Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space

We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym_>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym_>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role.     

Vibrational spectroscopic studies and DFT calculations on tribromoacetate and tribromoacetic acid in aqueous solution

Aqueous solutions of sodium tribromoacetate (NaCBr3CO2) and its corresponding acid (CBr3COOH) have been studied using Raman and infrared spectroscopy. The spectra of the species in solution were assigned according to symmetry Cs. Characteristic bands of CBr3CO2-(aq) and the tribromoacetic acid, CBr3COOH(aq), are discussed. For the hydrated anion, the CO2 group, the symmetric CO2 stretching mode at 1332 cm(-1) and the asymmetric stretching mode at 1651 cm(-1) are characteristic while the CO mode at 1730 cm(-1) is characteristic for the spectra of the acid. The stretching mode, νC-C at 912cm(-1) for CBr3CO2-(aq) is 10 cm(-1) lower in the anion compared with that of the acid. These characteristic modes are compared to those in acetate, CH3CO2-(aq). Coupling of the modes are fairly extensive and therefore DFT calculations have been carried out in order to compare the measured spectra with the calculated ones. The geometrical parameters such as bond length and bond angles of the tribromoacetate, and tribromoacetic acid have been obtained and may be compared with the ones published for other acetates and their conjugated acids. CBr3COOH(aq) is a moderately strong acid and the pKa value derived from quantitative Raman measurements is equal to -0.23 at 23°C. The deuterated acid CBr3COOD in heavy water has been measured as well and the assignments were given.     

Analytical Approach for Piecewise Linear Coupled Map Lattices

A simple construction is presented which generalizes piecewise linear one-dimensional Markov maps to an arbitrary number of dimensions. The corresponding coupled map lattice, known as a simplicial mapping in the mathematical literature, allows for an analytical investigation. In particular, the spin Hamiltonian which is generated by the symbolic dynamics is accessible. As an example, a formal relation between a globally coupled system and an Ising mean-field model is established. The phase transition in the limit of infinite system size is analyzed and analytical results are compared with numerical simulations.     

The influence of the flexoelectric effect on the electrohydrodynamic instability in nematics

A rigorous three-dimensional linear analysis of the electrohydrodynamic instability in nematic liquid crystals including the flexoelectric effect is presented for the case of an applied d.c. voltage. The flexoelectric effect leads to an appreciable reduction of the threshold and to the appearance of oblique rolls at threshold for the standard material MBBA. We discuss the influence of a magnetic field and test several approximations against the rigorous results     

Direct potentiometric immunoelectrodes

Summary TK pencil fine mines type Faber Castell Super polymer from Staedtler are used as graphite based material for an atrazine-sensitive immunoelectrode. Eighteen procedures are tested to immobilize antibodies against atrazine but only one turned out to be useful for the preparation of a graphite based immunoelectrode. It incorporates binding of glutardialdehyde to active sites of the graphite surface and subsequent cross-linking of the antibodies with glutardialdehyde. The method has been carefully optimized with respect to selectivity and the cross-sensitivity is determined for such an electrode based on atrazine-rabbit and atrazine-sheep antibodies as well. Due to the structural difference of the used immunogens, the sign of the potential response and the cross-sensitivities are different. The application of such an electrode to a natural water sample is demonstrated and shows that the useful concentration range in a final solution is between 20 and 250 ng/l. Of course, interference from ions must be excluded in future analytical applications.Dedicated to Professor Dr. Wilhelm Fresenius on the occasion of his 80th birthdayPart II: Fresenius J Anal Chem