Archive for power

battery recharged!

Posted in pictures, Travel with tags , , , , , on September 10, 2022 by xi'an

midnight run

Posted in Running, Travel with tags , , , , , , , , on December 8, 2019 by xi'an

Le Monde puzzle [#1114]

Posted in Kids, R with tags , , , , , , on October 16, 2019 by xi'an

Another very low-key arithmetic problem as Le Monde current mathematical puzzle:

32761 is 181² and the difference of two cubes, which ones? And 181=9²+10², the sum of two consecutive integers. Is this a general rule, i.e. the root z of a perfect square that is the difference of two cubes is always the sum of two consecutive integers squared?

The solution proceeds by a very dumb R search of cubes, leading to


The general rule can be failed by a single counter-example. Running

  if (sol) 

which is based on the fact that, if z is the sum of two consecutive integers squared, a² and (a+1)² then

2 a²<z<2 (a+1)²

Running the R code produces

x=14, y=7

as a counter-example. (Note that, however, if the difference of cubes of two consecutive integers is a square, then this square can be written as the sum of the squares of two different integers.) Reading the solution in the following issue led me to realise I had missed the consecutive in the statement of the puzzle!

How many subjects? [not a book review]

Posted in Books, pictures, Statistics with tags , , , , , , on September 24, 2018 by xi'an

take those hats off [from R]!

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , on May 5, 2015 by xi'an

from my office, La Défense & Bois de Boulogne, Paris, May 15, 2012This is presumably obvious to most if not all R programmers, but I became aware today of a hugely (?) delaying tactic in my R codes. I was working with Jean-Michel and Natesh [who are visiting at the moment] and when coding an MCMC run I was telling them that I usually preferred to code Nsim=10000 as Nsim=10^3 for readability reasons. Suddenly, I became worried that this representation involved a computation, as opposed to Nsim=1e3 and ran a little experiment:

> system.time(for (t in 1:10^8) x=10^3)
utilisateur     système      écoulé
     30.704       0.032      30.717
> system.time(for (t in 1:1e8) x=10^3)
utilisateur     système      écoulé
     30.338       0.040      30.359
> system.time(for (t in 1:10^8) x=1000)
utilisateur     système      écoulé
      6.548       0.084       6.631
> system.time(for (t in 1:1e8) x=1000)
utilisateur     système      écoulé
      6.088       0.032       6.115
> system.time(for (t in 1:10^8) x=1e3)
utilisateur     système      écoulé
      6.134       0.029       6.157
> system.time(for (t in 1:1e8) x=1e3)
utilisateur     système      écoulé
      6.627       0.032       6.654
> system.time(for (t in 1:10^8) x=exp(3*log(10)))
utilisateur     système      écoulé
     60.571        0.000     57.103

 So using the usual scientific notation with powers is taking its toll! While the calculator notation with e is cost free… Weird!

I understand that the R notation 10^6 is an abbreviation for a power function that can be equally applied to pi^pi, say, but still feel aggrieved that a nice scientific notation like 10⁶ ends up as a computing trap! I thus asked the question to the Stack Overflow forum, getting the (predictable) answer that the R code 10^6 meant calling the R power function, while 1e6 was a constant. Since 10⁶ does not differ from ππ, there is no reason 10⁶ should be recognised by R as a million. Except that it makes my coding more coherent.

> system.time( for (t in 1:10^8) x=pi^pi)
utilisateur     système      écoulé
     44.518       0.000      43.179
> system.time( for (t in 1:10^8) x=10^6)
utilisateur     système      écoulé
     38.336       0.000      37.860

Another thing I discovered from this answer to my question is that negative integers are also requesting call to a function:

> system.time( for (t in 1:10^8) x=1)
utilisateur     système      écoulé
     10.561       0.801      11.062
> system.time( for (t in 1:10^8) x=-1)
utilisateur     système      écoulé
     22.711       0.860      23.098

This sounds even weirder.

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