Intrepid twenty on speed, getting information and understanding the gears?
Help Me Understand My Bicycle's Gears
Install suspension forks on non-suspension bike How to prove this for positive real numbers? $$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$  I tried AM-GM, CS inequality but all failed. Let $P$ denote the quotient space obtained by the action of $\mathbb{Z}\backslash2\mathbb{Z}$ by the antipodal map $z\mapsto\frac{1}{z}$ on the riemann sphere $\hat{\mathbb{C}}$ (identified here with $\mathbb{C}\cup\left\{\infty\right\}$). I identify $P$ with the set:  $\left\{ z\in\mathbb{C}:\left|z\right|&lt;1\right\} \cup\left\{ e^{it}:0\leq t\leq\pi\right\} $   that is to say, I use elements of the above set as the representatives for the equivalence classes in $P$. I am looking for formula for a function $f:P\times P\rightarrow[0,\infty)$ such that $f$ is a metric on $P$. Specifically, I would like a formula for $f$ that I can evaluate by plugging in representative elements in the above set (or something like that, more or less).  Thanks in advance.  Edit: forgive me for sounding desperate, but I cannot make due with an explanation of how to obtain such a formula. I want the formula.   An analogy for you if I have yet to make myself clear: suppose I was asking for the area of a square with side length $s$. The answers I have received so far for my question are akin to saying "multiply $s$ by itself" or "use the area formula for a square". The answer I am looking for is akin to saying "$s^{2}$". I need the formula. And please, no expressions with differentials, nor matrices, or any of that. I need to know how to compute the metric by using the complex numbers in the indicated set that I have identified with $P$. Displaying mysql data in html table PHP Is normality testing 'essentially useless'? Arduino Negative Voltage Reading with Positive Ground I am trying to measure voltage &amp; current off a solar charge controller from panel, battery &amp; load simultaneously with my arduino. I have managed to achieve this by tying all the negatives together and using it as GND, however, when I do that the charge controller doesn't function properly and when I switch to off-mode the loads still run. After checking the instruction manual further, I found out that the positives are shared common between Pan, Bat, Load. This was confirmed with a multimeter.   So now what I am thinking of doing is using the positive from the battery as the GND reference and then measuring all other voltages (through voltage dividers with respect to the GND). The voltages should then range between -17V to 0V.   However, I have read that the analog input pins are not able to read negative voltages, and with respect to the positive GND, the Analog inputs will all be negative.   EDIT: I was thinking of using an amplifier to invert positive ground to a negative ground, meaning I could measure all the positive sides of the voltages. But after getting answers from you guys it seems I may not have to use an amplifier and a voltage divider may suffice. I have currently included a 2 resistor voltage divider between each analog input, +5V source, Vmeasure which you can see in the diagram below.  According to my calculations, if this works, @-17V I should see 0.6V @ analog input. @0V I should see 4V at analog input.   I'd appreciate feedback on this circuit, and below in the second image I've also included a piece of another method suggested by @jonk (thanks), and would be interested in knowing what the benefits are of method B) over method A)  Thanks!  Ps. the ameters are there in place of ACS712 current sensors which also go into analog input pins.   Method A)        &ndash; Schematic created using   Method B (EDIT: This method has worked, I am still yet to test method A) thanks @Jonk)) How can I fix a 404 Error when using a PPA or updating my package lists? Difference between shutdown and shutdownNow of Executor Service $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes Consider the following snippet:  $$\text{d}_{H}(A,B)  = \max\left\{ \sup_{a\in A} \inf_{b\in B} \text{d}(a,b),\sup_{b\in B} \inf_{a\in A}\text{d}(a,b)\right\}$$     In the output, the text under the inf's is significantly smaller then the one under the sup's. The same happens when I have it in an align environment.  You can see the same problem in the wiki definition of the . How can I have the typeset under the inf's and sup's in the same size? What exactly is weakly informative prior? Is there a precise definition of weakly informative prior?  How is it different from a subjective prior with broad support? How can I compile, run and decompile C# code in Ubuntu terminal? When wouldn't I use LASSO for model selection? Transferring Fallout 4 Saves (PS4) how to show textview and edittextview on same line? This is my first question in /photography. I'm a very amateur photographer, doing things wrong and by eye with lots of trial-and-error all the time.  I currently own a reflex Nikon D70 my uncle gave us after he bought a better one. It has some nice settings, and he bought a a Nikkor 28-80mm F/3.5-5.6 zoom lens for it. It behaves nicely, but it's getting old (it has those old cartridges before SD cards existed) and every new charge takes longer than the last.  I'm planning to upgrade to a better camera, and there are some neat offers on a nearby superstore (MediaMarkt), this one catching my eye:  Sony DSC-H400 63x, Sensor Super HAD 1/2/3 CCD 20.1MP -> for less than 300€ taxes included  I don't have any type of extra information aside from that, all I can get from the official site is clinical data from other cameras that I suspect it's not 100% accurate, and there's no photography expert on the store to ask.  The common things I use my D70 for are: taking pictures of scenery (castles, towns, forests, waves hitting rocks, etc...), nearby (&lt;5m) persons and objects, and well, things where I want to save a memory in digital format, not for professional purposes at all.  Being that my objective is to have a decent, good-zooming (superzoom looks awesome), amateur (but with manual settings) camera, is this Sony a good replacement? Will I miss any features? Is there any better, similarly-priced alternative you know of?  Basically, I want to know whether I will get better quality photos than I currently do, given the examples, with any averagely-priced bridge camera similar to that Sony, and if it's worth it.  examples of current camera results  Full zoom-out of town at night, bad quality I know; notice the pink blur at the left.    Maximum zoom on that pink blur from almost the same spot:    Daylight, cloudy, modified with Adobe Lightroom, scenery:    Same day, same beach, different settings:    Same day, same beach, rocks: How can I do case insensitive string comparison in Python?  I would like to encapsulate comparison of a regular strings to a repository string using in a very simple and Pythonic way. I also would like to have ability to look up values in a dict hashed by strings using regular python strings. I've got a capture the flag arena. One issue we keep running into during play, is that of clutter.  We have /gamerule keepInventory false, so the pitch gets rather cluttered with armor, weapons and building materials, such that it becomes difficult to pick the flag up, due to full inventories.  I've discovered the following command to remove all drops from the arena:  /kill @e[type=Item]   However, this removes all the floating items from the ground. I'd like, if possible, to preserve the flag from being destroyed.  I have this command for detecting the flag:  /testfor @e[type=Item,x=40,y=1,z=40,dx=40,dy=27,dz=80] {Item:{id:"minecraft:banner",Damage:4s}}   But I'm not sure how to combine the two in a way that it will destroy everything except for the flag.  Any ideas? Why is this series of square root of twos equal $\pi$? Wikipedia claims this but only cites an offline proof:  $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$  for $n$ square roots and one minus sign. The formula is not the "usual" one, like Taylor series or something like that, so I can't easily prove it. I wrote a little script to calculate it and it's clearly visible, but that's not a proof. How do I add a kernel boot parameter? I need to add the boot parameter foo=bar to my kernel. How can I do this once for testing, and permanently if testing was a success?