The Betti numbers of forests

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs

Positivity constraints on initial spin observables in inclusive reactions

For any inclusive reaction of the type $A_1({spin} 1/2)+ A_2({spin} 1/2) \to B + X$, we derive new positivity constraints on spin observables and study their implications for theoretical models in view, in particular, of accounting for future data from the polarized $pp$ collider at BNL-RHIC. We find that the single transverse spin asymmetry $A_N$ for several processes, in the central region for example jet production, direct photon production, lepton-pair production, must be such that |A_N| \lappeq 1/2, rather than the usual bound $|A_N| \leq 1$.Comment: 6 pages, version to be published in Phys.Rev.Let

The Betti numbers of forests

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs