Walter Wyss

Fractional diffusion and wave equations

Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.

The fractional diffusion equation

In one space—and in one time—dimension a diffusion equation is solved, where the first time derivative is replaced by the λ‐fractional time derivative, 0<λ≤1. The solution is given in closed form in terms of Fox functions.

The Field algebra and its positive linear functionals

We show that positive linear functionals on the field algebra are necessarily continuous and can be represented by conical measures. Furthermore extension theorems for continuous linear functionals, defined on a subspace of the field algebra, to positive linear functionals are discussed.

Poincaré gauge in electrodynamics

The gauge presented here, which we call the Poincare gauge, is a generalization of the well‐known expressions φ = −r⋅E0 and A = 1/2 B0×r for the scalar and vector potentials which describe static, uniform electric and magnetic fields. This gauge provides a direct method for calculating a vector potential for any given static or dynamic magnetic field. After we establish the validity and generality of this gauge, we use it to produce a simple and unambiguous method of computing the flux linking an arbitrary knotted and twisted closed circuit. The magnetic flux linking the curve bounding a Mobius band is computed as a simple example. Arguments are then presented that physics students should have the opportunity of learning early in their curriculum modern geometric approaches to physics. (The language of exterior calculus may be as important to future physics as vector calculus was to the past.) Finally, an appendix illustrates how the Poincare gauge (and others) may be derived from Poincare’s lemma relatin...

Bures distance and relative entropy

We have previously constructed an entropy functional which characterizes statistical inference from partial measurement by maximum relative entropy. Here we discuss the mathematical properties of this concept in greater detail and establish its relation to the Bures distance and the Uhlmann transition probability.

Statistical Inference and Entropy

We construct an entropy function such that statistical inference with respect to a partial measurement and a given a priori distribution is characterized by maximal entropy.

A note on relative entropy

The quantum-mechanical concept of relative entropy is discussed from an information-theoretic point of view. We show that not all definitions found in the recent literature are equally suitable for the purpose of statistical inference by entropy maximization.

The quasi-multiplication on rings and algebras

On a ring or an algebra we introduce the quasi-multiplication a ⨂ b = ab + a + b, taking into account the multiplication and addition simultaneously. An example shows that this quasi-multiplication has remarkable and fundamental properties.

Taylor's theorem for analytic functions of operators

We discuss analytic functions on a Banach algebra into itself. In particular expressions for derivatives are given as well as convergent Taylor expansions.

Cumulants and moments

We introduce a ∗-algebra of terminating sequences. The set of normalized linear functionals on this algebra is then equipped with a quasi-multiplication and thus with a multiplication. A state corresponds to a sequence of moments. With it we associate its sequence of cumulants. The quasi-multiplication relates cumulants and moments in a very easy way. We give necessary and sufficient conditions for a sequence to be a sequence of moments and necessary and sufficient conditions for a sequence to be a sequence of cumulants.