Horrocks Correspondence on a Quadric Surface

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.Comment: 19 page

Horrocks Correspondence on ACM Varieties

We describe a vector bundle \sE on a smooth $n$-dimensional ACM variety in terms of its cohomological invariants H^i_*(\sE), $1\leq i \leq n-1$, and certain graded modules of "socle elements" built from \sE. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems where we construct vector bundles from these invariants and uniqueness theorems where we show that these data determine a bundle up to isomorphisms. The cases of the quadric hypersurface in $\mathbb P^{n+1}$ and the Veronese surface in $\mathbb P^5$ are considered in more detail.Comment: 18 pages, not figure

The Bis(ferrocenyl)phosphenium Ion Revisited

The bis(ferrocenyl)phosphenium ion, [Fc2P]+, reported by Cowley et al. (J. Am. Chem. Soc. 1981, 103, 714–715), was the only claimed donor‐free divalent phosphenium ion. Our examination of the molecular and electronic structure reveals that [Fc2P]+ possesses significant intramolecular Fe⋅⋅⋅P contacts, which are predominantly electrostatic and moderate the Lewis acidity. Nonetheless, [Fc2P]+ undergoes complex formation with the Lewis bases PPh3 and IPr to give the donor–acceptor complexes [Fc2P(PPh3)]+ and [Fc2P(IPr)]+ (IPr=1,3‐bis(2,6‐diisopropylphenyl)imidazole‐2‐ylidene)

Rank two bundles on P^n with isolated cohomology

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair of values $(k,n-k)$. We show that on $\mathbb P^8,$ if $H^3_*(\mathcal E)=H^4_*(\mathcal E)=0$, then $\mathcal E$ must be decomposable. More generally, we show that for $n\geq 4k$, there is no indecomposable bundle $\mathcal E$ for which all intermediate cohomology modules except for $H^1_*, H^k_*, H^{n-k}_*, H^{n-1}_*$ are zero.Comment: 14 pages, no figure

On the traction problem for steady elastic oscillations equations: the double layer potential ansatz

The three-dimensional traction problem for steady elastic oscillations equations is studied. Representability of its solution by means of a double layer potential is considered instead of the more usual simple layer potential

On the double layer potential ansatz for the n-dimensional helmholtz equation with Neumann condition

In the present paper we consider the Neumann problem for the ndimensional Helmholtz equation. In particular we deal with the problem of representability of the solutions by means of double layer potentials

The water supercooled regime as described by four common water models

The temperature scale of simple water models in general does not coincide with the natural one. Therefore, in order to make a meaningful evaluation of different water models a temperature rescaling is necessary. In this paper we introduce a rescaling using the melting temperature and the temperature corresponding to the maximum of the heat capacity to evaluate four common water models (TIP4P-Ew, TIP4P-2005, TIP5P-Ew and Six-Sites) in the supercooled regime. Although all the models show the same general qualitative behavior, the TIP5P-Ew appears as the best representation of the supercooled regime when the rescaled temperature is used. We also analyze, using thermodynamic arguments, the critical nucleus size for ice growth. Finally, we speculate on the possible reasons why atomistic models do not usually crystalize while the coarse grained mW model do crystallize.Comment: 8 pages, 8 figure