Other

Under construction - (A selection of random interesting things)

Interesting books

 

Interesting charts

  • FB relationships and betweenness-centrality fb2.pdf

 

Interesting software

  • vowpal wabbit (online machine learning)
  • cplex - gurobi - ortools (combinatorial optimization)
  • porta (integer polytope analysis)
  • tesseract (an old-fashioned low level ocr)
  • postscript (the postfix-notation programming language)

 

Interesting paradoxes

Some of my favourite "mathematical" paradoxes

  • Counter-intuitive, but nonetheless true 
    • Braess : How adding an additional road (at a badly selected place) increases capacity, but can make traffic worse (non optimal Nash equilibrium).
    • Newcomb : Intuitively correct versus obviously correct: when a problem is (deliberately) ambiguously underspecified.
    • Simpson : Treatment A is better than B for group 1, and also separately for group 2, but the opposite is true for groups 1 and 2 combined. Can made to happen easily by adding imbalance between the two groups for each treatment.
    • Harmonic mean : People use the average (a+b)/2 for situations where the correct way is to use the harmonic mean (the average of 1/a and 1/b, inverted). The harmonic is the correct way to average km/hour, miles per hour, dollars per kilo, miles per litre (gallon) etc, because in reality, one ought to be averaging (with the arithmetic mean) the hours/km, hours per mile, kilos per dollar, and litres or gallons per mile.
    • Benford : the first non-zero digit of physics constants in any units, or numbers generated among [1,..,N] where N varies, has a 30% chance of being 1, 17% chance of being 2, etc, and only 4% chance of being 9. 
    • Friendship : On average, a person's friends will have more friends than the person.
    • Spurious calculations - limitations of floating number precision : Eg prove that sum_n=0^infinity (floor((e^pi - pi)n) / 3^n ) does not equal 14.50000. NB the fist non-zero decimal digit after 14.5 is at the ~531th decimal digit.
  • Hidden mistake paradoxes 
    • Various division by zero paradoxes (0=1 paradoxes)
    • All horses are the same colour paradox (mistake in the induction)
  • Logical or semantic paradoxes
    • Russel paradox (set of all sets etc)
    • Hilbert hotel paradox (double of infinity is infinity)
    • Monty Hall paradox (both answers are correct, depending on assumptions)