When does wrong grammar become right?
Will grammar errors become correct after enough people use them for enough time?
How does the unit circle work for trigonometric ratios of obtuse angles? I know that the x coordinate is $cos (\theta )$ and the y coordinate is $sin (\theta )$. But I understand these in context of only acute angles? I don't understand why the unit circle definition works for other than acute angles? Somebody please provide me some good intuition. What is the value of $1^i$? $\,$ Why must a computer have a BIOS? The  software is built into the PC, and is the first code run by a   PC when powered on ('boot firmware'). When the PC starts up, the first   job for the BIOS is the power-on self-test, which initializes and   identifies system devices such as the CPU, RAM, video display card,   keyboard and mouse, hard disk drive, optical disc drive and other   hardware. The BIOS then locates boot loader software held on a   peripheral device (designated as a 'boot device'), such as a hard disk   or a CD/DVD, and loads and executes that software, giving it control   of the PC.[2] This process is known as booting, or booting up, which   is short for bootstrapping.   Which requisites determine that a computer system must have a BIOS installed on it?  Why can't the Operational System actually do all of the BIOS work? Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$ Can md5 be used for encrypting data? I know that md5 shouldn't be used for password hashing because of collisions and possibility of making dictionary attacks e.g. using rainbow tables. But what about other uses? Can I use md5 for encrypting data?  Say, attacker gains access to database where sensitive data was stored using md5. He wants to know just this data that was hashed, not another data that gives this hash. Can he do it?  I know that there are better cryptographic hash functions so don't suggest to use them if md5 is enough. Limit at infinity of $x^{1/x}$ I encountered a problem: calculate $\lim\limits_{x\to\infty} x^{1/x}$  My solution is quite weird and I find no document explaining whether I am right or wrong!  My solution:  As $\lim\limits_{x\to\infty} 1/x = 0$  then $\lim\limits_{x\to\infty} x^{1/x} = \lim\limits_{x\to\infty} x^0  = 1$  My question is: Whether I am right or wrong, and could you please explain why? While I was investigating a problem I had with lexical closures in Javascript code, I came along this problem in Python:  flist = []  for i in xrange(3):     def func(x): return x * i     flist.append(func)  for f in flist:     print f(2)   Note that this example mindfully avoids lambda. It prints "4 4 4", which is surprising. I'd expect "0 2 4".   This equivalent Perl code does it right:  my @flist = ();  foreach my $i (0 .. 2) {     push(@flist, sub {$i * $_[0]}); }  foreach my $f (@flist) {     print $f-&gt;(2), "\n"; }   "0 2 4" is printed.  Can you please explain the difference ?    Update:   The problem is not with i being global. This displays the same behavior:  flist = []  def outer():     for i in xrange(3):         def inner(x): return x * i         flist.append(inner)  outer() #~ print i   # commented because it causes an error  for f in flist:     print f(2)   As the commented line shows, i is unknown at that point. Still, it prints "4 4 4". Video editing software options? What video editing software would you recommend on Ubuntu (or Linux in general) and why?  It can be for either beginners or experienced users.  Please only place one piece of software per response and include details to what makes it great! "Images with street signs" CAPTCHA impossible to solve I just got challenged (over on ) by the new "Mark all images with street signs" CAPTCHA.  I must have tried a dozen times to solve it by checking all images that had street signs, but I never succeeded.  Of note is a text input field without instructions of what to enter there. I suspect that is the problem: It is expecting input, but doesn't tell me what to input.  It's either broken or I am missing something.  I am using Chrome on an iPhone in "full site" mode. If a two vectors $\bf u$ and $\bf v$ in linearly independent, can their images be linearly dependent given they were transformed by a linear transformation $T$? I think not. My reasoning If the images are linearly dependent, then $c_1T({\bf u})+c_2T({\bf v})={\bf 0}$ for scalars $c_1$ and $c_2$, not both $0$. Then $c_1T({\bf u})+c_2T({\bf v})={\bf 0}$ $T(c_1{\bf u}+c_2{\bf v})=T({\bf 0})$ $c_1{\bf u}+c_2{\bf v}={\bf 0}$ Therefore, ${\bf u}$ and $\bf v$ are linearly dependent. My reservation A homework problem seems to contradict the above. Here's the problem: Let $\mathbb R^n\rightarrow\mathbb R^m$ be a linear transformation. Suppose $\{{\bf u},{\bf v}\}$ is a linearly independent set, but $\{T({\bf u}),T({\bf v}\})$ is a linearly dependent set. Show that $T({\bf x})={\bf 0}$ has a nontrivial solution. I have come across these three definitions of the limit superior (or upper limit) and the limit inferior (or lower limit) of a sequence of real numbers and I wonder how to establish the equivalence of these.    Walter Rudin: PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition:  Definition 3.16: Given a sequence $\{s_n\}$ of real numbers, let $E$ be the set of numbers $x$ (in the extended real number system) such that $s_{n_k} \to x$ for some subsequence $\{s_{n_k}\}$. This set contains all subsequential limits ..., plus possibly the numbers $+\infty$, $-\infty$. We now ... put  $$s^{*} = \sup E,$$ $$s_{*} = \inf E.$$ The numbers $s^{*}$, $s_{*}$ are called the upper and lower limits of $\{s_n\}$; we use the notation  $$ \lim_{n \to \infty} \sup s_n = s^{*}, \, \, \, \lim_{n \to \infty} \inf s_n = s_{*}.$$ Tom M. Apostol: MATHEMATICAL ANALYSIS, 2nd edition:  Sec. 8.3: Definition 8.2: Let $\{a_n\}$ be a sequence of real numbers. Suppose there is a real number $U$ satisfying the following conditions:    i} For every $\epsilon &gt; 0$ there exists an integer $N$ such that $n &gt; N$ implies $$a_n &lt; U + \epsilon.$$ ii) Given $\epsilon &gt; 0$ and given $m &gt; 0$, there exists an integer $n&gt; m$ such that $$a_n &gt; U - \epsilon.$$ Then $U$ is called the limit superior (or upper limit) of $\{a_n\}$, and we write $$U = \lim_{n \to \infty} \sup a_n.$$ Statement (i) implies that the set $\{a_1, a_2, a_3, \ldots \}$ is bounded above. If this set is not bounded above, we define $$\lim_{n\to\infty}\sup a_n = +\infty.$$ If the set is bounded above but not bounded below and if $\{a_n\}$ has no finite limit superior, then we say $\lim \sup_{n\to\infty} a_n = -\infty$. The limit inferior (or lower limit) of $\{a_n\}$ is defined as follows: $$\lim_{n\to\infty}\inf a_n = -\lim_{n\to\infty}\sup b_n,$$ where $b_n = -a_n$ for $n= 1, 2, 3, \ldots$.    Robert G. Bartle and Donald R. Sherbert: INTRODUCTION TO REAL ANALYSIS, 3rd edition: Exercises for Section 3.3: Problem 10: Let $(x_n)$ be a bounded sequence of real numbers, and for each $n\in \mathbb{N}$ let $s_n \colon= \sup \{x_k \colon k \geq n\}$ and let $t_n \colon= \inf \{x_k \colon k \geq n\}$. Prove that $(s_n)$ and $(t_n)$ are monotone and convergent. Also prove that if $\lim (s_n) = \lim (t_n)$, then $(x_n)$ is convergent. [One calls $\lim (s_n)$ the limit superior of $(x_n)$ and $\lim (t_n)$ the limit inferior of $(x_n)$.]   Now how can one show that the above three definitions are equivalent (i.e. these three definitions are of the same pair of numbers)? How did Walter White poison that character? I've seen Breaking Bad complete series 2-3 times, but I never found out how Walter White was able to poison Brock.  I mean did he himself directly feed the Valley of the Lilly berries to Brock or did he let Saul do the work? I can't figure it out. Shortcuts in Objective-C to concatenate NSStrings I don't really remember the plot but I think the a couple of alien girls (look human) crash land on Earth and are helped by human boy (18-19 ish). They get back to their ship but the older girl, with white hair, decides to stay on Earth to be with the boy.  At the beginning of the TV movie the boy wakes up and uses a ham radio and then goes to work on a ranch/farm(?).  I think it might have been one of these old Disney Sunday night movies. I've got a list of Python objects that I'd like to sort by an attribute of the objects themselves.  The list looks like:  &gt;&gt;&gt; ut [&lt;Tag: 128&gt;, &lt;Tag: 2008&gt;, &lt;Tag: &lt;&gt;, &lt;Tag: actionscript&gt;, &lt;Tag: addresses&gt;,  &lt;Tag: aes&gt;, &lt;Tag: ajax&gt; ...]   Each object has a count:  &gt;&gt;&gt; ut[1].count 1L   I need to sort the list by number of counts descending.  I've seen several methods for this, but I'm looking for best practice in Python. What does an icon on the tag mean, for example the "rg" before the sql-server tag as pictured below? An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$. I want to prove:     For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$   I tried to change the values and define $y=a+x$ so that $dy=dx$ and the limits of the integrals are as we want, but I'm not sure how to use the fact that $f(x)$ is periodic.  Thanks a lot! I'm trying to insert a figure into my LaTeX file using the following syntax:  \begin{figure}   \centering   \includegraphics [width=3in]{fig1.png}   \caption{Calculation of shortest path betweenness}   \label{Fig.3:Shortest path betweenness} \end{figure}   It worked for me before but this time I'm getting the following error:   LaTeX Error: \begin{figure} on input line xxx ended by \end{document}   Also, hovering over the red cross symbol indicating error in Eclipse, I see this message    paragraph ended before Gin@iii was completed.   It's obvious that I have the \end{figure} to end my \begin{figure} so I don't understand how I still get that error.  Can someone please tell me what the problem is?  Edit: Add more details for debugging. The problem went away if I remove '[width=3in]'. Here is the detail error:  pdflatex&gt; Runaway argument? pdflatex&gt; width=3in]{fig1.png} \caption {Caption} \label {Fig.1}  pdflatex&gt; ! Paragraph ended before \Gin@iii was complete. pdflatex&gt; &lt;to be read again&gt;  pdflatex&gt;                    \par  pdflatex&gt; l.40  pdflatex&gt;        pdflatex&gt;  pdflatex&gt; Underfull \vbox (badness 10000) has occurred while \output is active pdflatex&gt; (./document.bbl pdflatex&gt;  pdflatex&gt; Package natbib Warning: Empty `thebibliography' environment on input line 3.   Added sample code:  \documentclass[10pt,psfig,letterpaper,twocolumn]{article} \usepackage{geometry} \usepackage{graphics} \usepackage{setspace} \usepackage{natbib} \usepackage{verbatim} \usepackage[scaled=0.9]{helvet} \singlespacing \paperwidth 8.5in \paperheight 11in \oddsidemargin 0in \headsep 1.3cm  \geometry{left=0.75in,top=0.75in,right=0.75in,bottom=1in} \textwidth 7in  \textheight 9.25in \columnsep 0.4in \footskip 0in  \renewcommand{\bibname}{REFERENCE} \pagestyle{empty}  \begin{document} \bibliographystyle{acm}   \title{\fontfamily{phv}\selectfont{\huge{\bfseries{Title here}}}} \author{ {\fontfamily{ptm}\selectfont{\large{\bfseries{author name}}}}\thanks{contact info }}  \maketitle \thispagestyle{empty}  \section*{\fontfamily{phv}\selectfont{\normalsize{\bfseries{Section name}}}}  \begin{figure} \centering \includegraphics[width=3in]{fig1.png} \caption{Caption} \label{Fig.1} \end{figure}  \pagebreak[4] \vspace*{5.52in} \bibliography{ref} \end{document} Official reasons for "Software caused connection abort: socket write error" We know the optimum angle for greatest horizontal displacement when launching an object with projectile motion is 45 degrees.  How to solve the angle when it is real long distance around the earth where we cannot assume that gravity is constant but is given by $F=\frac {-GMm}{r^2}$. I use differetial equation to solve it and obtain 45 degrees.   First, for the vertical distance $y$, change $v=u-gt$ to $y'=u-y''t$. Then solve it to obtain a function $y(t)$. When $y'(t)=0$ we have the time for maximum vertical distance. Then insert the value to horizontal equation.   Can anyone guide me through this problem because I am not confident with my working