Infinite cartesian product from Herbert Enderton's Elements of set theory
Definition of the Infinite Cartesian Product
How to Truncate a string in PHP to the word closest to a certain number of characters? I have a code snippet written in PHP that pulls a block of text from a database and sends it out to a widget on a webpage.  The original block of text can be a lengthy article or a short sentence or two; but for this widget I can't display more than, say, 200 characters.  I could use substr() to chop off the text at 200 chars, but the result would be cutting off in the middle of words-- what I really want is to chop the text at the end of the last word before 200 chars. Dimensional Tolerance for US Standard NEMA 1-15 Plugs I am trying to find the dimensional tolerance on US electrical plugs. I have tried looking online, but I have only been able to find that they are called , and have found the picture below showing length and height for the male connector (but not width of the male prong), and I haven't been able to find dimensional tolerances anywhere.     I ask here because I'm assuming there's a US code that governs electrical plugs, but I can't seem to find the standard anywhere. Also show "Add a comment for the author" selection when deleting/voting to delete an answer I'm aware that this only affects high rep users and mods, but nevertheless it would be nice if the following selection would appear when a user casts a delete vote on an answer.    If I am not mistaken, this only appears when casting delete votes from the review queues. I don't see a reason why this shouldn't also appear when casting a vote directly to the answer, instead of the (java script) alert with "Delete this post?".  Or in other words, the selection should always appear when a user does some kind of 'delete answer' action. What is an example of Gâteaux differentiable but not Fréchet differentiable at a point in a finite-dimensional space? Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional.  Let $E$ open in $\mathbb{K}$ and $p\in E$.  Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$.  Is $f$ necessarily Fréchet-differentiable at $p$ in this case?  I think this is not true in general, but cannot find a counterexample. What would be a counterexample? In university last semester I was asked to prove that $\sin1$ (1 radian that is) is irrational,  and ended up simply using the Taylor Series Expansion. This method provides a very quick solution, but I am curious as to whether anyone has a method for proving this without making use of the Taylor Series Expansion. I feel as though doing so must be possible using some number theory, but am low on ideas as to an alternative approach to the question.  Note:  If anyone is interested in my solution using Taylor Series Expansion (although it is not the focus of my question), here it is :      From    $$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} -  \dots$$   We see that    $$ \alpha = \sin 1 = 1 - \frac{1^3}{3!} + \frac{1^5}{5!} -  \dots $$   Given integers $a$ and $b$, if $\alpha = \frac{a}{b}$ then it follows that $b!\alpha \in \mathbb{Z}$, and $b!\alpha = C + D$ where $C \in \mathbb{Z}$ and we have :   $$ D = \begin{cases} \pm(\frac{1}{b+1} - \frac{1}{(b+1)(b+2)(b+3)} + \dots) \text{    if $b$ is even}\\ \pm(\frac{1}{(b+1)(b+2)} - \frac{1}{(b+1)\dots(b+4)} + \dots ) \text{ if $b$ is odd}\\ \end{cases} $$   In each case we can see that $0 &lt; D &lt; 1$, giving us a contradiction.    Thus, we have that $\sin 1 $ is irrational.    $$ \blacksquare $$ At the end of my animation, the animation pauses a bit then loops again, how can I stop the slow pause, and let the loop work fluently? Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. Is there geometric or visual (but rigorous) proof for the equation of hyperbola whose foci are on the $x$-axis using only the traditional definition of hyperbola: a hyperbola is a set of points such that the difference of distances from each point to focus is constant $(|r_1 - r_2| = 2a)$?  Also, can you maybe geometrically explain asymptotes of hyperbola (why do they exist)? Do eyeballs exhibit chromatic aberration? Fairly straightforward question. If not, why not?  I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity. In GKP's Concrete Mathematics, there is a useful and interesting feature I would like to see in more books. The bibliography entries include:   a - the page numbers where the book or article is cited in the document. b - a few lines of review of the cited work.   It is a kind of mix between bibliography, index and review.  Are there stable LaTeX packages/BibTeX styles to do (a) and (b) ?  EDIT: I wrote (b) initially as "comments about the work cited". It could be understood as "gathering comments made in the book about the work and reproducing them in the bibliography".  This is not what I meant but this could be something useful for others. Is it appropriate to down-vote posts containing SQL injection vulnerabilities? In the early days of C++ when it was bolted on top of C, you could not use NULL as it was defined as (void*)0. You could not assign NULL to any pointer other than void*, which made it kind of useless. Back in those days, it was accepted that you used 0 (zero) for null pointers.  To this day, I have continued to use zero as a null pointer but those around me insist on using NULL. I personally do not see any benefit to giving a name (NULL) to an existing value - and since I also like to test pointers as truth values:  if (p &amp;&amp; !q)   do_something();   then using zero makes more sense (as in if you use NULL, you cannot logically use p &amp;&amp; !q - you need to explicitly compare against NULL, unless you assume NULL is zero, in which case why use NULL).  Is there any objective reason to prefer zero over NULL (or vice versa), or is all just personal preference?  Edit: I should add (and meant to originally say) that with RAII and exceptions, I rarely use zero/NULL pointers, but sometimes you do need them still. How are virtual functions and vtable implemented? We all know what virtual functions are in C++, but how are they implemented at a deep level?  Can the vtable be modified or even directly accessed at runtime?  Does the vtable exist for all classes, or only those that have at least one virtual function?   Do abstract classes simply have a NULL for the function pointer of at least one entry?  Does having a single virtual function slow down the whole class? Or only the call to the function that is virtual?  And does the speed get affected if the virtual function is actually overwritten or not, or does this have no effect so long as it is virtual. Why do corticosteroids harm COVID-19 patients? states that steroids could do more harm than good in treating coronavirus (COVID-19), referring to .      Understanding the evidence for harm or benefit from corticosteroids in   2019-nCoV is of immediate clinical importance.   It's unclear to me how exactly the use of corticosteroids (which is a common treatments for Asthma patients) can do harm other than the vague explanation of "steroids also impair the immune system's ability to fight viruses".   How exactly does the use of steroids harm the body in case of fighting a virus such as SARS-CoV-2 (2019-nCoV) that causes COVID-19? In my project, I use a lot of geodata from different directories. Now I want to give the QGIS project to a colleague (eg on a CD).  Is it possible to copy the shapes from different directories automatically to one directory? Suppose every subsequence of X converges to 0. Show that lim(X)=0 Question: Suppose that every subsequences of $X = (x_n)$ has a subsequence that converges to $0$. Show that $\lim(X) = 0$   My attempt:   Suppose that $\lim(X) =L \neq 0$  Let $\epsilon &gt; 0$, and let $K(\epsilon)$ be s.t. $n \ge K(\epsilon)$   then $|x_n-L| &lt; \epsilon$  let $n_k \ge k$, and then $k \ge K(\epsilon)$ and $n_k \ge k \ge K(\epsilon)$ s.t. $|x_{n_{k}}-L| &lt; \epsilon$, which is a contradiction. therefore $\lim(x) =0$ Houston, we have a porblem Since  in titles, and there's not even a way  (or ) we all know users just get around that filter by typing various permutations of the word "problem".  The filter has . The bottom line there was:     "Hacking around" the filter is in fact very very rare.   I beg to disagree. I tried searching Stack Overflow for the following variations of the title:  problme probelm probeml probmle probmel prolbem prolbme prolebm prolemb prolmbe prolmeb proeblm proebml proelbm proelmb proembl proemlb   These are just a couple of permutations from 5040 possible ones. I also tried searching for variations with left out letters, like proble, or prblm. Here's just a graph of a subset from maybe 20 searches—but in reality, there might be more ways to get around the filter:    Was this ever intended? I know it saves a lot of bad titles from being created, but on the other hand it enforces even worse titles as well. 3% based on the  aren't much, but they're 3% crap the filter actively provokes.  Now, my questions up for discussion are:   What are we going to do about this large number of porblematic posts? It seems almost impossible to fix all those. We now have a really good review system that allows people to dig through low quality posts the entire day, so that'd be a way to address this of course. Are there any plans of getting rid of the filter again? This, at least, would give you a good indicator of poor quality. If a post title contains the word "problem" you instantly know something's off. Why not have it reviewed? I understand that some people will choose a better title after being blocked, but I don't even want to start looking at the number of those who merely did a s/problem/issue, or even worse, just used "problems" (because that's not blocked, just sayin').   In the end I'm just thinking if there was a way to clean up the back log on this, or how to prevent more posts of this sort in the future, even with a title filter in place if it has to be. Using X11 in OSX 10.8.3 : "ld: library not found for -lX11" Behavior of $\sum_{n=1}^\infty n^{-1}z^n$ on the circle of convergence Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the values of $z\in C:=\lbrace z\in\mathbb{C}: |z|=1\rbrace$, the circle of convergence, for which the given series converges. How do I run a GUI app via SSH?