4,687 research outputs found
Divisors on Rational Normal Scrolls
Let be the homogeneous coordinate ring of a rational normal scroll. The
ring is equal to the quotient of a polynomial ring by the ideal
generated by the two by two minors of a scroll matrix with two rows and
catalecticant blocks. The class group of is cyclic, and is infinite
provided is at least two. One generator of the class group is ,
where is the ideal of generated by the entries of the first column of
. The positive powers of are well-understood, in the sense that the
ordinary power, the symmetric power, and the
symbolic power all coincide and therefore all three powers are
resolved by a generalized Eagon-Northcott complex. The inverse of in the
class group of is , where is the ideal generated by the entries of
the first row of . We study the positive powers of . We obtain a
minimal generating set and a Groebner basis for the preimage in of the
symbolic power . We describe a filtration of in which all of
the factors are Cohen-Macaulay -modules resolved by generalized
Eagon-Northcott complexes. We use this filtration to describe the modules in a
finely graded resolution of by free -modules. We calculate the
regularity of the graded -module and we show that the symbolic
Rees ring of is Noetherian.Comment: 32 page
Phase Boundary of the Boson Mott Insulator in a Rotating Optical Lattice
We consider the Bose-Hubbard model in a two dimensional rotating optical
lattice and investigate the consequences of the effective magnetic field
created by rotation. Using a Gutzwiller type variational wavefunction, we find
an analytical expression for the Mott insulator(MI)-Superfluid(SF) transition
boundary in terms of the maximum eigenvalue of the Hofstadter butterfly. The
dependence of phase boundary on the effective magnetic field is complex,
reflecting the self-similar properties of the single particle energy spectrum.
Finally, we argue that fractional quantum Hall phases exist close to the MI-SF
transition boundaries, including MI states with particle densities greater than
one.Comment: 5 pages,3 figures. High resolution figures available upon reques
A study of singularities on rational curves via syzygies
Consider a rational projective curve C of degree d over an algebraically
closed field k. There are n homogeneous forms g_1,...,g_n of degree d in
B=k[x,y] which parameterize C in a birational, base point free, manner. We
study the singularities of C by studying a Hilbert-Burch matrix phi for the row
vector [g_1,...,g_n]. In the "General Lemma" we use the generalized row ideals
of phi to identify the singular points on C, their multiplicities, the number
of branches at each singular point, and the multiplicity of each branch.
Let p be a singular point on the parameterized planar curve C which
corresponds to a generalized zero of phi. In the "Triple Lemma" we give a
matrix phi' whose maximal minors parameterize the closure, in projective
2-space, of the blow-up at p of C in a neighborhood of p. We apply the General
Lemma to phi' in order to learn about the singularities of C in the first
neighborhood of p. If C has even degree d=2c and the multiplicity of C at p is
equal to c, then we apply the Triple Lemma again to learn about the
singularities of C in the second neighborhood of p.
Consider rational plane curves C of even degree d=2c. We classify curves
according to the configuration of multiplicity c singularities on or infinitely
near C. There are 7 possible configurations of such singularities. We classify
the Hilbert-Burch matrix which corresponds to each configuration. The study of
multiplicity c singularities on, or infinitely near, a fixed rational plane
curve C of degree 2c is equivalent to the study of the scheme of generalized
zeros of the fixed balanced Hilbert-Burch matrix phi for a parameterization of
C.Comment: Typos corrected and minor changes made. To appear in the Memoirs of
the AM
P-band in a rotating optical lattice
We investigate the effects of rotation on the excited bands of a tight
binding lattice, focusing particulary on the first excited (p-) band. Both the
on-site energies and the hopping between lattice sites are modified by the
effective magnetic field created by rotation, causing a non-trivial splitting
and magnetic fine structure of the p-band. We show that Peierls substitution
can be modified to describe p-band under rotation, and use this method to
derive an effective Hamiltonian. We compare the spectrum of the effective
Hamiltonian with a first principles calculation of the magnetic band structure
and find excellent agreement, confirming the validity of our approach. We also
discuss the on-site interaction terms for bosons and argue that many-particle
phenomena in a rotating p-band can be investigated starting from this effective
Hamiltonian.Comment: 7 pages, 4 figures, new discussion of effective Hamiltonian,
references adde
On the use of CrN/Cr and CrN interlayers in hot filament chemical vapour deposition (HF-CVD) of diamond films onto WC-Co substrates
CrN/Cr-based films were deposited using PVD-arc technique onto Co-cemented tungsten carbide (WC-Co) substrates and, then, seeded with diamond powder suspension or mechanically treated by Fluidized Bed Peening (FBP) of brittle diamond powders. Multilayered coatings were obtained from the superimposition of 4 mu m-thick diamond coatings, deposited on the PVD interlayer using hot filament chemical vapour deposition (HFCVD). The effectiveness of fluidized bed peened CrN/Cr interlayers on the adhesion enhancement of diamond on WC-Co substrates was studied and compared to diamond coated WC-Co substrates with unpeened CrN/Cr or CrN interlayers, or pre-treated with two-step chemical etching (Murakatni's reagent and Caro's acid, MC-treatment). In particular, growth, morphology, wear endurance and adhesion of the CVD deposited diamond films onto peened CrN/Cr interlayer were looked into. Diamond coatings on peened CrN/Cr interlayers exhibited a rougher surface morphology than as-prepared CrN/Cr films as a result of the surface roughening of the ductile Cr layer produced by the repeated impacts on it of the diamond powders during FBP. FBP was found to be a necessary step in improving the scarce adhesion of CVD diamond onto CrN/Cr-interlayer. However, the use of FB peened CrN/Cr interlayer did not represent the best way to pre-treat WC-Co substrates, as the unpeened single-layer CrN, or the use of MC pretreatment, was found to ensure better adhesion and wear endurance. (C) 2008 Published by Elsevier B.V
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