PGSA Colloquium
Travis Hoppe
NIH, NIDDK, LBG,
(former Drexel grad. student)
print "Hello World"
Hello World
Function points where the input is identical to the output
It's not as simple as you think...
print 'Hello world'
> Hello world
print 'print \'Hello world\''
> print 'Hello world'
print 'print \'print \'Hello world\'\''
> print 'print \'Hello world\''
Some languages allow for the trivial case of empty code
No reading the code from the file
def quine(source):
quote = '"'*3
print source + '(' + quote + source + quote + ')'
quine("""def quine(source):
quote = '"'*3
print source + '(' + quote + source + quote + ')'
quine""")
> def quine(source):
quote = '"'*3
print source + '(' + quote + source + quote + ')'
quine("""def quine(source):
quote = '"'*3
print source + '(' + quote + source + quote + ')'
quine""")
Create a function, that when called,
outputs the input and the function scaffolding
Once built, we can add any arbitrary code into the quine!
def quine(source):
quote = '"'*3
x = 1
y = 2**4
print source + '(' + quote + source + quote + ')'
quine("""def quine(source):
quote = '"'*3
x = 1
y = 2**4
print source + '(' + quote + source + quote + ')'
quine""")
A direct result of Kleen's recursion theorem, says among other things, that a quine is possible in any language
Programs that fail, but the error message is valid code (which happens to be the original source!)
Highly version and even system specific
Chain of quines: output is fed back in times
Not fixed points, but cycles:
Chain of quines: output of one language is fed into another
Not fixed points, but cycles of different functions:
Usually implies minimization of a global utility
May not be fair
May only be possible with outside help
The ratio of utilitarian to egalitarian ,
or best global average to the most fair
No shortcut
Drivers spread out evenly on both routes
This is a Nash equilibrium.
With shortcut
Drivers only take route top/bottom
This is a Nash equilibrium.
A divergent geometric series ... hopeless?
Take the limit of the arithmetic means
Thus Grandi's series is "Cesaro" summable to 1/2
Take the series
Consider the power series
If it converges in , then take limit
Partial sums visit every natural integer!
Cauchy product of two Grandi series
Not Cesaro summable, but an Abel summation gives 1/4
Can also be solved with
(not covered today, but they give 1/4!)
For this is the Riemann zeta function (super important)
Let be our series and (let's pretend)
that everything will be OK at
is not Abel summable, but it can be zeta regularized when we analytically continue onto the complex plane
It is a shadow of the original function, but it is finite...
Consider the expectation value of the zero-point energy for all standing waves of an E&M field in a cavity
This sum clearly diverges ...
for mortals
Two metal plates of area distance apart
Zeta normalized, take limit
The force scales as
This is real and can be measured!
This has a radius of convergence of 1/2 hence it is not convergent at 1. However there is a unique analytic continuation onto the complex plane with 1/2 deleted.