Memory problem when solving a system of modular equations
Factorizing polynomials over fields other than $\mathbb{C}$
Convert a router to a switch? How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$? I'd like to find out why  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align}  I tried to rewrite it into a geometric series  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=0}^{\infty} \Big(\frac{1}{2}\Big)^nn^2 \end{align}  But I don't know what to do with the $n^2$. How can author deal with late reviews? Another problem I personally struggle with in peer-review publication process is the situation when review takes really, really, really long. The precise definition of "long" is probably very area-specific: for example a paper in Maths will probably took longer to review than one in Physics. Many well-established journals and major publishers, sadly, prefer not to specify the "rules of the game" precisely, particularly how long the review process should normally take. Fortunately, I quite often serve as a referee with the very same journals and know the amount of time I am given to write a review. Unfortunately, when I am an author, I do not observe these deadlines to be always met. For example, a notable U.S. publisher would ask a reviewer to send a review in 2 months; but in practice a first review of my paper once took as long as 8 months.  As a rule of thumb, I usually start writing to editors with questions about "the status of my paper" in X+1 months time, when X is a deadline time for a reviewer to submit a review. This strategy is not particularly successful &mdash; what I usually get is a recommendation to be more patient. Since there is no "official" deadline time announced, there is no formal ground for complaints here.  Of course, such delays lead to a whole bunch of funny situations, particularly if a preprint is put online (e.g., on arXiv) and the results are presented in some conferences. Follow-ups with no official paper to cite, grant applications in which you can not justify by a solid publication, good research but bad bibliometrics or CV, to name just a few.   Could you recommend an efficient way to deal with this problem? Install Ubuntu from ISO image directly from hard disk of a system running Linux? Why is there a B in my flair? This was originally noticed and fixed back around the beginning of July. As of August 1, this problem seems to have returned. New bug report is here:   I just happened to notice that my SO flair has some sort of a B icon right before my username (instead of the expected site icon) Snapshot of the unknown icon:    My current (live) flair for monitoring:  It currently showing up like this for all users on all sites, but not on Combined or Meta flairs. For example Jon Skeet flair on programmers:    Live flair: What is the list of keys and their corresponding wins?  I have it listing that the highest number of wins I've had in arena is a Champion key, but it doesn't say how many wins it was. Do parts of a hash carry the properties of the entire hash? When I need to generate unique id's based on some information hashing is typical choice. However, sometimes that id needs to be of a particular size.  I've seen a lot of schemes (HMAC-MD5-96 in SSH, CGA in SeND for IPv6) that use a portion of a hash, so I'm thinking it might be alright to use it that way.    Which properties of a hash are preserved and which do not apply anymore?  Obviously, with less bits, the chance of collision goes up, but can I still rely on the partial hash for uniform character distribution?  What about the ?  I'm guessing that if a hash modifies the entire hash upon the smallest change to the input, that would imply that a portion of that hash would be equally well changed.  Am I thinking about this correctly?  Also, would these behaviors be different for different hashing algorithms? Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$, and $f$ be a function from $U$ into $U$.   Finally, let $F : P(U) \to P(U)$ be the function defined by  $$F(X) = \{f(x) | x \in X\}$$  For instance, if $f(x) = 13 − x$, and $X = ${1, 3, 4, 7}, then $$F(X) = \{f(1), f(3), f(4), f(7)\} = \{12, 10, 9, 6\}$$  Show that $F(A−B)$ is not necessarily a subset of $F(A)−F(B)$.    I am having trouble showing an example. Let's say:  $A = \{1,3\} , B = \{2,3\}$  $F(A) = \{f(1),f(2)\} = \{12,11\}$  $F(B) =\{f(2),f(3)\} = \{11,10\}$  Now, $F(A-B) = \{12,10\}$  $F(A)-F(B) = \{12\}$.  Hence, $\{12,10\} \nsubseteq \{12\}$ &lt;-- hmm..  By definition if $A \nsubseteq B$, there exist an $x \in A$ such that $x \notin B$. From my example, we have $x=10 \in A $ but $\notin B$. Is that sufficient?     I'm confused between the difference of $F(A-B)$ and $F(A)-F(B)$. Does Hard reboot damage my Ubuntu Berlin Public Transport Ticket Validity I have purchased an AB zone Berlin 7 day transport ticket and validated it on Sunday 17th April so hopefully it will be valid until Sunday 24th.  I validated the ticket at about 10:00am so I guess 1000 is the time I validated? But what are the numbers 15 and 6 and letters 'So' correspond to? How (if checked by a guard) can they tell if the ticket is valid? And how can I as a customer tell when it is validated to? Best way to include pdfs in an app I am trying to grab the capital letters of a couple of words and wrap them in span tags. I am using  for extract and wrapping purposes, but it's not outputting anything.  preg_replace("/[A-Z]/", "&lt;span class=\"initial\"&gt;$1&lt;/span&gt;", $str) How do I turn off equation auto numbering SQL function as default parameter value? I tried changing a default parameter value with this:  ALTER PROCEDURE [dbo].[my_sp] @currentDate datetime = GETDATE()   and all the SQL pre-compiler gave me was this error:     Msg 102, Level 15, State 1, Procedure my_sp, Line 8 Incorrect syntax near '('.   I have already created the procedure. (I'm not sure if that's relevant.) I was using a null default value and checking for that later, but that doesn't seem proper. Can I do this in one line?   Update: I was going off of :      [ = default ] Is a default value for the parameter. If a default value is defined, the function can be executed without specifying a value for that parameter.            Note:     Default parameter values can be specified for CLR functions except for the varchar(max) and varbinary(max) data types.         When a parameter of the function has a default value, the keyword DEFAULT must be specified when the function is called to retrieve the default value. This behavior is different from using parameters with default values in stored procedures in which omitting the parameter also implies the default value.   Am I reading this wrong?  Many thanks. How to calculate a confidence interval for estimating a proportion when the population is very small? I plan to execute a little study on whether dentists tend to grant privately insured patients (PRIV) a sooner appointment compared with publicly insured patients (PUB). I want to restrict this study to the dentists of a single town in Germany.  In this town there are about 500 dental practices and I am willing to call maximum 100 practices (twice, once as a PUB and once as PRIV). (A sample without replacement of size up to 100)  So the result of this examination would be a list of up to 100 practices with an associated value of 1 - if the appointment for the PRIV was more than two days sooner - and a 0 - if not. The proportion of 1s is what I want to estimate.  Now the sources I have been reading always assumed a much larger population with a finite population correction close to 1. This is why I doubt that blindly using a conservative or approximate confidence interval would lead to a statistically meaningful result.   So my first question is, of course, whether it is reasonable at all trying to estimate the proportion using a confidence interval in this case? And if yes, then what statistical technology/method (leading to a CI) would be a good choice for this use case?   One idea that comes to mind is using a sample size of 25 f.x., in that case the FPC would be 0.95 - "close enough" to 1. But it seems stupid trying to improve the quality of an estimation by reducing the information used for the estimation. Motivated by , can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group is an exponent for every element in the group?)  The simplest proof I can think of uses the coset proof of Lagrange's theorem in disguise and goes like this: take $a \in G$ and consider the map $f\colon G \to G$ given by $f(x)=ax$. Consider now the orbits of $f$, that is, the sets $\mathcal{O}(x)=\{ x, f(x), f(f(x)), \dots \}$. Now all orbits have the same number of elements and $|\mathcal{O}(e)| = o(a)$. Hence $o(a)$ divides $|G|$.  This proof has perhaps some pedagogical value in introductory courses because it can be generalized in a natural way to non-cyclic subgroups by introducing cosets, leading to the canonical proof of Lagrange's theorem.   Has anyone seen a different approach to this result that avoids using Lagrange's theorem? Or is Lagrange's theorem really the most basic result in finite group theory? I'd like to see if somebody has been trying to log-in by brute-force into my Ubuntu 12.04 server over SSH. How can I see if such activities have been taking place? Is there any method to close connection similar to  for  in WordPress?  Is it not necessary to close connection for global $wpdb? Custom function won't refresh as inputs are changed Can an infinite set A be larger than an infinite set B but still have the same cardinality? Definition. Two sets have the same cardinality iff they can be put into one-to-one correspondence; or,     $$A \simeq B \iff |A| = |B| $$    This definition applies to infinite as well as to finite sets. It follows from the last three definitions that set $A$ has a larger cardinality than set $B$ iff both    a proper subset of $A$ and the whole of $B$ can be put into one-to-one correspondence the whole of $A$ cannot be put into one-to-one correspondence with any proper subset of $B$.   From: A Crash Course in the Mathematics Of Infinite Sets Peter Suber, Philosophy Department, Earlham College  The last part specifically refers to proper subsets of $B$.  That excludes at least one subset of $B$, namely $B$.  So, is it possible that the whole of A cannot be put into one-to-one correspondence with any proper subset of $B$ but may with $B$ itself? Why did he not say "any subset of $B$"?