Symmetric transition probabilities in convex model of quantum mechanics

The relation between the geometry and the statistics of the quantum theory is most explicit in the convex models of quantum mechanics. A class of models is investigated where the manifold S of all states is a convex set with a smooth boundary. It is shown that for these models the simple assumption about symmetry of the transition probability implies an ellipsoidal shape of S, thus leading to the “spherical” geometries described in [7].     

Completeness properties of perturbed sequences

If S is an arbitrary sequence of positive integers, let P(S) be the set of all integers which are representable as a sum of distinct terms of S. Call S complete if P(S) contains all large integers, and subcomplete if P(S) contains an infinite arithmetic progression. It is shown that any sequence can be perturbed in a rather moderate way into a sequence which is not subcomplete. On the other hand, it is shown that if S is any sequence satisfying a mild growth condition, then a surprisingly gentle perturbation suffices to make S complete in a strong sense. Various related questions are also considered.     

The coordinate transformations of the absolute space-time theory

In the light of our recently performed experiments, revealing the anisotropy of light velocity in any frame moving with respect to absolute space, we show that the Lorentz transformation, where the relativity of light velocity is given implicitly through the relativity of the time coordinates, must be treated from an absolute point of view if one seeks to preserve its adequacy to physical reality. Then we propose a new transformation (which is to be considered as a legitimate companion of the Lorentz transformation) wherein the relativity of light velocity is given explicitly and the time coordinates are absolute.     

The conformational dependence of vicinal 13C13C spin–spin coupling constants in alicyclic compounds

A series of alicyclic compounds with dihedral angles of 0°, 60°, 90°, 120° and 180° between a ¹³C-labelled carbon atom and a carbon atom separated by three bonds from the label has been synthesized. The vicinal ¹³C¹³C spin coupling constants were measured, and from the results a Karplus-type relationship between ¹³C¹³C spin coupling and dihedral angle is proposed.     

Normal Forms for Nonautonomous Differential Equations

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λ_j−∑ⁿ_i=1 ?_iλ_i≠0 for eigenvalues is generalized to the new nonresonance condition λ_j∩∑ⁿ_i=1 ?_iλ_i=∅ for spectral intervals.