Exact Question: Invent a single binary operator $*$ such that for every real numbers $a$ and $b$, the operations $a + b$, $a - b$, $a \times b$, $a \div b$ can be created by applying $*$ (multiple times), starting with only $a$'s and $b$'s  From my interpretation, you can apply $*$ recursively some number of times with carefully selected parameters to produce the desired outcome.  I thought the operator should be a combination of $a - b$ and $a \cdot b^{-1}$ Since $-$ and $\div$ can produce $+$ and $\times$ respectively  Please do not tell me the full answer. Give me a hint to point me towards the right path
I remember reading a while back about arithmetics with only a single operator that did all the work of the familiar $+ - \times /$ operators. If there is such a thing, could someone point me to it?  As background: I have a side project where I'm playing with neural nets and genetic search algorithms. Right now it's the good old kind, with matrix multiplication and addition. I thought it would be interesting to implement a sort of neural network that used a combined or "all in one" operator.  I've googled like heck and can't pick up the trail of where I thought I saw that before.  Is there study in that area, or am I mis-remembering, or worse, just making it up? I'd appreciate any pointers.
Why the pressure of atmosphere doesn't crush you when you e.g. walk outside? Why the pressure of atmosphere doesn't crush you when you e.g. walk outside? I mean the density of air is $1.26 kg/m^3$, so with $100 km$ above us, it exerts much pressure on you when you walk outside. Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof. How do you remove duplicates from a list whilst preserving order? How can I search for a keyword with special characters in Google Search? My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :) How to delete GRUB entirely from GRUB rescue and boot Windows only How can I visualize which folders and files are taking up all of the space on my hard drive?  I'm getting some conflicting reports on the size of hard drive contents. Namely what is and isn't there and what folders are actually using the space.   I need to know which of the files or folders the culprits behind all this hidden bloat. Also there should be a print option to get it on paper. Many companies offer index funds and ETFs. If two funds or ETFs track the same fund, what differentiates the two? Is it common for one index fund to outperform another? How can this happen if the funds/ETFs are tracking the same index?  I found . When forced to choose between them, index ETFs look like the better option (under the right circumstances), but what differentiates index funds/ETFs when compared to similar vehicles? An incorrect proof by exhaustion I refer to , which I understand says that the branch points of a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are not necessarily compact, form a discrete subset of range $Y$ of the map $F$. (Not sure if 'connected' is part of the definition of Riemann surfaces in other textbooks, but it is here.) Question 1: Do I understand correctly, and is this indeed true? What else I understand:  $A$ is a discrete subspace/subset (I just assume everything here is the usual subspace topology) of $B$ if and only if every singleton subset of $A$ is open in $A$ if and only if every point of $A$ is isolated.  This overflow question () seems to suggest this is not true but is true if $F$ is proper. However, I'm not sure the definitions of Riemann surface are the same (maybe some answers use definitions where Riemann surfaces are not necessarily connected or something).  2.1. I know $F$ is 'discrete' (meaning that its fibres are discrete; here, i refer to a ): see definition before Lemma 3.1, on p. 7), by Proposition II.3.12, and open, by open mapping theorem, which is Proposition II.3.8, but I'm not sure these help show $F$ is proper or are otherwise helpful. Ostensibly 'discrete' maps don't map discrete subsets to discrete subsets.   This stackexchange question () seems to suggest this is true.  I think this is true if $F$ is injective or at least something like locally injective. (Please don't make me type up all the stuff I did for this part.) However based on the question in (3) and based on the , which I think uses the term 'branch points' to mean the same thing as what Miranda means by 'ramification points' (and thus is different from Miranda's 'branch points'), it seems that no neighbourhood $U$ of a ramification point $p \in U \subseteq X$ of $F$ is such that the restriction $F|_U$ is injective.  Kinda related:    This page () says    Now, the ramification and branch points must form a discrete set  I don't think the page assumes anything like $F$ is proper.  The  text in (2.1) actually seems to assume proper based on the '3.7. Proper holomorphic maps' part, but said part refers to 'Lemma 3.17', so I think proper can be replaced with closed. Maybe $F$ is closed or something.  Kinda related (see Question 2 below):     says    If $F$ is proper then the image $\Delta = F(R)$ is discrete in $Y$.   This seems to have all the same assumptions as with Miranda. Ostensibly, either Miranda is wrong or there's some higher level machinery that allows us to omit $F$ proper.  Question 2: To possibly generalise this, what are the minimum requirements on $F, X$ and $Y$ to make $F$ map discrete subsets to discrete subsets? I mean, I'm not sure we really need 'holomorphic' here. I have a feeling this applies perhaps to maps that are just open/closed, continuous and discrete and spaces that are just Hausdorff and locally compact or something.  (again) Kinda related: If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$? Have $n$ vectors in $\mathbb{R}^n$.  If the $n$ vectors are linearly independent, can we conclude that their span is $\mathbb{R}^n$? Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth till the two trains meet and it gets squashed. How far did the fly travel before it got squashed?  The question is relatively simple - the trains take an hour to collide, and in an hour the fly would travel 75 km. However, the question got me wondering: Lets say you are given a time $t$ in hours(where $t&lt; 1$) How many &quot;rounds&quot; would the fly make in $t$ time? A round  is defined as moving from one train to another and back. How would I model the equation? mknod /tmp/oracle.pipe p  sqlplus / as sysdba &lt;&lt; _EOF set escape on  host nohup gzip -c &lt; /tmp/oracle.pipe &gt; /tmp/out1.gz \&amp; spool /tmp/oracle.pipe select * from employee; spool off  _EOF  rm /tmp/oracle.pip   I need to insert a trailer at the end of the zipped file out1.gz , I can count the lines using  count=zcat out1.gz |wc -l   How do i insert the trailer  T5 (assuming count=5)   At the end of out1.gz without unzipping it. I want to fill a part of a picture with more or less evenly and randomly distributed circles which should not overlap. It would be okay if circles are only partly inside the shape.  The density should be like what you see on this page  with values 80-100. (But I do not need so many circles, 30 should be enough, I could repeat the pattern).   I cannot use a grid and "wobble" the circles a bit: The circles are too large and it simply does not look randomly.   Currently I am thinking about a random attempt: Get some random point, measure its distance to existing circles, either paint or throw the point away. Try 100 times and hope that you get the right number of circles (and that it does not take too long). But perhaps someone has a better idea.   The example draws the circles simply manually:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{backgrounds} \begin{document}   \begin{tikzpicture}[framed,gridded,radius=0.5cm]  \draw[red](0,0) rectangle (5,5);   \draw (1,1) circle;   \draw (2.3,1.1) circle ;   \draw (4.5,0.8) circle ;   \draw (5.1,1.8) circle ;   \draw (0.4,3.3) circle ;   \draw (2.1,2.8) circle ;   \draw (3.8,3.5) circle ;   \draw (4.8,4.2) circle ;   \draw (0.8,4.9) circle ;   \draw (2.1,4.1) circle ;   \draw (3.8,2.0) circle ;   \draw (3.5,0.6) circle ;   \draw (3.0,5.0) circle ;      \draw (4.1,5.1) circle ;             \draw (0.9,2.1) circle ;  \end{tikzpicture}  \end{document} This is maybe a duplicate like , but the error still appears.    When I try to click "mark as duplicate" a HTTP 404 error occur.  I'm using Firefox 16.0.2 with Firebug on Mac OS X 10.5.8 How do I blend a black-background fire image in Photoshop? This fire image:    As you can see, it has a black background, which I can't delete (because it would probably destroy the smooth-looking fire effect). I want to place some fire in a photo.  However, I remember it was possible to take such images and do something in photoshop so that when you place them over a photo, the black is gone and the fire is perfectly blended there. But I can't remember what feature or option was it. Any ideas?  Photoshop CS5. motion tracking : how to extract 3d position from selected 2d tracker I already solve my camera and created 3d scene successfully. Now i want to do 2d track the original clip and by using solving data, i want to extract that specific point into 3d coordinate (as empty object or plain axes). How can i do this ?  Thanks Why is it bad practice to test for item names? I've heard that testing for item names and lores in Minecraft JE is bad practice. For example, the following commands are not good: execute if entity @e[nbt={Inventory:[{tag:{display:{Name:'[{&quot;text&quot;:&quot;Test Thing&quot;}]'}}}]}] clear @p minecraft:diamond{display:{Name:'{&quot;text&quot;:&quot;Shiny…&quot;}'}} 0 clear @p minecraft:netherite_sword{display:{Lore:['{&quot;text&quot;:&quot;A Mysterious Sword&quot;}']}} 0  Why are these commands considered bad practice? Are they prone to failure and may not work? And what can I do to fix them? Note that this only applies to reading item names, not setting them. The following are not considered bad practice and in fact are the only ways to accomplish these tasks: give @p minecraft:diamond{display:{Name:'{&quot;text&quot;:&quot;Shiny…&quot;}'}} replaceitem @p weapon.mainhand minecraft:netherite_sword{display:{Lore:['{&quot;text&quot;:&quot;A Mysterious Sword&quot;}']}} Why does a free-falling body experience no force despite accelerating? Whenever light rays are entering a denser medium they bend towards the normal.  Why do rays choose the path towards the normal? Why cannot they choose the path away form the normal?