There are circular (trig) functions which determine all the points on a unit circle:  and which relate to the area swept out by an angle subtended on the circle. -- These functions can of course be extended to relations to ellipses as well.  There are also hyperbolic functions which determine all the points on a hyperbola:   My question is why there are no analogs of these functions for parabolas (the other type of conic section):  Here I have defined $\mathrm {sinp}(\theta)$ and $\mathrm {cosp}(\theta)$ to be the x- and y-coordinates of points on a "unit parabola".  Is there any good reason why we should have these extremely useful transcendental functions (sin, cos, sinh, cosh, etc), but we can't (or don't) define analogous functions for parabolas?    NOTE: I recommended that this post get deleted because Henning's response in  explained theoretically why a "parabolic trigonometric function" is different than the circular and hyperbolic trigonometric functions.  However johannesvalks' answer to this was very interesting, as well, and probably shouldn't be deleted.
The parametric equation  $$\begin{align*} x(t) &amp;= \cos t\\ y(t) &amp;= \sin t \end{align*}$$  traces the unit circle centered at the origin ($x^2+y^2=1$).   Similarly,   $$\begin{align*} x(t) &amp;= \cosh t\\ y(t) &amp;= \sinh t \end{align*}$$  draws the right part of a regular hyperbola ($x^2-y^2=1$).  The hyperbolic trigonometric functions are very similar to the standard trigonometric function.  Do similar functions exist that trace parabolas (because it is another conic section) when set up as parametric equations like the above functions?  If so, are they also similar to the standard and hyperbolic trigonometric functions?
How do I remove the space between inline/inline-block elements? Proving Cauchy condensation test Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence Would someone please advance/discuss some real-life situations falsities $1, 2$? I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I haven't naturalised these concepts yet. Thus, I beg leave to ask about other real-life examples which require less or no computations.  I tried and would appreciate less numerical examples than  and , and  Examples 1.22 and 1.23 on P39 of Introduction to Pr by Bertsekas.  $1.$ Pairwise Independence $\require{cancel} \cancel{\implies}$ (Mutual) Independence.      $2.$ Pairwise Independence $\require{cancel} \cancel{\Longleftarrow}$ (Mutual) Independence.         P38 defines (Mutual) Independence :    For all $S \subseteq \{1, ..., n - 1, n\} $ and events $A_i$, $Pr(\cap_{i \in S} A_i) = \Pi_{i \in S} A_i.$ Error Code: 1215. Cannot add foreign key constraint (foreign keys) How do I make animations a steady speed? Is it possible to create a geocoding and reverse geocoding service based on the shapefiles data Search which man sections contain a given entry Prove that $371\cdots 1$ is not prime Prove that $371\cdots 1$ is not prime.   I tried mathematical induction in order to prove this, but I am stuck.  My partial answer:  To be proved is that $37\underbrace{111\cdots 1}_{n\text{ ones}}$ is never prime for $n\geq 1$. Let $P(n)$ be the statement that $37\underbrace{111\cdots 1}_{n\text{ enen}}$ is not prime.  For $n=1$, we can write $371=7\cdot 53$, and therefore $P(1)$ is true. Let $P(k)$ be true for $k&gt;1$. Then we now have to prove that $P(k+1)$ is true.  I found that $37\underbrace{111\cdots 1}_{k\text{ ones}}$ can be written as $37\cdot 10^k+10^{k-1}+10^{k-2}+\cdots +10^0$ and that $37\underbrace{1111\cdots 1}_{k+1\text{ ones}}$ can be written as $37\cdot 10^{k+1}+10^{k}+10^{k-1}+\cdots+10^0$.  I think I'm pretty close to the answer know, but I don't know how to proceed. Configuring dpkg stops processing when calling an HTTP request to Adobe Flash Player \textcite macro produces "Author (year)" correctly but colours first paren with hyperref $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes Can we some how use color labels on icons like OS X?  Something like this: What is the "show desktop" keyboard shortcut? I just upgraded to 11.10 from 11.04. Previously the shortcut to show the desktop was Super +D (same as in Windows), But now it is not working. What is the new shortcut? In indirect speech we generally backshift the tense. For ex - Direct speech : "I am the most powerful person." Indirect speech : "He said that he was the most powerful person."  What if someone reports like -  He denied the charges claiming that he is/was the most powerful person. So here which is correct- is or was ?  Thanks in advance. Why can't humans throw objects at bullet speed when the force involved seems so small? The force that propels a bullet should at least be similar in intensity to the force experienced as recoil and the force exerted on the target. But these forces seem really low (the equivalent of a shove).  If all it takes to make a small object achieve bullet speed is the equivalent of a shove, why can't humans throw objects at these speeds? Given this HTML:  &lt;div&gt;foo&lt;/div&gt;&lt;div&gt;bar&lt;/div&gt;&lt;div&gt;baz&lt;/div&gt;   How do you make them display inline like this:     foo bar baz   not like this:     foo   bar   baz Why don't the zoom buttons work on my DSLR? I am a newbie to the new camera I just bought. It is a Nikon D5200 it has the original lens that came on the camera a 18-55 lens on it. My question is when I turn the lens to the 55 and then push the plus and minus buttons at the bottom of my camera to zoom in it appears to be zooming in and then I take the photo and the picture is not zoomed in at all.    I have a new Canon T3i and was able to take and capture zoomed in photos with the lens that came with the camera (EF-S 18-55mm) as well as a Canon zoom lens EF 75-300mm 1:4-5.6. Now when I zoom in with either lens and take a photo, when I review it, it shows original image before zooming. Is it a defect in the camera or inexperience of the operator?    (Two separate questions from separate users but with the same root cause.) I often use Cycles' built in Glass BSDF shader to achieve realistic refraction effects, but in many cases, it seems that Cycles' glass caustics creates an unfortunately large number of fireflies (pixels that are abnormally bright relative to their surroundings). In some models of mine, even when I render for 3000 or more samples/pixel, my final image is still covered in fireflies. While I can improve this in many cases by increasing the total amount of light available in the scene, this loses a lot of the potency of the refractive effects I'm looking for, especially in scenes with low light levels.  So my question is, how can I reduce the amount of fireflies while using Cycles' caustics without sacrificing the quality of the refractive effects?  Note: I am aware that Cycles' caustics will perform much better once Cycles has full support for bidirectional path-tracing, but it looks unlikely that that will become , unfortunately. How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay? Why didn't Harry Potter tell Aunt Petunia he was rich to improve how he was treated by Dursleys? This is a follow-up question to   One reason Aunt Petunia mistreated Harry was that she did not know he was rich. If she knew, she would naturally have treated this golden goose nicely. Her retirement would be secured if Harry financially supported her in old age.  Why didn't Harry tell Aunt Petunia he was rich? Surely every boy wants to be treated nicely by his family.