13,711 research outputs found

    On tt-extensions of the Hankel determinants of certain automatic sequences

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    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 tt-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~tt. Then we prove that the tt-extension of each Hankel determinant of the period-doubling sequence is a polynomial in tt, 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 tt counts the number of fixed points of a permutation. Finally, we prove that all the tt-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in tt of degree less than or equal to 33

    The effects of KSEA interaction on the ground-state properties of spin chains in a transverse field

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    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 XYXY chain in a transverse field (XYXYTF) 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 XZY−YZXXZY-YZX 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 XZY−YZXXZY-YZX type of three-site interaction, the gapless chiral phases also disappear.Comment: 7 pages, 3 figure
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