The determination of the nuclear ground state and transition charge density from measured electron scattering data

The evaluation of elastic and inelastic electron scattering cross sections is discussed in terms of the Fourier-Bessel expansion of the nuclear ground state and transition charge density, respectively. The method allows one to deduce the charge distributions and the moments as well as the corresponding errors, which originate on the one hand from the uncertainties in the measured data and on the other hand from the lack of knowledge about the large-q behaviour of the form factors; these two contributions are determined separately. The method is described and proved with an evaluation of pseudodata and then applied to ²⁰⁸Pb cross sections. For this nucleus, detailed results concerning the possible structure of the charge density are presented.     

Long range substituent-induced chemical shifts in the 13C NMR spectra of N-benzylidenebenzylamines

Carbon-13 chemical shifts for N-benzylidenebenzylamines are affected by substituents up to eleven bonds away from the substituent group. The data correlate well with Hammett constants.     

X-rays from antiprotonic helium in helium gas

X-ray transitions to the 4F, 3D, and 2P atomic levels of p?He have been observed with antiprotons stopped in He gas at 4 and 1.1 atm NT. The population by radiative transitions of the 3D level in gas Y_M(4atm) = (28±14)% and Y_M(1.1 atm) = (43±22)% exceeds by more than one order of magnitude that measured in liquid He. The annihilation width of the 3D level Γ^a_3D = 2.8±1.0 × 10^?3eV is determined from the ratio between the numbers of X-rays feeding and depopulating the 3D level. The strong-interaction shift of the 2P level ε(2P) = ?14±6 eV is obtained by inputting the $\text{p}\text{He}$ experimental X-ray yields into a cascade calculation, the results of which are in good agreement with well-established data from muonic, pionic, and kaonic helium.     

Model independent evaluation of the inelastic e− scattering cross sections from the 3− (2.615 MeV) level in 208Pb

The radial transition charge density and the reduced transition probability of the 3^? level in ²⁰⁸Pb are determined model independently from inelastic e^? scattering cross sections. The results are compared with (α, α') experiments and theoretical calculations.     

Convexity and a certain propertyP m

The propertyP _m (directly analogous to Valentine’s propertyP ₃) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyP _m and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets. (b) Any closed connected set with propertyP _m is polygonally connected. (c) A closed connected setS with propertyP _m is anL _m−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS). (d) A finite-dimensional set with propertyP _m is anL _{2m − 3} set. A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures.     

Near infrared transitions between Rydberg states of nitric oxide: The 4f-F2Δ band at 1.71 μm

Several near infrared transitions between highly excited Rydberg (R) states of nitric oxide have been recorded at high resolution (0.15 cm^?1) with a SISAM interferometer. The spectra all show a simple rotational structure, characteristic of R-R transitions. A full analysis has been performed of the previously unobserved 4f(v = 0) → F²Δ(v = 0) band at 1.71 μm. The band shows a peculiar structure, characteristic of Hund's Case (d) behaviour for one of the states. The relative branch intensities are calculated from first principles, and the agreement with the observed spectrum is excellent. This work shows the potential of interferometers for studying excited electronic states of simple molecules at high resolution.     

A Behavioural Approach to Obtaining a Doubly Constrained Trip Distribution Model

An alternative method of producing a doubly constrained trip distribution similar to that of a gravity model is described. This alternative model differs from conventional models in that it has an explicit, though naive, behavioural rationale. It also produces a doubly constrained distribution more rapidly than these conventional models, partly because it involves fewer iterations and partly because each successive iteration involves fewer zones than that preceding it. In the one full-scale test for which data are available, the new model efficiently simulates a survey derived distribution, while tests with hypothetical systems show that it produces a distribution which can be closely identified with that produced by a conventional model.     

On a sharp lower bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation with initial condition in the energy space and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in a sharp and stable upper bound on the blow-up rate: .

In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is,

In this paper, we prove the sharp lower bound

by exhibiting the dispersive structure in the scaling invariant space for this log-log regime. In addition, we will extend to the pure energy space a dynamical characterization of the solitons among the zero energy solutions.