Vertex Corrections in Gauge Theories for Two-dimensional Condensed Matter Systems

We calculate the self-energy of two-dimensional fermions that are coupled to transverse gauge fields, taking two-loop corrections into account. Given a bare gauge field propagator that diverges for small momentum transfers q as 1/qη, 1<η≤ 2, the fermionic self-energy without vertex corrections vanishes for small frequencies ω as Σ(ω)∝ ωγ with γ=2/(1+η)<1. We show that inclusion of the leading radiative correction to the fermion-gauge field vertex leads to Σ(ω)∝ωγ [1-aη ln (ω0/ω)], where aη is a positive numerical constant and ω0 is some finite energy scale. The negative logarithmic correction is consistent with the scenario that higher order vertex corrections push the exponent γ to larger values.

Maximal inequalities of Kahane–Khintchine's type in Orlicz spaces

Several maximal inequalities of Kahane–Khintchine's type in certain Orlicz spaces are proved. The method relies upon Lévy's inequality and the technique established in [14] which is obtained by Haagerup–Young–Stechkin's best possible constants in the classical Khintchine inequalities. Moreover by using Donsker's invariance principle it is shown that the numerical constant in the inequality deduced by the method presented is nearly optimal: If is a Bernoulli sequence, and ‖ · ‖ψ denotes the Orlicz norm induced by the function then the following inequality is satisfied:for all a1,…, an and all n ≥ 1, and the best possible numerical constant which can take the place of lies in the interval ]. Sharp estimates of this type are also deduced for some other maximal inequalities in Orlicz spaces discovered in this paper.

On: “Ionospheric induced very low frequency electric field wavetilt changes” by D. V. Thiel and I. J. Chant (GEOPHYSICS, January, 1982, p. 60–62)

The paper by Thiel and Chant reports the well‐known sunrise and sunset effect which in micropulsation studies is referred to as the dawn and dusk “chorus.” However, we disagree with the interpretation of these authors. Their choice of wavetilt as the diagnostic function is slightly misleading since they are essentially measuring at the surface of the earth despite the unfortunate location of their magnetic sensor at a height of 4 m and presumably in a building. In this case the wavetilt [Formula: see text] according to their equation (1) is equivalent to [Formula: see text] where [Formula: see text] is the impedance of the earth and [Formula: see text] is the impedance of free space. This result is independent of the mode of propagation but certainly at their distance of 4000 km is predominantly the sky wave except during the dawn and dusk periods. The turbulence due to the formation or breakup of the D‐layer in the ionosphere virtually destroys the ionospheric reflected component which is the dominant contributor to the field incident on the earth. We suggest that during most of the day the authors are measuring the impedance of the earth scaled by a numerical constant.