photoshop. Also on the second screenshot it indicates that the apps are using just 1.13 GB and so I can't understand the situation. Also what is the "Other"? Encrypt home drive after installation I declined to encrypt to my home drive whilst installing ubuntu.  Is it possible to encrypt it after installation?  if so, how? I have Ubuntu server 12.04 installed, so I have no GUI. When I do the command ifconfig, I cannot find my internal IP address. It says inet addr: 127.0.0.1. Here is the output of ifconfig -a: eth0   link encap:Ethernet  HWaddr 00:06:4f:4a:66:f0     BROADCAST MULTICAST  MTU:1500  Metric:1     RX packets:0 errors:0 dropped:0 overruns:0 frame:0     TX packets:0 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:1000     RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)  eth1   link encap:Ethernet  HWaddr 00:16:ec:05:c8:9c      BROADCAST MULTICAST  MTU:1500  Metric:1     RX packets:0 errors:0 dropped:0 overruns:0 frame:0     TX packets:0 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:1000     RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)  lo     Link encap:Local Loopback     inet addr 127.0.0.1  Mask:255.0.0.0     inet6 addr:  ::1/128 Scope:Host     UP LOOPBACK RUNNING MTU:65536  Metric:1     RX packets:1800 errors:0 dropped:0 overruns:0 frame:0     Tx packets:1800 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:0     RX bytes:143896 (143.b KB)  TX bytes:143896 (143.8 KB)  here are the contents of etc/network/interfaces: # The loopback network interface auto lo iface lo inet loopback  # The primary network interface auto eth0 iface eth0 inet dhcp  If someone could edit this for me, the contents of etc/network/interfaces should be on separate lines. The output of host askubuntu.com was: ;; connection timed out; no servers could be reached.  I set up owncloud and webmin a few months ago and was using them for a month with no problems. I think the power went off one day 2 months ago and I never turned the server back on until yesterday. I haven't done anything that would have affected the internet setup So i'm not sure why it doesn't work anymore. As far as my network topology goes, I have a pci-e network card for the pc. The ethernet line goes from the network card to a switch, and then to a modem/router from there. Is the universe fundamentally deterministic? Total number paths between two nodes in a complete graph Try to do something vs. Try doing something Is there any differences between      Try to do   And     Try doing   Thank you in advence. I have found that the question I want to ask someone had asked, here is the website:    Here is my question: Why $\dim(\hom(V,W))=\dim V * \dim W $?    Thanks for explanation~! What does the big intersection or union sign of a set mean? In some cases, we will have to consider the union or the intersection of several, even infinitely many sets, defined in the obvious way. For example, if for every positive integer $n$, we are given a set $S_n$, then   $$\bbox[border:1px solid red]{\bigcup_{n=1}^\infty S_n}=S_1\cup S_2\cup\cdots=\{x\mid x\in S_n\text{ for some }n\},$$   and   $$\bigcap_{n=1}^\infty S_n=S_1\cap S_2\cap\cdots=\{x\mid x\in S_n\text{ for all }n\}.$$   Two sets are said to be disjoint if their intersection is empty. More generally, several sets are said to be disjoint if no two of them have a common element. A collection of sets is said to be a partition of a set $S$ if the sets in the collection are disjoint and their union is $S$.   Normally what I know is that you can make a union or an intersection between only two sets. In this expression, there is a big union of sets. I'm asking about the meaning of this expression – what does it mean? What does the infinity sign do at the top?  Things are even more complicated with De Morgan's laws, which use the same expression:     Two particularly useful properties are given by De Morgan's laws which state that   $$\left(\bigcup_nS_n\right)^c=\bigcap_nS_n^c,\quad\quad\quad\quad\left(\bigcap_nS_n\right)^c=\bigcup_nS_n^c.$$   Anyone who can explain to me the expression or De Morgan's laws would be much appreciated. Does mathematics become circular at the bottom? What is at the bottom of mathematics? I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other? Rotating hatch labels based on orientation changes in Data Driven Pages Strip maps? I've created a strip map for a linear feature and have it loaded as a data driven page.  I want to show stationing for the line.  The stationing displays properly on the first data driven page but on each subsequent page the stationing labels get messed up as the map orientation changes. I want the stationing labels perpendicular to the linear feature as shown in the first image.  The tick marks seem to rotate and stay perpendicular to the line but the labels won't.  How can I fix this? Anti-Matter Black Holes Assuming for a second that there were a pocket of anti matter somewhere sufficiently large to form all the type of object we can see forming from normal matter - then one of these objects would be a black hole.  Question is, would there be a difference between an anti matter black hole and a normal matter black hole - in terms of how would the matter/anti-matter make the black hole different, or would they be the same?  I would expect the answer to be that the black hole formed from anti matter would retain the anti-matter properties in such a way that if it was to merge with a black hole at same size formed from normal matter that they would annihilate  each other con convert into pure energy - Would that be a correct understanding? For example:     Ever wish you could share information broadly   Could it be rewritten to:     Ever wish you could broadly share information   Are there any rules for the position of the adverbs. Do descriptive statistics have p-values?