Proton stopping in C+C, d+C, C+Ta and d+Ta collisions at 4.2A GeV/c

The shape of proton rapidity distributions is analysed in terms of their Gaussian components, and the average rapidity loss is determined in order to estimate the amount of stopping in C+C, d+C, C+Ta and d+Ta collisions at 4.2A GeV/c. Three Gaussians correspond to the nuclear transparency and describe well all peripheral and also C+C central collisions. Two-component shape is obtained in case of d+C and C+Ta central collisions. Finally one Gaussian, found in d+Ta central collisions, corresponds to the full stopping. The calculated values of the average rapidity loss support the qualitative relationship between the number of Gaussian components and the corresponding stopping power. It is also observed, in central collisions, that the average rapidity loss increases with the ratio of the number of target and the number of projectile participants.Comment: 9 pages REVTeX, 1 PS figure replaced, to be published in Phys.Rev.

STRUCTURE FOR REGULAR INCLUSIONS

We study pairs (C,D) of unital C∗-algebras where D is an abelian C∗-subalgebra of C which is regular in C in the sense that the span of {v 2 C : vDv∗ [ v∗Dv D} is dense in C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D whose restriction to D is the identity on D. We show that the left kernel of E, L(C,D), is the unique closed two-sided ideal of C maximal with respect to having trivial intersection with D. When L(C,D) = 0, we show the MASA D norms C in the sense of Pop-Sinclair-Smith. We apply these results to significantly extend existing results in the literature on isometric isomorphisms of norm-closed subalgebras which lie between D and C. The map E can be used as a substitute for a conditional expectation in the construction of coordinates for C relative to D. We show that coordinate constructions of Kumjian and Renault which relied upon the existence of a faithful conditional expectation may partially be extended to settings where no conditional expectation exists. As an example, we consider the situation in which C is the reduced crossed product of a unital abelian C∗-algebra D by an arbitrary discrete group acting as automorphisms of D. We charac- terize when the relative commutant Dc of D in C is abelian in terms of the dynamics of the action of and show that when Dc is abelian, L(C,Dc) = (0). This setting produces examples where no conditional expectation of C onto Dc exists. In general, pure states of D do not extend uniquely to states on C. However, when C is separable, and D is a regular MASA in C, we show the set of pure states on D with unique state extensions to C is dense in D. We introduce a new class of well behaved state extensions, the compatible states; we identify compatible states when D is a MASA in C in terms of groups constructed from local dynamics near an element 2 ˆD. A particularly nice class of regular inclusions is the class of C∗-diagonals; each pair in this class has the extension property, and Kumjian has shown that coordinate systems for C∗-diagonals are particularly well behaved. We show that the pair (C,D) regularly embeds into a C∗-diagonal precisely when the intersection of the left kernels of the compatible states is trivial

Uniquely D-colourable digraphs with large girth

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of Mathematic

D-Branes and Spin^c Structures

It was recently pointed out by E. Witten that for a D-brane to consistently wrap a submanifold of some manifold, the normal bundle must admit a Spin^c structure. We examine this constraint in the case of type II string compactifications with vanishing cosmological constant, and argue that in all such cases, the normal bundle to a supersymmetric cycle is automatically Spin^c.Comment: 9 pages, LaTe