On the diophantine equation x 2 + 2 a · 3 b · 11 c = y n

In this note, we find all the solutions of the Diophantine equation x ² + 2 ^a · 3 ^b · 11 ^c = y ⁿ , in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.     

Syntheses and characterization of two copper pyridine-dicarboxylate compounds containing water clusters

A new homonuclear and heterodinuclear pyridine-2,6-dicarboxylate complexes, formulated as (enH₂)[Cu(dipic)₂]·2.5H₂O (1) and [Cu(μ-dipic)₂Zn(H₂O)₅]·2H₂O (2) (en = ethylenediamine, dipic = pyridine-2,6-dicarboxylate) were synthesized according the reactions between the three species of copper(II) and zinc(II) nitrate, ethylenediamine and pyridine-2,6-dicarboxylic acid. Complexes have been characterized by the methods of elemental, spectroscopic (IR and UV–Vis), thermal (TG/DTG, DTA) analysis, magnetic measurement and single crystal X-ray diffraction. Correlation coefficient, activation energies, E∗; pre-exponential factor, A; entropies, S∗; enthalpies, H∗ and Gibbs free energies, G∗ of the thermal decomposition reactions have been calculated under the derivations from thermogravimetric (TG) and differential thermogravimetric (DTG) curves, using the Coats–Redfern and Horowitz–Metzger methods. Complex 1 belonged to a six-coordinate behavior with a distorted octahedral geometry around Cu(II), that the structure contains two pyridine-2,6-dicarboxylate species as a tridentate ligands and ethylenediaminium cation as a counter ion as well as 2.5 uncoordinated water molecules. On the other hand complex 2 contains six-coordinated Cu(II) and Zn(II) ions, which are linked by two O atoms of the same carboxyl group from dipicolinic acid. The dipicolinate dianions again behave as tridentate ligands in 2. Two uncoordinated water molecules are also present in the structure. The structure units of 1 and 2 are mutually held by the hydrogen bonds and π?π interactions. There is also a C–O?π interaction in 2. The Cu(II) complexes are connected to one another via O–H?O hydrogen bonds, forming water clusters, which play an important role in the stabilization of the crystal structure. In the water clusters, the water molecules are trapped by the cooperative association of coordination interactions as well as hydrogen bonds.     

Determination of 4-nonylphenol in water samples using 4-(2,6-dimethylhept-3-yl)phenol as new internal standard

A new method for determining the endocrine disrupting substance 4-nonylphenol (technical grade = mixture of isomers, 4-NP) from water samples has been developed by using 4-(2,6-dimethylhept-3-yl)phenol (4-sec-NP) as model compound. This branched monoalkylphenol is shown to serve as internal standard (IS) for the determination of technical 4-nonylphenol. To the best of our knowledge, 4-(2,6-dimethylhept-3-yl)phenol (racemic mixture) is a newly synthesized 4-nonylphenol isomer and has not been described elsewhere. Recoveries have been determined by analyzing spiked water samples from distilled water, river water and wastewater. Following acetylation, the compounds were enriched via solid phase extraction (SPE). Analyses of the compounds were performed by capillary column gas chromatography/mass spectrometry (GC/MS), operating in selected ion-monitoring (SIM) mode. The recovery of technical 4-NP using either the newly prepared 4-sec-NP or 4-n-nonylphenol (4-n-NP) as IS have been compared. 4-sec-NP showed slightly better results. However, in the first series of experiments using wastewater, the yields for the derivatization of the two standard compounds were remarkably different. The yield for derivatization of 4-n-NP was approximately 20%, probably due to the difficult matrix of the wastewater. In contrast, the yield for the derivatization of 4-sec-NP was considerably higher (approximately 63%). This problem can be solved by increasing the concentration of the reagent used for derivatization. For better control of the clean-up process, we recommend application of 4-sec-NP as internal standard, at least in water samples with complex matrices (e.g., high content of hydroxylated compounds).     

An alternative globalization strategy for unconstrained optimization

We propose a new globalization strategy that can be used in unconstrained optimization algorithms to support rapid convergence from remote starting points. Our approach is based on using multiple points at each iteration to build a sequence of representative models of the objective function. Using the new information gathered from those multiple points, a local step is gradually improved by updating its direction as well as its length. We give a global convergence result and also provide the parallel implementation details accompanied with a numerical study. Our numerical study shows that the proposed algorithm is a promising alternative as a globalization strategy.     

Die bestimmung kinetischer parameter endothermer zersetzungsreaktionen unter nicht-isothermen bedingungen

By using the kinetic parameters calculated from non-isothermal measurements according to Freeman and Carroll a computer programm is given, which permits the determination of the more probable reaction mechanism by use of 17 different kinetic equations. The computer programm includes the kinetic equation for chemical reaction, nucleation, phase boundary reaction and diffusion and can easily be extended to other equations. The evaluation of the experimental values by a distinct kinetic equation is quantitatively characterized by means of the correlation coefficient.The kinetic parameters are calculated on the basis of a regression analysis. By dividing the thermogravimetric curve in different reaction intervals a differential calculation is possible.Decomposition of CaCO₃, MgCO₃ and CaSO₄·2H₂O is discussed from the results of the computer calculation. It is seen that the kinetic parameters and the reaction mechanism are not constant during the reaction. Further experiments for a physical interpretation are being carried out.     

On the frame bounds of iterated filter banks

We investigate the frame bounds of iterated non-perfect reconstruction filter banks. We provide frame bounds valid for iterated FBs with an arbitrary number of stages using the frame bounds of the underlying frame on the real line. Conversely, given the frame bounds of the iterated FB, we derive bounds for the underlying wavelet frame.