Kamienny's criterion and the method of Coleman and Chabauty

This paper gives a new proof of Kamienny's Criterion using the method of Coleman and Chabauty.

     

Global Geometry of 3-Body Motions with Vanishing Angular Momentum Ⅰ

Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）.
In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.     

Assessment of the Nature of Depolarization Current in Thermally Treated Kapton Polyimide

Pyrolysis of normally insulating aromatic polyimide is known to impart electrical conductivity to the polymer due to the formation of carbonized regions in an insulating matrix with a concomitant change in the polymer’s structural arrangement. The wholly pyrolyzed polyimide is potentially useful for specific applications in certain types of semiconductor devices because of the polyimide’s insulator/conductor transition which creates a barrier type conduction. Pyrolysis, however, degrades the required mechanical integrity of the polyimide for construction of such devices. In order to evaluate the fundamental aspects of barrier conduction by high voltage electron transfer from metal contact that can still produce measurable current in thermally treated non-pyrolyzed polyimide, the nature of depolarization in Kapton was assessed by the thermally stimulated depolarization current (TSDC) technique. The results show that thermal treatment of polyimide without pyrolysis and therefore without loss of mechanical integrity offers a viable means of steady electron conduction for semiconductor operation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM _−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM _−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM _−L can be regarded as the compactified moduli scheme of vector bundles onY with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves onY. The relation to Seshadri-Nagaraj conjecture is studied.     

Anchoring of sulfur‐containing alkylphosphonic and semifluorinated alkylphosphonic molecules on a polycrystalline aluminum substrate

Self‐assembly on a polycrystalline aluminum substrate of two sulfur‐containing alkylphosphonic acids, CH₃? (CH₂)₁₁? S? (CH₂)₂? PO(OH)₂, and CF₃? (CF₂)₇? (CH₂)₂? S? (CH₂)₂? PO(OH)₂, has been compared with CH₃? (CH₂)₁₅? PO(OH)₂. The reaction of the phosphonic head groups with the hydroxyls at the alumina surface to form phosphonates was studied with X‐ray photoelectron spectroscopy (XPS) and polarization modulation infrared reflection‐absorption spectroscopy (PM‐IRRAS). Barrier effects of the resulting layers was assessed by electrochemical polarization curves. With the conditions used in the present work for the self‐assembly reaction, it appears that the sulfur‐containing molecules do not perform as well as CH₃? (CH₂)₁₅? PO(OH)₂ in terms of film quality. Questions are raised about the possibility that the sulfur‐containing molecules could undergo cleavage during surface modification. Copyright © 2008 John Wiley & Sons, Ltd.     

In response to the article by Ivan Kausz: A discriminant and an upper bound for w 2 for hyperelliptic arithmetic surfaces

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.     

Structural transitions of NaAlH4 under high pressure by first-principles calculations

First-principles calculations have been performed on NaAlH₄ using the generalized gradient approximation pseudopotential method. The predicted β-NaAlH₄ (α-LiAlH₄-type) structure is energetically more favorable than α-NaAlH₄ for pressures over 15.9 GPa, which is apparently correlated with the experimental transition pressure 14 GPa. This transition is identified as first-order in nature with volume contractions of 1.8%. There is no pressure-induced softening behavior from our calculated phonon dispersion curves near the phase transition pressure. Based on the Mulliken population analysis, the β-NaAlH₄ structure is expected to be the most promising candidate for hydrogen storage.     

参数曲线的全局凸性判别条件

基于平面曲线的二次微商,导出了二重点的判别条件,结合参数曲线的局部凸性条件,得到了参数闭曲线的充要条件。给出了参数曲线的拐点判别条件,从而得到了参数曲线局部凸的充要条件。     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.