Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

Taylor expansion of an Eisenstein series

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight for . Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke -functions.

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

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.