[VA] SRC #007 - 2020 April 1st Ramblings

[Valentin Albillo]

      
Hi all, welcome to my new SRC #007 - 2020 April 1st Ramblings:

As usual, every time April 1^st comes by I tend to indulge in assorted math/computing ramblings, 9 of which I'll presently share with you. Also, you surely own a favorite HP calc which you're proud to show off to everyone and their uncle, and you have no trouble solving whatever math problems or computing tasks come your way using it, so this will be an ideal opportunity to put it to work for good. Let's begin:

      Note: In what follows, [x] is the integer part of x, {x} is the fractional part of x, and Ceil(x) is the ceiling of x, e.g.: [3.14] = 3, {3.14} = 0.14, Ceil(3.14) = 4.


1) Solving a system of N plain-vanilla linear equations in N unknowns is dead easy with most HP calculators but as soon as you introduce some very minor changes things aren't that easy anymore. For instance I wonder what the solution is for this system:

       u  + [v] + {w} = 200.0
      {u} +  v  + [w] = 190.1
      [u] + {v} +  w  = 178.8


2) In the same vein, we're used to the fact that a plain-vanilla 2^nd-degree equation has exactly two roots, real or complex, and our HP calcs have no problem finding them. But I wonder about the real roots of this slightly-modified "2^nd-degree" equation (how many roots, their values ...):

      x² - 10 [x] + 12.75 = 0


3) Now, here's another 2^nd-degree equation in X:

      X² + 2 X + 5 I = 0

with the caveat that this time X is not just some scalar value but a square matrix and I is the corresponding Identity matrix. Matricial equations can have any number of roots, including an infinity of them or none at all, and it would be nice to find some roots for this equation, if they do exist.


4) Also, after dealing with finding some roots of the above matricial equation I then considered the converse problem, i.e.: to find an N^th-degree equation which has a given NxN square matrix as a root. For instance, I wonder what 3^rd-degree matricial equation (if any) would have the following 3x3 matrix as a root:

       2     3     5
       7    11    13
      17    19    23


5) The N^th Fibonacci number (belonging to the well-known Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 ...) is given by an expression involving one or two exponential functions a^N, where a is the Golden Ratio, 1.618+. Since the hyperbolic functions are also combinations of exponential functions e^x, where e is 2.718+, I wonder if there's some simple way to express the N^th Fibonacci number using them, perhaps as simple as a few lines of RPN/RPL code or a short single-line user-defined function, used like this:

      FNF(1) = 1, FNF(2) = 1, FNF(3) = 2, ..., FNF(10) = 55, ...


6) The following expression (where N > 0 is an integer and log2 is the natural logarithm of 2 = 0.693+):

      Ceil(2/(2^1/N - 1)) = [2*N/log2]

seems to be an identity for all integer values of n. For instance for N = 1 both sides equal 2, for N = 5 both sides equal 14, and for N = 2020 both sides equal 5828. Now I wonder if there are any exceptions at all and, if yes, whether a short program (say 4 lines of code) would be able to very quickly find the first three.


7) I'm a fan of Star Trek since always and yesterday I came up with the following hypothetical, idealized scenario (assume all times and distances are exact and lightspeed is exactly 300,000 Km per second):

      Captain Kirk is in command of the USS Enterprise and has to carry out a most critical mission: an unstable star is going supernova exactly 6 weeks from the present moment, destroying several inhabited planets and causing massive loss of life. However, the explosion can be inhibited using certain exotic ore available in some other nearby uninhabited system located just 504 lighthours away, so the USS Enterprise is ordered to travel there, beam the ore onboard, then return to the unstable star and use the ore to stop the supernova explosion from ever happening.

      Now, the USS Enterprise can use two different types on engine. The main one is the warp engine, which is used for interstellar travel and can achieve faster-than-light speeds depending on the cube of the warp factor engaged, i.e., warp 2 achieves 2³ = 8x lightspeed, while warp 10 achieves 10³ = 1,000x lightspeed. The other engine type is the impulse engine, which is used while orbiting some planet or coasting to a starbase and other such relatively low-speed navigation as well as for emergencies, achieving all speeds from 0 at rest to just shy of lightspeed at full impulse.

      As (pretty bad) luck would have it, the very moment the USS Enterprise starts her journey to reach the ore the warp engine suddenly fails utterly and further Scotty reports that without the help of the warp field the impulse engine can only achieve half impulse (= 150,000 Km per second) so that's what Kirk orders and the Enterprise goes for the ore at half impulse while warp engine repairs are underway on the double.

      Nevertheless, despite the unexpected setback Kirk's not especially worried as he knows his starship is capable of sustaining speeds up to warp 14 (2,744x lightspeed) if needed for the return leg of the journey and fortunately by the time the USS Enterprise arrives at the ore and beams it aboard (which takes no significant time, assume instantly) the warp engine is back to fully operational status and thus Kirk sets for immediate return to the unstable star to try and arrive still within the inescapable deadline.

      That said, the question I pondered is this: What is the minimum warp factor Kirk should engage in order to meet the deadline and avoid massive loss of life ?


8) And speaking of Kirk, Picard's Little Theorem says that there's at most one value which an entire function does not assume. For instance, exp(z) is analytic in the whole complex plane and so is representable by an everywhere convergent power series, thus it's an entire function and the only value it omits is 0 because exp(z) is never 0 for any finite complex argument z, so it complies with Picard's LT alright. So far so good.

Now consider the function exp(exp(z)), which is also an entire function, with a power series convergent everywhere and all that jazz. Being of the form exp(something) it omits 0 because exp(z) does. But then it also omits 1 ( = exp(0) ), because its argument is exp(z), which omits 0. Thus there are two values it doesn't take (0 and 1), contradicting Picard's LT. So I wonder: what gives ?


9) As a bonus, a final spread of misc ramblings o'mine:

      - the equation Gamma(x) = Gamma(y) has trivial solutions x = y but surely nontrivial solutions can be found as well ?

      - the Gamma function grows faster than the exponential function, and I wonder at what positive value of x do their graphics cross for the last time.

      - at such value you'll have Gamma(x) = exp(x) for some x and at some other related value you'll have Gamma(Log(y)) = y for some y.

      - Gamma(Pi) and Gamma(-Pi/2) are surprisingly close when rounded to 2 decimal places ...

      - ... but slightly changing Pi to Pi+(2/5²)² above makes both expressions agree to no less than 8 places when truncated.

      - the equation  sin(x)+(7*x+1)*cos(x+1) = x  has a surprising root in the interval [12, 14].

      - and to top it all, some nice square roots:

            Sqrt(95888) = 309.657875727390470000000975517...

            Sqrt(22008840) = 4691.3580123456789961013...

            Sqrt(40850970) = 6391.4763552719179248943697799317869709544850492715609207148954163663
                                  8031627876551744428575854110967050931635948387731415926546989...


I'll eventually post some additional comments on all 9 ramblings above but let's see your comments first (if any).

Have a nice week and, if you're keeping confinement, I hope this humble effort of mine will offer you some hopefully welcome diversion for a little while. Take care !  
V.