Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator \(\mathcal {L}\) with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.     

Novel mixed-valence manganese cluster with two distinct Mn3(II/III/II) and Mn3(III/II/III) trinuclear units in a pseudocubane-like arrangement

The reaction between Mn(ClO 4) 2 and di-(2-pyridyl)-ketone in the presence of the sodium salt of propanediol as a base in MeOH leads to the formation of a hexanuclear manganese cluster. This cluster has been characterized by the formula [Mn(II) 3Mn(III) 3O(OH)(CH 3pdol) 3(Hpdol) 3(pdol)](ClO 4) 4 ( 1). Molecular conductance measurements of a 10 (-3) M solution of compound 1 in CH 3CN, DMSO, or DMF give Lambda m = 529, 135, or 245 muS/cm, respectively, which suggests a 1:4 cation/anion electrolyte. The crystal structure of hexanuclear manganese cluster 1 consists of two distinct trinuclear units with a pseudocubane-like arrangement. The trinuclear units show two different valence distributions, Mn(II)/Mn(III)/Mn(II) and Mn(III)/Mn(II)/Mn(III). Additional features of interest for the compound include the fact that (a) two of the Mn(III) ions show a Jahn-Teller elongation, whereas the third ion shows a Jahn-Teller compression; (b) one bridge between Mn(III) atoms is an oxo (O (2-)) ion, whereas the bridge between Mn(II) and Mn(III) is a hydroxyl (OH (-)) group; and (c) the di-(2-pyridyl)-ketone ligand that is methanolyzed to methyl-Hpdol and R 2pdol (R = CH 3, H) acts in three different modes: methyl-pdol(-1), Hpdol(-1), and pdol(-2). For magnetic behavior, the general Hamiltonian formalism considers that (a) all of the interactions inside the two "cubanes" between Mn(II) and Mn(III) ions are equal to the J 1 constant, those between Mn(II) ions are equal to the J 2 constant, and those between the Mn(III) ions are equal to the J 3 constant and (b) the interaction between the two cubanes is equal to the J 4 constant. The fitting results are J 1 = J 2 = 0.7 cm (-1), J 3 approximately 0.0, J 4 = -6.2 cm (-1), and g = 2.0 (fixed). According to these results, the ground state is S = 1/2, and the next excited states are S = 3/2 and 5/2 at 0.7 and 1.8 cm (-1), respectively. The EPR spectra prove that the spin ground state at a low temperature is not purely S = 1/2 but is populated with the S = 3/2 state, which is in accordance with the susceptibility and magnetization measurements.     

Experimental and theoretical study of the antisymmetric magnetic behavior of copper inverse-9-metallacrown-3 compounds

Use of PhPyCNO (-)/X (-) "blends" (PhPyCNOH = phenyl 2-pyridyl ketoxime; X (-) = OH (-), alkanoato, ClO 4 (-)) in copper chemistry yielded trinuclear clusters that have been characterized as inverse-9-metallacrown-3 compounds and accommodate one or two guest ligands. The magnetic behavior showed a large antiferromagnetic interaction and a discrepancy between the low-temperature magnetic behavior observed experimentally and that predicted from a magnetic model. The discrepancy between the Brillouin curve and the experimental result provides clear evidence of the influence of the antisymmetric interaction. Introducing the antisymmetric terms derived from the fit of the susceptibility data into the magnetization formula caused the simulated curve to become nearly superimposable on the experimental one. The EPR data indicated that the compound [Cu 3(PhPyCNO) 3(mu 3-OH)(2,4,5-T) 2] ( 1), where 2,4,5-T is 2,4,5-trichlorophenoxyacetate, has isosceles or lower magnetic symmetry (delta not equal 0), that antisymmetric exchange is important ( G not equal 0), and that Delta E > hnu. The structures of the complexes 1 and [Cu 3(PhPyCNO) 3(mu 3-OH)(H 2O)(ClO 4) 2] ( 2) were determined using single-crystal X-ray crystallography. Theoretical calculations based on density functional theory were performed using the full crystal structures of 1, 2, [Cu 3(PhPyCNO) 3(OH)(CH 3OH) 2(ClO 4) 2] ( 3), and [Cu 3(PhPyCNO) 3(mu 3-OMe)(Cl)(ClO 4)] ( 4). The geometries of the model compounds [Cu 3(kappa (3) N, N, O-HNCHCHNO) 3(mu 3-OH)(mu 2-HCOO)(HCOO)] ( 5), [Cu 3(kappa (3) N, N, O-HNCHCHNO) 3(mu 2-HCOO)(HCOO)] (+) ( 6), [Cu 3(kappa (3) N, N, O-HNCHCHNO) 3(mu 3-O)] (+) ( 7), and [Cu 3(kappa (3) N, N, O-HNCHCHNO) 3] (3+) ( 8) were optimized at the same level of theory for both the doublet and quartet states, and vibrational analysis indicated that the resulting equilibrium geometries corresponded to minima on the potential energy surfaces. Both e g and t 2g magnetic orbitals seem to contribute to the magnetic exchange coupling. The latter contribution, although less important, might be due to overlap of the t 2g orbitals with the p-type orbitals of the central triply bridging oxide ligand, thereby affecting its displacement from the Cu 3 plane and contributing to the antiferromagnetic coupling. The crucial role of the triply bridging oxide (mu 3-O) ligand on the antiferromagnetic exchange coupling between the three Cu(II) magnetic centers is further evidenced by the excellent linear correlation of the coupling constant J with the distance of the mu 3-O ligand from the centroid of the Cu 3 triangle.     

Merlon-type density waves in a compartmentalized conveyor system

Multi-particle flow through a cyclic array of K connected compartments with a preferential direction is known to be able to organize itself in the form of density waves [Kanellopoulos, Van der Meer, and Van der Weele, Phys. Rev. E 92, 022205 (2015)]. In this brief note we focus on the intriguing shape these waves take when K is even, in which case they travel through alternatingly dense and diluted compartments. We call them “merlon waves”, since the sequence of high and low densities is reminiscent of the merlons and crenels on the battlements of medieval castles.     

Structural features of mono- and tri-nuclear Zn(II) complexes with a non-steroidal anti-inflammatory drug as ligand

The interaction of Zn(II) with the non-steroidal anti-inflammatory drug tolfenamic acid leads to the formation of the structurally characterized trinuclear [Zn(3)(tolfenamato)(6)(CH(3)OH)(2)] complex. In the presence of the N,N'-donor heterocyclic ligands 1,10-phenanthroline and 2,2'-bipyridine at a range of ratios, the mononuclear Zn complexes of the general formulae [Zn(tolfenamato)(N,N'-donor)Cl] and [Zn(tolfenamato)(2)(N,N'-donor)] have been isolated and structurally characterized by X-ray crystallography. The deprotonated tolfenamato ligands are coordinated to the Zn(II) ion through carboxylato oxygen atoms. Tolfenamic acid and its complexes exhibit good binding propensity to human or bovine serum albumin protein having relatively high binding constant values.     

Random Motion on Simple Graphs

Consider a stochastic process that lives on n-semiaxes joined at the origin. On each ray it behaves as one dimensional Brownian Motion and at the origin it chooses a ray uniformly at random (Kirchhoff condition). The principal results are the computation of the exit probabilities and certain other probabilistic quantities regarding exit and occupation times.