Control Systems and Number Theory

Control Systems and Number Theory

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, th...

Integrable systems and number theory

where is the derivative of with respect to , was derived to describe water waves in shallow channels [KdV95]. It bears the names of Korteweg and de Vries and has become the prototype integrable nonlinear partial differential equation since the work in [Miu68, MGK68, SG69, Gar71, KMGZ70, GGKM74]. The KdV equation is for the field of integrability what the harmonic oscillator is for quantum mecha...

Number theory, dynamical systems and statistical mechanics

We shortly review recent work interpreting the quotient ζ(s− 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group Gk := (Z/2Z), with Z/2Z = ({0, 1},+). we set inductively h0 := 1, hk+1(σ, 0) := hk(σ) and hk+1(σ, 1) := hk(σ) + hk(1− σ), (1) ...

Ubiquitous Systems and Metric Number Theory

We investigate the size and large intersection properties of Et = {x ∈ R d | x − k − x i < r i t for infinitely many (i, k) ∈ I µ,α × Z d }, where d ∈ N, t ≥ 1, I is a denumerable set, (x i , r i) i∈I is a family in [0, 1] d × (0, ∞) and I µ,α denotes the set of all i ∈ I such that the µ-mass of the ball with center x i and radius r i behaves as r i α for a given Borel measure µ and a given α >...

Control Theory and Systems Biology