[...]... physical limit of any physical measurement: the finite resolution (C) The papers by M N Mihaila (M) and P Handel (H) remind us that the question of the origin of 1/f noise is as old as electronics and is very universal Nonlinear interactions between the lattice phonons and thermal phonons are clearly involved (M) Besides in the quantum 1/f effect, the basic nonlinear system is the charged particle interacting... the diophantine type (as it is the case for resonances in celestial mechanics) and are calculated by truncating the continued fraction expansion of the frequency ratio of the oscillators at the mixer inputs The standard map introduced by B Chirikov in 1979 is an alternative way to describe the nonlinear coupling between two oscillators as shown in the paper by P Moussa (M) It allows one to express the... The method consists in stretching the formulas to their extreme consequences, resorting to some internal feeling of coherence and harmony They are obvious pitfalls in such methods, and only experience can tell you cartier@ihes.fr M Planat (Ed.): LNP 550, pp 6–67, 2000 c Springer- Verlag Berlin Heidelberg 2000 Mathemagics 7 that for the Dirac δ-function an expression like xδ(x) or δ (x) is lawful,... numbers started as a shorthand for repeated additions, for instance 7 times 3 (or rather “seven taken three times”) is the sum of three terms equal to 7 7 × 3 = 7 + 7 + 7 3 times In the same vein 73 (so denoted by Viete and Descartes) means 7 × 7 × 7 3 factors There is no difficulty to define x2 as xx or x3 as xxx for any kind of multiplication (numbers, functions, matrices ) and Descartes uses interchangeably... to a and h D to a But life is not so easy If we take two matrices A and B and calculate exp(A + B) and exp A exp B by expansion we get 1 1 exp(A + B) = I + (A + B) + (A + B)2 + (A + B)3 + · · · 2 6 1 exp A exp B = I + (A + B) + (A2 + 2AB + B 2 ) 2 1 + (A3 + 3A2 B + 3AB 2 + B 3 ) + · · · 6 (14) (15) If we compare the terms of degree 2 we get 1 1 (A + B)2 = (A2 + AB + BA + B 2 ) 2 2 (16) in (13) and. .. fact that v ⊗ w is bilinear in v, w, but nothing more Taking V and W to be the space C ∞ (I) of the functions defined and indefinitely derivable in an interval I of R, we define the operators D1 and D2 in C ∞ (I) ⊗ C ∞ (I) by D1 (f ⊗ g) = Df ⊗ g, D2 (f ⊗ g) = f ⊗ Dg (27) The two operators D1 D2 and D2 D1 transform f ⊗ g into Df ⊗ Dg, hence ¯ D1 and D2 commute Define D as D1 + D2 hence ¯ D(f ⊗ g) = Df ⊗ g... deduction, and not simply the mathematical theories, have evolved over the centuries In modern times, theories like General Topology or Lebesgue’s Integration Theory represent an almost perfect model of precision, flexibility and harmony, and their applications, for instance to probability theory, have been very successful My thesis is: there is another way of doing mathematics, equally successful, and the... 11)/10 with periodicity (2, 1, 1, 2) in the cfe and so on Getting the whole theory is a formidable task which is attempted in the paper by S Perrine in Part III The first-order theory was obtained by Markoff in 1880 It predicts Markoff numbers at Ai = ai (9a2 − 4)1/2 where the ai ’s are 1, 2, 5, 13, 29 and satisfy i the algebraic condition a2 + b2 + c2 = 3abc and are the traces of matrices in a subgroup... topological terms it is a punctured torus as it is nicely explained in Gutzwiller’s book (Chaos in classical and quantum mechanics, Springer, 1990)] Frequency fluctuations of the oscillators are often characterized in the frequency domain from the power spectral density or in the time domain from 4 Michel Planat the Allan deviation (this averages the mean frequency deviation between two consecutive samples,... play the dominant role in the dynamics Since the psd of |1/ζ(s)| is a “white noise” at the critical line s = 1 + it and transforms into a “1/f 2 noise” close to it, it is tempting to expect that ζ(s) should play a role in the unification of physics Can we find an algebraic definition of randomness? This is attempted in the paper by J.P Allouche in Part III restricting the study to infinite sequences taking . Conditions and Real or Complex Brjuno Functions Pierre Moussa, Stefano Marmi 324 Part III Algebraic Randomness Algebraic and Analytic Randomness Jean-Paul Allouche 345 From Symbolic Dynamics to. (Ed. ) Noise, Oscillators and Algebraic Randomness FromNoiseinCommunicationSystems to Number Theory Lectures of a School Held in Chapelle des Bois, France, April 5–10, 1999 13 Editor Michel Planat LaboratoiredePhysiqueetMetrologie des. function F (x, y) of two variables into the one-variable function F (x, x) hence f(x)g(y) into f(x)g(x) as it should. Formula (2 4) reads now ∂ ∂x (f(x)g(x )) = ( ∂ ∂x + ∂ ∂y )f(x)g(y)| y=x . (3 1)