496 research outputs found
Rational divisors in rational divisor classes
We discuss the situation where a curve C, defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When C has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where C does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class
The arithmetic of hyperelliptic curves
We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve
Identities for hyperelliptic P-functions of genus one, two and three in covariant form
We give a covariant treatment of the quadratic differential identities
satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of
genera 1, 2 and 3
Serre's "formule de masse" in prime degree
For a local field F with finite residue field of characteristic p, we
describe completely the structure of the filtered F_p[G]-module K^*/K^*p in
characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of
F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's
mass formula in degree p. We also determine the compositum C of all degree p
separable extensions with solvable galoisian closure over an arbitrary base
field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in
the case of the local field F. Our method allows us to compute the contribution
of each character G\to\F_p^* to the degree p mass formula, and, for any given
group \Gamma, the contribution of those degree p separable extensions of F
whose galoisian closure has group \Gamma.Comment: 36 pages; most of the new material has been moved to the new Section
The proportion of failures of the Hasse norm principle
For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field
Testing Hardy nonlocality proof with genuine energy-time entanglement
We show two experimental realizations of Hardy ladder test of quantum
nonlocality using energy-time correlated photons, following the scheme proposed
by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)].
Unlike, previous energy-time Bell experiments, these tests require precise
tailored nonmaximally entangled states. One of them is equivalent to the
two-setting two-outcome Bell test requiring a minimum detection efficiency. The
reported experiments are still affected by the locality and detection
loopholes, but are free of the post-selection loophole of previous energy-time
and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure
Cross-plane electronic and thermal transport properties of p-type La0.67Sr0.33MnO3/LaMnO3 perovskite oxide metal/semiconductor superlattices
Lanthanum strontium manganate (La0.67Sr0.33MnO3, i.e., LSMO)/lanthanum manganate (LaMnO3, i.e., LMO) perovskite oxide metal/semiconductor superlattices were investigated as a potential p-type thermoelectric material. Growth was performed using pulsed laser deposition to achieve epitaxial LSMO (metal)/LMO (p-type semiconductor) superlattices on (100)-strontium titanate (STO) substrates. The magnitude of the in-plane Seebeck coefficient of LSMO thin films (/K) is consistent with metallic behavior, while LMO thin films were p-type with a room temperature Seebeck coefficient of 140 mu V/K. Thermal conductivity measurements via the photo-acoustic (PA) technique showed that LSMO/LMO superlattices exhibit a room temperature cross-plane thermal conductivity (0.89 W/m.K) that is significantly lower than the thermal conductivity of individual thin films of either LSMO (1.60 W/m.K) or LMO (1.29 W/m.K). The lower thermal conductivity of LSMO/LMO superlattices may help overcome one of the major limitations of oxides as thermoelectrics. In addition to a low cross-plane thermal conductivity, a high ZT requires a high power factor (S-2 sigma). Cross-plane electrical transport measurements were carried out on cylindrical pillars etched in LSMO/LMO superlattices via inductively coupled plasma reactive ion etching. Cross-plane electrical resistivity data for LSMO/LMO superlattices showed a magnetic phase transition temperature (T-P) or metal-semiconductor transition at similar to 330 K, which is similar to 80K higher than the T-P observed for in-plane resistivity of LSMO, LMO, or LSMO/LMO thin films. The room temperature cross-plane resistivity (rho(c)) was found to be greater than the in-plane resistivity by about three orders of magnitude. The magnitude and temperature dependence of the cross-plane conductivity of LSMO/LMO superlattices suggests the presence of a barrier with the effective barrier height of similar to 300 meV. Although the magnitude of the cross-plane power factor is too low for thermoelectric applications by a factor of approximately 10(-4)-in part because the growth conditions chosen for this study yielded relatively high resistivity films-the temperature dependence of the resistivity and the potential for tuning the power factor by engineering strain, oxygen stoichiometry, and electronic band structure suggest that these epitaxial metal/semiconductor superlattices are deserving of further investigation. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4754514
Rational approximation and arithmetic progressions
A reasonably complete theory of the approximation of an irrational by
rational fractions whose numerators and denominators lie in prescribed
arithmetic progressions is developed in this paper. Results are both, on the
one hand, from a metrical and a non-metrical point of view and, on the other
hand, from an asymptotic and also a uniform point of view. The principal
novelty is a Khintchine type theorem for uniform approximation in this context.
Some applications of this theory are also discussed
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
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