Supramolecular dimer formation through hydrogen bond extensions of carboxylate ligands – Path for magnetic exchange

A new complex of unusual composition [Cu(3-O₂Nbz)₂(nia)(H₂O)₂] (1) (nia = nicotinamide, 3-O₂Nbz = 3-nitrobenzoate) has been prepared and studied together with two other complexes of composition [Cu(4-O₂Nbz)₂(nia)₂(H₂O)₂] (2) and [Cu(4-O₂Nbz)₂(nia)₂]?(4-O₂NbzH)₂ (3) (4-O₂Nbz = 4-nitrobenzoate). The composition of all complexes has been determined by elemental analysis, the complexes have been studied by electronic, infrared and EPR spectroscopy, as well as by magnetization measurements over the temperature range 1.8–300 K, and their structures have been solved. The structure of complex (1) consists of molecules, where Cu(II) atom is monodentately coordinated by the pair of 3-nitrobenzoato anions in trans  -positions together with water and nicotinamide molecules, forming nearly tetragonal basal plane, and by another water molecule in axial position of tetragonal-pyramidal coordination polyhedron. The neighboring molecule coordination polyhedron basal planes are coplanar and allow formation of supramolecular dimers with strong H-bonds between hydrogen atoms from equatorially coordinated water molecules and uncoordinated carboxylate oxygen atoms thus giving the nearest Cu?$?$Cu distance of 4.886(2) Å. Magnetization measurements showed that complex (1) exhibits maximum of magnetic susceptibility at 6.5 K and a fit to Bleaney-Bowers equation gave singlet–triplet energy gap 2J = −6.25 cm⁻¹, and zJ′ = −0.03 cm⁻¹. This might be an experimental proof that the carboxylate bridges extended with hydrogen bonds are the pathway of the spin–spin interactions. The temperature dependence of changes in EPR spectra of (1) and the spectrum at 4.2 K have confirmed its hydrogen bonded dimeric structure. The calculated Cu?$?$Cu distance 4.8 Å is in very good agreement with the value obtained from crystal structure. The complexes (2) and (3) at 300 K exhibit magnetic moment μ_eff = 1.98 B.M. and μ_eff = 1.84 B.M., respectively. These values practically do not change with lowering the temperature up to 5 K and only small drops to μ_eff = 1.87 B.M. (for (2)) and μ_eff = 1.79 B.M. (for (3)) at 1.8 K have been observed. The EPR spectra of complex (2) at room temperature as well as at 77 K are of axial type with g_⊥ = 2.062 and g_|| = 2.285 and exhibit resolved parallel hyperfine splitting with A_|| = 160 Gauss. The EPR spectra of complex (3) at room temperature as well as at 77 K are of axial type with g_⊥ = 2.065 and g_|| = 2.235 and exhibit unresolved parallel hyperfine splitting. EPR spectra of (2) and (3) are consistent with the X-ray structure.     

[Cu(X-salicylato)2(N,N-diethylnicotinamide)2(H2O)2] complexes: conformational polymorphism and its consequence in supramolecular hydrogen-bonding networks formation

Novel copper(II) X-salicylate complexes with N,N-diethylnicotinamide (dena) of the formula [Cu(RCOO)₂(dena)₂(H₂O)₂] (RCOO = 3-methylsalicylate anion (3-Mesal, 1), 4-methylsalicylate anion (4-Mesal, 2), 5-methylsalicylate anion (5-Mesal, 3), 5-methoxysalicylate anion (5-MeOsal, 4) or 4-methoxysalicylate anion (4-MeOsal, 5)), and complex [Cu(3-MeOsal)₂(dena)₂(H₂O)₂]∙2H₂O (3-MeOsal = 3-methoxysalicylate anion (6)) have been prepared in the crystalline forms and characterized by spectroscopic methods (IR, Vis–UV, EPR). All the compounds according to their composition (1–5) seem to possess octahedral copper(II) stereochemistry. The complex 1 has been prepared in two different forms. X-ray analyses of the complexes 1, 4, and 5 were carried out and they featured a tetragonal-bipyramidal geometry around the copper atoms. The tetragonal planes are created by X-salicylate anions bonded to the copper(II) atoms via unidentate carboxylate oxygen atoms and the pyridine ring nitrogen atoms of the neutral ligand N,N-diethylnicotinamide, while in axial positions are water molecules. The two forms of complex 1 present conformation polymorphs and supramolecular isomers.     

The jump of the Milnor number in the X 9 singularity class

The jump of the Milnor number of an isolated singularity f ₀ is the minimal non-zero difference between the Milnor numbers of f ₀ and one of its deformations (f _s ). We prove that for the singularities in the X ₉ singularity class their jumps are equal to 2.     

Application of inverse linear parametric models in the identification of rail track irregularities

Requirements for current trains to be increasingly available have created the need to develop systems that can predict the quality of both trains and infrastructure components. The paper presents a new approach to the detection of rail truck irregularities, based on the measurements of bearing box acceleration during the operation of rail vehicles. The proposed procedure is based on an inverse problem solution, estimating track irregularities from measured acceleration of the applied model of vehicle dynamics. The simulation study of the proposed method, as well as its implementation, is presented. The method has been successfully applied for the identification of rail irregularities on a typical Polish railroad and vehicle.     

Boundary Value Problems with a Parameter of Differential Equations with Deviated Arguments

The aim of the present paper is to study boundary value problems with a parameter of differential equations with deviated arguments. The existence, uniqueness and continuous dependence of the solution on the right-hand side of our problem are considered.     

A class of B-(p,r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-(p,r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-(p,r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-(p,r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.