Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.  Then how do you check if his series is really completely random?  If we only check whether the $n$-th digit is evenly distributed, then he can cheat using:     $0,0,0,0,...$   $1,1,1,1,...$   $0,0,0,0,...$   $1,1,1,1,...$   $...$   If we check whether any given sequence is distributed evenly, then he can cheat using:     $(0,)(1,)(0,0,)(0,1,)(1,0,)(1,1,)(0,0,0,)(0,0,1,)...$   $(1,)(0,)(1,1,)(1,0,)(0,1,)(0,0,)(1,1,1,)(1,1,0,)...$   $...$   I may give other possible checking processes but as far as I can list, each of them has flaws that can be cheated with a prepared regular series.  How do we check if a series is really random? Or is randomness a philosophical concept that can not be easily defined in Mathematics?
Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random?  (By reasonable I mean to exclude silly things like «generate a looong list and see that it is periodic»)  Later: Notice I am looking for a test that fails on standard generators.
Drawing a square root spiral File1:  .tid.setnr := 1123  .tid.setnr := 3345  .tid.setnr := 5431 .tid.setnr := 89323   File2:  .tid.info := 12 .tid.info := 3 .tid.info := 44 .tid.info := 60   Output file:  .tid.info := 12 .tid.setnr := 1123 .tid.info := 3 .tid.setnr := 3345 .tid.info := 44 .tid.setnr := 5431 .tid.info := 60 .tid.setnr := 89323 Diagonalizable Operators and characteristic polynomials Let $A$ be an $n\times n$ diagonal matrix with characteristic polynomial $(x-c_1)^{d_1}\cdots(x-c_k)^{d_k}$, where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n\times n$ matrices $B$ such that $AB=BA$. Prove that the dimension of $V$ is $d_1^2+\cdots+d_k^2$. How to change Facebook username? Do I have to re-register a new Facebook account to change my facebook.com/USERNAME?  Or are there any ways to change it? Are there any rules related to the creation of a ?   I am aware of the creation method for the amulet. But I'm asking about creating the actual construct. When I visit the reputation tab of my profile, I see a lot of stuff. I can make out some of it, but not all.   What do all the event types mean? Why do some events show an odd reputation gain or not show any reputation change at all? Why are some events sometimes highlighted in yellow? My reputation suddenly dropped, but I don't see anything. What happened? Why do some events disappear from my reputation history? Why use a prime number in hashCode? In the Prisoner of Azkaban, Harry and Lupin could practice the patronus on the boggart, but when Harry was in the final task in the Goblet of Fire, he tried to use the patronus on the boggart, it didn't work and he then moved on to use ridikkilus.  Why could he use a patronus against the boggart, but then not in another? Using (or not) the word "to" in a sentence ending with "be able" Which of these is grammatically correct?   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able.   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able to. There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. Better extensible arrows Dealing with "java.lang.OutOfMemoryError: PermGen space" error Are human chromosomes connected or separate molecules? Example of set which contains itself I am trying to understand   How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself. Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)? Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").   Yet no one uses these. Why is this? From where do the permanent magnets get energy from? I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently.  Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy? Why is the definite article used in expressions like the more and the less? For example,  The more you study, the more you know. The less you study, the less you know. Render semi-transparent volume with cycles Batch jobs error emails report:      The transaction log for database 'WebDB' is full.   No, it isn't. Its 90% free. The transaction log backups run normally. My hardware guy tells me the hardware is running perfectly, with plenty of space. SQL Server 2008 SP3, but not R2.  I'm a developer, we no longer have a DBA. So it falls to me to try and fix this. Do I have to clear customs when transit via the US on my way to / coming from somewehere else? I shall arrive at Washington Dulles Airport (IAD) by Emirates flight to catch a united airlines flight to Detroit after 3.5 hours. What steps shall I have to go through? Is the connection time enough? I have US F1 visa.