13,540 research outputs found
On -extensions of the Hankel determinants of certain automatic sequences
In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse
sequence, and proved that all the Hankel determinants of the period-doubling
sequence are odd integral numbers. We speak of -extension when the entries
along the diagonal in the Hankel determinant are all multiplied by~. Then we
prove that the -extension of each Hankel determinant of the period-doubling
sequence is a polynomial in , whose leading coefficient is the {\it only
one} to be an odd integral number. Our proof makes use of the combinatorial
set-up developed by Bugeaud and Han, which appears to be very suitable for this
study, as the parameter counts the number of fixed points of a permutation.
Finally, we prove that all the -extensions of the Hankel determinants of the
regular paperfolding sequence are polynomials in of degree less than or
equal to
The effects of KSEA interaction on the ground-state properties of spin chains in a transverse field
The effects of symmetric helical interaction which is called the Kaplan,
Shekhtman, Entin-Wohlman, and Aharony (KSEA) interaction on the ground-state
properties of three kinds of spin chains in a transverse field have been
studied by means of correlation functions and chiral order parameter. We find
that the anisotropic transition of chain in a transverse field (TF)
disappears because of the KSEA interaction. For the other two chains, we find
that the regions of gapless chiral phases in the parameter space induced by the
DM or type of three-site interaction are decreased gradually with
increase of the strength of KSEA interaction. When it is larger than the
coefficient of DM or type of three-site interaction, the gapless
chiral phases also disappear.Comment: 7 pages, 3 figure
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