No abstract uploaded!

Within a well-known decay model describing a particle confined initially within a spherical potential shell, we consider the situation when the initial state has an unusual energy distribution decaying slowly as k; the simplest example corresponds to a wavefunction constant within the shell. We show that the non-decay probability as a function of time then behaves in a highly irregular, most likely fractal way.

Among approaches to defining the Feynman integral, various sequential methods are probably the most popular ones. Here we consider that one based on product formulae, the original idea of which belongs to Nelson71'. This method is powerful in the sense that it enables to prove the strict version of Feynman's basic dynamical formula for a wide class of poten tials^^/including time-dependent ones/4/in comparison with other known methods (see, eg, refs./5, 6/). 0n the other, hand, the method does not correspond exactly to Feynman's heuristic prescription because it replaces the exact action along poly gonal paths by its Riemannian approximation. Moreover, one can regard as an unsatisfactory feature that it employs the" equidistant" time-interval partitions only. In the present paper we show that the last mentioned sour ce of arbitrariness can be removed: the product formula for perturbations of propagators by

We show that solutions of Thurston's equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston's equation and a commutative ring associated to triangulated 3-manifolds. It is shown that solutions to homogeneous equations over the real numbers are critical points of an entropy function.

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) , where f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) stands for the density of unique measure-valued solution f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) satisfying\[\int| v|^{2k-2+ } dF_0 (v)<\infty, 0\leq < 2, k= 2, 3, 4,\cdots\] provided that f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) is not a single Dirac mass.