Three-loop corrections in a new variational treatment ofλφ 4

We present the three-loop calculation of the effective potential forλφ ⁴ in 3+1 dimensions, based on a new variational approach to quantum field theory. This formulation is based on a 2PPI loop expansion of an effective action for local composite operators and yields non-perturbative results. Renormalisation is carried out for both the trivial and the precarious phase using dimensional regularisation. The three-loop result is seen to be similar to the one-loop result (which coincides with the G.E.P.) and only one numerical constant in the effective potential has to be changed.     

Weierstrass points on trigonal curves I: The ramification points

LetC be a smooth curve of genusg≥5. Assume thatP is a Weierstrass point onC which first non-gap is equal to 3. The gap sequence atP is completely determinated by numbersn and ε satisfying (g−1)/3≤n≤g/2 and ε is 1 or 2 as follows. Given suchn and ε, the corresponding gap sequence is (1, 2, 4, 5,…, 3n−2, 3n−1, 3n+ε, 3n+3+ε, …, 3(g−n−1)+ε). We say thatP is of then-th kind andP is of type I (resp. II) if ε=1 (resp. 2). Because a curve of genusg≥5 has at most one linear systemg1/3, it follows that the Weierstrass points onC with first non-gap equal to 3 are of the same kind.     

Transverse beam spin asymmetries at backward angles in elastic electron-proton and quasielastic electron-deuteron scattering

We have measured the beam-normal single-spin asymmetries in elastic scattering of transversely polarized electrons from the proton, and performed the first measurement in quasielastic scattering on the deuteron, at backward angles (lab scattering angle of 108°) for Q2 = 0.22 GeV2/c2 and 0.63 GeV2/c2 at beam energies of 362 and 687 MeV, respectively. The asymmetry arises due to the imaginary part of the interference of the two-photon exchange amplitude with that of single-photon exchange. Results for the proton are consistent with a model calculation which includes inelastic intermediate hadronic (πN) states. An estimate of the beam-normal single-spin asymmetry for the scattering from the neutron is made using a quasistatic deuterium approximation, and is also in agreement with theory.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Representations of epi-Lipschitzian sets

A closed subset M of a Banach space E is epi-Lipschitzian, i.e., can be represented locally as the epigraph of a Lipschitz function, if and only if it is the level set of some locally Lipschitz function f:E→R, for which Clarke’s generalized gradient does not contain 0 at points in the boundary of M, i.e., such that:

     

Influence of the Al source and synthesis of ordered Al-SBA-15 hexagonal particles with nanostairs and terraces

We investigated the effect of various aluminum sources on the morphology and microstructure of nanoporous Al-SBA-15. With aluminum tri-sec-butoxide (Al-TSB), very large, ordered hexagonal rods of Al-SBA-15 covered by nanostairs and terraces were synthesized. Such a morphology was not obtained with an inorganic Al source (sodium aluminate) or with organic sources seemingly similar to Al-TSB, such as aluminum tri-tert-butoxide, aluminum tri-n-butoxide, or aluminum isopropoxide. The results obtained from X-ray diffraction, scanning electron microscopy/transmission electron microscopy, N(2) adsorption/desorption, and high-performance (129)Xe NMR suggest that preparing an organic Al-Si precursor by premixing liquid organic Al and Si sources (Al-TSB and tetraethyl orthosilicate) is crucial in obtaining highly ordered mesoporous Al-SBA-15 materials with a well-defined morphology.