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.     

On CP,LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.     

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.     

Revealing an Interconnected Interfacial Layer in Solid‐State Polymer Sodium Batteries

Replacing the commonly used nonaqueous liquid electrolytes in rechargeable sodium batteries with polymer solid electrolytes is expected to provide new opportunities to develop safer batteries with higher energy densities. However, this poses challenges related to the interface between the Na‐metal anode and polymer electrolytes. Driven by systematically investigating the interface properties, an improved interface is established between a composite Na/C metal anode and electrolyte. The observed chemical bonding between carbon matrix of anode with solid polymer electrolyte, prevents delamination, and leads to more homogeneous plating and stripping, which reduces/suppresses dendrite formation. Full solid‐state polymer Na‐metal batteries, using a high mass loaded Na₃V₂(PO₄)₃ cathode, exhibit ultrahigh capacity retention of more than 92 % after 2 000 cycles and over 80 % after 5 000 cycles, as well as the outstanding rate capability.     

Contribution to poly(m-xylylene adipamide) characterization: Hydrolysis,condensation, and oxidation in the melt

PAmXD,6 [poly(m-xylylene adipamide)] was processed using a Brabender plastograph at 30 rpm and at 265 ± 5°C. The evolution of the M_n versus mixing time determined from solution viscosity and chain end concentrations measurements shows that PAmXD,6 undergoes hydrolysis or condensation reactions depending on its initial molecular mass. Lower masses undergo condensation while higher masses undergo hydrolysis. The determination of water concentration in the melt during mixing time confirmed that this behavior is related to an equilibrium constant which was estimated at 500 ± 100. For a mixing period shorter than 20 min, a good correspondence was observed between M_n values determined either by chain end concentrations or by solution viscosity. However, for a mixing period longer than 20 min, discrepancy between M_n values using both methods witnesses the probable appearance of oxidative degradation of PAmXD,6. Mechanisms adapted from the established oxidation reactions of PA6 and PA6,6 are proposed to justify the PAmXD,6 behavior. Those reactions could explain the slight yellowing of PAmXD,6 during its processing. It was also shown that the PAmXD,6 Brabender plastograph mixing torque (30 rpm, 265 ± 5°C) is linearly related to M_n in a log/log diagram. © 1996 John Wiley & Sons, Inc.     

Linear Series on 4 – Gonal Curves

Let C be the general 4 – gonal curve of genus g > 6. We investigate the complete linear series on C and the varieties W^r_d(C), and we study the birational models of C in ℙ^r (r ≥ 2) of minimal degree.