A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

The Orevkov invariant of an affine plane curve

We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.

     

Electron-cluster interactions

Beam depletion spectroscopy has been used to measure absolute total inelastic electron-sodium cluster collision cross sections in the energy range fromE0.1 toE6 eV. The investigation focused on the closed shell clusters Na₈, Na₂₀, and Na₄₀. The measured cross sections show an increase for the lowest collision energies where electron attachment is the primary scattering channel. The electron attachment cross section can be understood in terms of Langevin scattering, connecting this measurement with the polarizability of the cluster. For energies above the dissociation energy the measured electron-cluster cross section is energy independent, thus defining an electron-cluster interaction range. This interaction range increases with the cluster size.     

Measurement of diffusion coefficients using a quick echo split NMR imaging technique

A new method for the ultrafast generation of diffusion-weighted images is reported. The technique combines a quick echo split NMR imaging sequence with the principle of Stejskal and Tanner. It allows to determine the diffusion constant with nearly the same accuracy as the conventional spin-echo technique, requiring only a fraction of the time. The determined values for water doped with 1 g Cu(NO₃)₂ per liter of H₂O and pure acetone were D_water = (1.95 ± 0.02) × 10⁻⁹ m²/s and D_acetone = (4.05 ± 0.02) × 10⁻⁹ m²/s at 18.5°C. They are in good agreement both with literature and our own reference measurements using a diffusion-weighted spin-echo sequence. In addition, the temperature dependence of D_water was measured in the range of 18.5–45.9°C and a good correspondence with reported data was found.     

Cumulants and moments

We introduce a ¹-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.     

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

On the Functional Equation of Periodic Lines

It will be proved that the functional equation of periodic lines has no solution for hyperbolic distances.