6,074 research outputs found
Thue-Morse at Multiples of an Integer
Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod
2), where s_2 is the sum of the bits in the binary representation of n. It is
well known that for any integer k>=1 the frequency of the letter "1" in the
subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for
any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be
chosen to have Hamming weight <=3. This is best in a twofold sense. First,
there are infinitely many k such that t_{kn}=1 implies that n has Hamming
weight >=3. Second, we characterize all k where the minimal n equals k, k+1,
k+2, k+3, or k+4. Finally, we present some results and conjectures for the
generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.Comment: 14 page
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves
This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute
First-Principles Calculation of Electric Field Gradients and Hyperfine Couplings in YBa2Cu3O7
The local electronic structure of YBa2Cu3O7 has been calculated using
first-principles cluster methods. Several clusters embedded in an appropriate
background potential have been investigated. The electric field gradients at
the copper and oxygen sites are determined and compared to previous theoretical
calculations and experiments. Spin polarized calculations with different spin
multiplicities have enabled a detailed study of the spin density distribution
to be made and a simultaneous determination of magnetic hyperfine coupling
parameters. The contributions from on-site and transferred hyperfine fields
have been disentangled with the conclusion that the transferred spin densities
essentially are due to nearest neighbour copper ions only with marginal
influence of ions further away. This implies that the variant temperature
dependencies of the planar copper and oxygen NMR spin-lattice relaxation rates
are only compatible with commensurate antiferromagnetic correlations. The
theoretical hyperfine parameters are compared with those derived from
experimental data.Comment: 14 pages, 12 figures, accepted to appear in EPJ
First principles study of local electronic and magnetic properties in pure and electron-doped NdCuO
The local electronic structure of Nd2CuO4 is determined from ab-initio
cluster calculations in the framework of density functional theory.
Spin-polarized calculations with different multiplicities enable a detailed
study of the charge and spin density distributions, using clusters that
comprise up to 13 copper atoms in the CuO2plane. Electron doping is simulated
by two different approaches and the resulting changes in the local charge
distribution are studied in detail and compared to the corresponding changes in
hole doped La2CuO4. The electric field gradient (EFG) at the copper nucleus is
investigated in detail and good agreement is found with experimental values. In
particular the drastic reduction of the main component of the EFG in the
electron-doped material with respect to LaCuO4 is explained by a reduction of
the occupancy of the 3d3z^2-r^2 atomic orbital. Furthermore, the chemical
shieldings at the copper nucleus are determined and are compared to results
obtained from NMR measurements. The magnetic hyperfine coupling constants are
determined from the spin density distribution
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