5,786 research outputs found
Hilbert-Kunz functions of cubic curves and surfaces
We determine the Hilbert-Kunz function of plane elliptic curves in odd
characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz
functions of nodal cubic curves. Together with results of K. Pardue and P.
Monsky, this completes the list of Hilbert-Kunz functions of plane cubics.
Combining these results with the calculation of the (generalized) Hilbert-Kunz
function of Cayley's cubic surface, it follows that in each degree and over any
field of positive characteristic there are curves resp. surfaces taking on the
minimally possible Hilbert-Kunz multiplicity.Comment: LaTex 2e with Xy-pic v3.2 for commutative diagram
Positive Semi-Definiteness and Sum-of-Squares Property of Fourth Order Four Dimensional Hankel Tensors
A positive semi-definite (PSD) tensor which is not a sum-of-squares (SOS)
tensor is called a PSD non-SOS (PNS) tensor. Is there a fourth order four
dimensional PNS Hankel tensor? Until now, this question is still an open
problem. Its answer has both theoretical and practical meanings. We assume that
the generating vector of the Hankel tensor is symmetric. Under this
assumption, we may fix the fifth element of at . We show that
there are two surfaces and with the elements of as variables, such that , is SOS if and only if
, and is PSD if and only if , where is the
first element of . If for a point , then there are no fourth order four dimensional PNS Hankel tensors
with symmetric generating vectors for such . Then, we
call such a point PNS-free. We show that a -degree planar closed convex
cone, a segment, a ray and an additional point are PNS-free. Numerical tests
check various grid points, and find that they are also PNS-free
Nonlinear Dirac equations on Riemann surfaces
We develop analytical methods for nonlinear Dirac equations. Examples of such
equations include Dirac-harmonic maps with curvature term and the equations
describing the generalized Weierstrass representation of surfaces in
three-manifolds. We provide the key analytical steps, i.e., small energy
regularity and removable singularity theorems and energy identities for
solutions.Comment: to appear in Annals of Global Analysis and Geometr
On the photofragmentation of SF: Experimental evidence for a predissociation channel
We report on the first observation of the photofragmentation dynamics of
SF. With the aid of state-of-the-art ab initio calculations on the
low-lying excited cationic states of SF performed by Lee et al. [J. Chem.
Phys. 125, 104304 (2006)], a predissociation channel of SF is evidenced
by means of resonance-enhanced multilphoton ionization spectroscopy. This work
represents a second experimental investigation on the low-lying excited
cationic states of SF. [The first one is the He I photoelectron spectrum
of SF reported by de Leeuw et al. three decades ago, see Chem. Phys. 34,
287 (1978).]Comment: 7 pages, 3 figures, submitted to JCP as a Not
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