Recent results for tree amplitudes for the N = 2 noncritical strings are presented and compared with the critical case. Arguments are given which indicate a certain discontinuity in passing from thê c < 1 model (in a Coulomb gas representation) to thê c = 1 critical case.

The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of [11], extending those results to the more general case and in some instances g...

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of β-expansions (with positive or negative β) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is pr...

We build up normal ordered products for fermionic open string coordinates consistent with boundary conditions. The results are obtained considering the presence of antisymmetric tensor fields. We find a discontinuity of the normal ordered products at string endpoints even in the absence of the background. We discuss how the energy momentum tensor also changes at the world-sheet boundary in such...

Research demonstrates that police reduce crime. We study this question by using a natural experiment in which a private university increased the number of police patrols within an arbitrarily defined geographic boundary. Capitalizing on the discontinuity in patrols at the boundary, we estimate that the extra police decreased crime in adjacent city blocks by 43–73%. Our results are consistent wi...

We show that the recently demonstrated absence of the van Dam-VeltmanZakharov discontinuity for massive spin 3/2 with a Λ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop. As a numerical check on the calculation, we rederive the vanishing of the oneloop beta function for D = 11 supergravity on AdS4×S level-by-level in the Kaluza-Klein tower. Typese...