On eigenvectors of the Pascal and Reed-Muller-Fourier transforms

In their paper at the International Symposium on Multiple-Valued Logic in 2017, C. Moraga, R. S. Stankovi´c, M. Stankovi´c and S. Stojkovi´c presented a conjecture for the number of fixed points (i.e., eigenvectors with eigenvalue 1) of the Reed-Muller-Fourier transform of functions of several variables in multiple-valued logic. We will prove this conjecture, and we will generalize it in two directions: we will deal with other transforms as well (such as the discrete Pascal transform and more general triangular self-inverse transforms), and we will also consider eigenvectors corresponding to other eigenvalues

Synthesis, molecular and solid state structures, and magnetic properties of sandwich lanthanide phthalocyanines lacking C-H bonds

A new class of sandwich phthalocyanine (Pc) compounds without C-H bonds was synthesized and characterized. They are the bis[octakis(perfluoro i-C3F7) octakis(perfluoro)phthalocyaninato(2−)]M(III) complexes, formulated as (F64Pc)2MH, M = Tb, Dy, Lu, Y. Single molecular magnetic (SMM) behavior in the (F64Pc)2TbH and (F64Pc)2DyH complexes was confirmed through slowed relaxation response of their magnetization in an applied time varying magnetic field during alternating current (AC) magnetic testing in the range 2 - 50 K. The energy barrier to magnetic reversal, Δ = 215 cm-1 and the pre-exponential factor, τ0-1 = 2.7 x 107-1 were estimated for the undiluted (F64Pc)2TbH with direct current bias magnetic field (Hdc) of 1000 Oe applied. Undiluted (F64Pc)2DyH gave estimated values of Δ = 31 cm-1 and τ0-1 = 1.6 x 105 s-1 at Hdc = 500 Oe. These respective values are within the range observed for the unsubstituted Tb and Dy analogues. Magnetic hysteresis testing at 0.04 K exhibited hysteresis in both of these complexes. Magnetic circular dichroism (MCD) tests on (F64Pc)2Tb frozen in acetone at 1.8-1.9 K exhibited a “butterfly shaped” hysteresis which appeared to depend on the oxidation state of the complex and which also provided evidence of quantum tunneling of magnetization. The chemical tuning of the Pc macrocycle of these complexes through the replacement of the H atoms with the electron withdrawing fluorine atom and i-C3F7 group has created additional functionalities. Thermal analysis in the forms of thermogravi metric analysis and differential scanning calorimetry showed the complexes to be stable up to 450°C in air and N2. Cyclic voltammetry revealed four reduced and one oxidized states, the oxidized state occurring at \u3e 1.2 V, showing the complexes’ resistance to oxidation. The (F64Pc)2MH compounds derive their uniqueness as [(F64Pc)2M]− sandwich phthalocyanine complexes with exceptional thermal and oxidative stability