The spectral theory of the vibrating periodic beam

We study the spectral theory of the fourth-order eigenvalue problem

Residual finiteness of infinite amalgamated products of cyclic groups

We classify the residual finiteness of infinite amalgamated free products of infinite cyclic groups. As a side result we also classify residual finiteness and subgroup separability of the subgroups of (Q,+).     

Complexes derived from the copper(II) perchlorate/maleamic acid/2,2′-bipyridine and copper(II) perchlorate/maleic acid/2,2′-bipyridine reaction systems: Synthetic,reactivity, structural and spectroscopic studies

A systematic investigation of the reactions of Cu(ClO₄)₂ · 6H₂O with maleamic acid (H₂L) in the presence of 2,2′-bipyridine (bpy) has been carried out. The chemical and structural identity of the products depends on the solvent, the absence or presence of external hydroxides in the reaction mixture and the molar ratio of the reactants. Various reaction schemes have led to the isolation of the complexes [Cu₂(HL)₂(bpy)₂(H₂O)₂](ClO₄)₂ (1), [Cu₂(HL)₂(bpy)₂(H₂O)₂](ClO₄)₂ · 2H₂O (1 · 2H₂O), [Cu(L′′)(bpy)]_n · 2nH₂O (2 · 2nH₂O), [Cu₂(L′′)(bpy)₂(H₂O)₂]_n(ClO₄)_2n · 0.5nH₂O (3 · 0.5nH₂O), [Cu₂(L′′)₂(bpy)₂] · 2MeOH (5 · 2MeOH), [Cu₂(L′)₂(bpy)₂(ClO₄)₂] (6) and [Cu(ClO₄)₂(bpy)(MeCN)₂] (7b), where L′′^2? and L′^? are the maleate(?2) and monomethyl maleate(?1) ligands, respectively. The HL^? ion has been transformed to L′′^2? and L′^? in the known compounds 2 · 2nH₂O and 6, respectively, via metal ion-assisted processes involving hydrolysis (2 · 2nH₂O) and methanolysis (6) of the primary amide group. The reaction that leads to 6 takes place through the formation of the mononuclear complex [Cu(ClO₄)₂(bpy)(MeOH)₂] (7a), whose structure was assigned on the basis of its spectral similarity with the structurally characterized complex 7b. The structures of the cations in 1 and 1 · 2H₂O consists of two Cu^II atoms bridged by the carboxylate groups of the two HL^? ligands, each exhibiting the less common η² coordination mode; a chelating bpy molecule and a H₂O ligand complete square pyramidal coordination at each metal centre. The structure of the dinuclear repeating unit in the 1D coordination polymer 3 · 0.5nH₂O consists of two Cu^II atoms bridged by two syn,syn η¹:η¹:μ₂ carboxylate groups belonging to two L′′^2? ions; each ligand bridged two neighboring [Cu^II,II₂] units thus promoting the formation of a helical chain. The structure of the dinuclear molecule of complex 5 · 2MeOH consists of two Cu^II atoms bridged by two η² carboxylate groups from two L′′^2? ligands; the second carboxylate group of each maleate(?2) ligand is monodentately coordinated to Cu^II, creating a remarkable seven-membered chelating ring. The L′^? ion behaves as a carboxylate-type ligand in 6, with the carboxylate group being in the familiar syn,syn η¹:η¹:μ₂ coordination mode; a chelating bpy molecule and a coordinated ClO₄^? complete five-coordination at each Cu^II centre. The crystal structures of the complexes are stabilized by various H-bonding patterns. Characteristic IR bands of the complexes are discussed in terms of the known structures and the coordination modes of the ligands.     

Magnetic anisotropy axis reorientation at ultrathin FePt films

Ultrathin FePt films (thickness between 1 nm and 5 nm) were studied for non‐volatile memories applications. The films were magnetron sputtered on monocrystalline MgO?001? substrates at 500 °C. The films are polycrystalline, except the 1 nm thick film which is not continuous. It is shown that films with thickness higher than 2.7 nm have L1₀ structure and perpendicular magnetic anisotropy, while a transition to in‐plane anisotropy occurs for thinner films. The out‐of‐plane coercivity drops from 16 kOe at the thicker film to 0.5 kOe at the thinner one.

Hysteresis cycles of FePt films as a function of film thickness.     

The linearized pressure Poisson equation for global instability analysis of incompressible flows

The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.     

A New Approach to Prepare Poly(ethylene terephthalate)/Silica Nanocomposites with Increased Molecular Weight and Fully Adjustable Branching or Crosslinking by SSP

Summary: In the present study, it has been unexpectedly found that solid‐state polycondensation (SSP) can act as a facile method to prepare poly(ethylene terephthalate)/silica (PET/SiO₂) nanocomposites with high molecular weight and an adjustable degree of branching or crosslinking. Fumed silica, with its surface silanol groups, seems to participate in some kind of reaction, probably esterification with the hydroxy end‐groups of PET, during SSP, to act as a multifunctional chain extender. Differential scanning calorimetry and FT‐IR spectroscopy reveal this ability of the silanol groups. The molecular weight increase depends on the used temperatures of SSP as well as on the amount of SiO₂ added. As the amount of silica increases the rate of increase of the intrinsic viscosity slows because of the higher extent of branching. At 5 wt.‐% SiO₂ the extensive branching produces a crosslinked polymeric material. Such polyesters with increased molecular weight and low silica content could be suitable for blown bottle production, while the high SiO₂ content and adjustable branching or crosslinking could make them ideal high‐melt‐strength resins suitable for the preparation of low‐density closed‐shell foams.

Schematic representation of PET/SiO₂ crosslinked macromolecules.     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

Substituent effect on formation of heterometallic molecular wheels: synthesis, crystal structure, and magnetic properties

The reaction of manganese(III) Schiff bases of the type salen(2-) (N,N'-ethylenebis(salicylideneaminato)) with X-substituted (X = CH(3), Cl) pyridinecarboxamide dicyanoferrite(III) [Fe(X-bpb)(CN)(2)](-) gave rise to a series of cyanide-bridged Mn(6)Fe(6) molecular wheels, [Mn(III)(salen)](6)[Fe(III)(bpmb)(CN)(2)](6) x 7H(2)O (1), [Mn(salen)](6)[Fe(bpClb)(CN)(2)](6) x 4H(2)O x 2CH(3)OH (2), [Mn(salen)](6)[Fe(bpdmb)(CN)(2)](6) x 10H(2)O x 5CH(3)OH (3), [Mn(5-Br(salpn))](6)[Fe(bpmb)(CN)(2)](6) x 24H(2)O x 8CH(3)CN (4), and [Mn(5-Cl(salpn))](6)[Fe(bpmb)(CN)(2)](6) x 25H(2)O x 5CH(3)CN (5). Compared with [Fe(bpb)(CN)(2)](-), which always gives rise to 1D or polynuclear species when reacting with Mn(III) Schiff bases, the introduction of substituents (X) to the bpb(2-) ligand has a driving force in formation of the novel wheel structure. Magnetic studies reveal that high-spin ground state S = 15 is present in the wheel compounds originated from the ferromagnetic Mn(III)-Fe(III) coupling. For the first time, the quantum Monte Carlo study has been used to modulate the magnetic susceptibility of the huge Mn(6)Fe(6) metallomacrocycles, showing that the magnetic coupling constants J range from 3.0 to 8.0 K on the basis of the spin Hamiltonian [Formula: see text]. Hysteresis loops for 1 have been observed below 0.8 K, indicative of a single-molecule magnet with a blocking temperature (TB) of 0.8 K. Molecular wheels 2-5 exhibit frequency dependence of alternating-current magnetic susceptibility under zero direct-current magnetic field, signifying the slow magnetization relaxation similar to that of 1. Significantly, an unprecedented archlike Mn(2)Fe(2) cluster, [Mn(5-Cl(salpn))](2)[Fe(bpmb)(CN)(2)](2) x 3H(2)O x CH(3)CN (6), has been isolated as an intermediate of the Mn(6)Fe(6) wheel 5. Ferromagnetic Mn(III)-Fe(III) coupling results in a high-spin S = 5 ground state. Combination of the high-spin state and a negative magnetic anisotropy (D) results in the observation of slow magnetization relaxation in 6.     

Non-steroidal antiinflammatory drug-copper(II) complexes: structure and biological perspectives

Copper(II) complexes with the non-steroidal antiinflammatory drug mefenamic acid in the presence of aqua or nitrogen donor heterocyclic ligands (2,2'-bipyridine, 1,10-phenanthroline, 2,2'-bipyridylamine or pyridine) have been synthesized and characterized. The crystal structures of [(2,2'-bipyridine)bis(mefenamato)copper(II)], 2, [(2,2'-bipyridylamine)bis(mefenamato)copper(II)], 4, and [bis(pyridine)bis(methanol)bis(mefenamato)copper(II)], 5, have been determined by X-ray crystallography. UV study of the interaction of the complexes with calf-thymus DNA (CT DNA) has shown that the complexes can bind to CT DNA and [bis(aqua)tetrakis(mefenamato)dicopper(II)] exhibits the highest binding constant to CT DNA. The cyclic voltammograms of the complexes in the presence of CT DNA solution have shown that the complexes can bind to CT DNA by the intercalative binding mode verified also by DNA solution viscosity measurements. Competitive studies with ethidium bromide (EB) indicate that the complexes can displace the DNA-bound EB suggesting strong competition with EB. Mefenamic acid and its complexes exhibit good binding propensity to human or bovine serum albumin protein having relatively high binding constant values. All the compounds have been tested for their antioxidant and free radical scavenging activity as well as for their in vitro inhibitory activity against soybean lipoxygenase showing significant activity.