Representability of Hom implies flatness

Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that the functor $Hom_{L^{-n},F}$ is representable for infinitely many positive integers $n$, then $F$ is flat over $S$. As a corollary, taking $X=S$, it follows that if $F$ is a coherent sheaf on $S$ then the functor $T\mapsto H^0(T, F_T)$ on the category of $S$-schemes is representable if and only if $F$ is locally free on $S$. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author's earlier result (see arXiv.org/abs/math.AG/0204047) that the automorphism group functor of a coherent sheaf on $S$ is representable if and only if the sheaf is locally free.Comment: 9 pages, LaTe

Planetary Nebula Abundances and Morphology: Probing the Chemical Evolution of the Milky Way

This paper presents a homogeneous study of abundances in a sample of 79 northern galactic planetary nebulae whose morphological classes have been uniformly determined. Ionic abundances and plasma diagnostics were derived from selected optical line strengths in the literature, and elemental abundances were estimated with the Ionization Correction Factor developed by Kingsbourgh & Barlow (1994). We compare the elemental abundances to the final yields obtained from stellar evolution models of low-and intermediate-mass stars, and we confirm that most Bipolar planetary nebulae have high nitrogen and helium abundance, and are the likely progeny of stars with main-sequence mass larger than 3 solar masses. We derive =0.27, and discuss the implication of such a high ratio in connection with the solar neon abundance. We determine the galactic gradients of oxygen and neon, and found Delta log (O/H)/Delta R=-0.01 dex/kpc$ and Delta log (Ne/H)/Delta R=-0.01 dex/kpc. These flat PN gradients do not reconcile with galactic metallicity gradients flattening with time.Comment: The Astrophysical Journal, in pres

Circumstellar Interaction Around Type Ib/c Supernovae and the GRB Connection

Radio observations of Type Ib/c supernovae suggest that circumstellar interaction takes place with a wide range of wind densities, comparable to that seen in Galactic Wolf-Rayet stars. Efficient production of magnetic field in the shocked region is needed. The X-ray emission observed from some Type Ib/c supernovae is higher than would be expected by the thermal or inverse Compton mechanisms; a synchrotron interpretation requires a flattening of the electron energy spectrum at high energies, as might occur in a cosmic ray dominated shock wave. The wind density variations that are indicated in two supernovae may be due to a binary companion, although variable mass loss from a single star remains a possibility. Other than the optical supernova radiation, the emission from the nearby SN 2006aj/GRB 060218 appears to be powered by a central engine, while that from SN 1998bw/GRB 980425 could be powered by either a central engine or the outer supernova ejecta.Comment: 8 pages, to appear in Circumstellar Media and Late Stages of Massive Stellar Evolution, RevMexA

A Note on the Expressiveness of BIP

We extend our previous algebraic formalisation of the notion of component-based framework in order to formally define two forms, strong and weak, of the notion of full expressiveness. Our earlier result shows that the BIP (Behaviour-Interaction-Priority) framework does not possess the strong full expressiveness. In this paper, we show that BIP has the weak form of this notion and provide results detailing weak and strong full expressiveness for classical BIP and several modifications, obtained by relaxing the constraints imposed on priority models.Comment: In Proceedings EXPRESS/SOS 2016, arXiv:1608.0269

ASSOCIATIONS BETWEEN EXERCISE-INDUCED FLATTENING OF FOOT ARCH AND FATIGUE OF EXTRINSIC AND INTRINSIC FOOT MUSCLES

We aimed to examine the associations between exercise-induced flattening of the foot arch and fatigue of extrinsic and intrinsic foot muscles. Fifteen male participants repeated 100 hops/set until they could no longer maintain the hop frequency or had completed 30 sets. The sagittal plane range of motion (ROM) of the midfoot during stance phase significantly decreased at the final set compared to the initial set (-8.8°). After the fatigue task, MRI T2 relaxation time in all measured extrinsic and intrinsic foot muscles significantly increased (17.2‒37.4%); however, only its increase in the tibialis posterior (TP) correlated with the relative change in midfoot ROM (r = 0.684). These results suggest that fatigue of TP is associated with the occurrence of flattening of the foot arch. This study provides a clinical implication that monitoring midfoot kinematics can be used to assess the condition of TP