Low frequency dielectric spectroscopy of two room temperature chiral liquid crystal mixtures

Dielectric spectroscopy of two room temperature chiral liquid crystal mixtures (W-96 and W-97) have been carried out in the frequency range of 100?mHz–10?MHz. Low frequency dielectric relaxation modes corresponding to collective behavior of molecules (Goldstone- and soft-modes) in the SmC* phase have been found to be masked by the ionic conductance. Two slow modes of dielectric relaxation due to the ionic conductance have been detected (below 15?Hz) in planar-aligned samples. It has been observed that the effect of ionic conductance decreases with the number of thermal annealing cycles on the materials. With large number of thermal annealing cycles it has been possible to wipe out the effects of ionic conductance and then to detect other weak modes of dielectric relaxation which are otherwise masked.     

Bessel-type inequalities in Hilbert C *-modules

Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombieri and Boas–Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert C *-modules.     

On the spectrum of linear combinations of two projections in C*-algebras

In this note, we study the spectrum and give estimations for the spectral radius of linear combinations of two projections in C*-algebras. We also study the commutator of two projections.     

An Approximation Theory for Operators Generated by Shifts

This paper develops a theory about the applicability of approximation methods to operators that can be represented as a function of a shift. An axiomatic approach is used, in which a small set of conditions that involve the operator and the method are proved to guarantee the applicability. All concrete methods we know for the underlying operators are subjected to this approach. We also study the behavior of the singular values and related questions. New results concerning Galerkin approximation methods for Mellin operators with piecewise continuous symbol are given to illustrate the application of the theory.     

Relaxation effects in the quantification of fat using gradient echo imaging

Quantification of fat has been investigated using images acquired from multiple gradient echoes. The evolution of the signal with echo time and flip angle was measured in phantoms of known fat and water composition and in 21 research subjects with fatty liver. Data were compared to different models of the signal equation, in which each model makes different assumptions about the T1 and/or T2* relaxation effects. A range of T1, T2*, fat fraction and number of echoes was investigated to cover situations of relevance to clinical imaging. Results indicate that quantification is most accurate at low flip angles (to minimize T1 effects) with a small number of echoes (to minimize spectral broadening effects). At short echo times, the spectral broadening effects manifest as a short apparent T2 for the fat component.     

Ferroelectric Liquid Crystals: Synthesis and Thermal Behavior of Optically Active,Three‐Ring Schiff Bases and Salicylaldimines

The chiral ferroelectric smectic C (SmC*) phase, characterized by a helical superstructure, has been well exploited in developing high‐resolution microdisplays that have been effectively employed in the fabrication of a wide varieties of portable devices. Although, an overwhelming number of optically active (chiral) liquid crystals (LCs) exhibiting a SmC* phase have been designed and synthesized, the search for new systems continues so as to realize mesogens capable of meeting technical necessities and specifications for their end‐use. In continuation of our research work in this direction, herein we report the design, synthesis, and thermal behavior of twenty new optically active, three‐ring calamitic LCs belonging to four series. The first two series comprise five pairs of enantiomeric Schiff bases whereas the other two series are composed of five pairs of enantiomeric salicylaldimines. In each pair of optical isomers, the configuration of a chiral center in one stereoisomer is opposite to that of the analogous center in the other isomer as they are derived from (3 S)‐3,7‐dimethyloctyloxy and (3 R)‐3,7‐dimethyloctyloxy tails. To probe the structure–property correlations in each series, the length of the n‐alkoxy tail situated at the other end of the mesogens has been varied from n‐octyloxy to n‐dodecyloxy. The measurement of optical activity of these chiral mesogens was carried out by recording their specific rotations. As expected, enantiomers rotate plane polarized light in the opposite direction but by the same magnitude. The thermal behavior of the compounds was established by using a combination of optical polarizing microscopy, differential scanning calorimetry, and powder X‐ray diffraction. These complementary techniques demonstrate the existence of the expected, thermodynamically stable, chiral smectic C (SmC*) LC phase besides blue phase I/II (BPI or BPII) and chiral nematic (N*) phase. However, as noted in our previous analogous study, the vast majority of the Schiff bases show an additional metastable, unfamiliar smectic (SmX) phase just below the SmC* phase. Notably, the SmC* phase persists over the temperature range ≈80–115 °C. Two mesogens chosen each from Schiff bases and salicylaldimines were investigated for their electrical switching behavior. The study reveals the ferroelectric switching characteristics of the SmC* phase featuring the spontaneous polarization (P_S) in the range 69–96 nC cm^?2. The helical twist sense of the SmC* phase as well as the N* phase formed by a pair of enantiomeric Schiff bases and salicylaldimines has been established with the help of circular dichroism (CD) spectroscopic technique. As expected, the SmC* and the N* phase of a pair of enantiomers showed mirror image CD signals. Most importantly, the reversal of helical handedness from left to right and vice versa has been evidenced during the N* to SmC* phase transition, implying that the screw sense of the helical array of the N* phase and the SmC* phase of an enantiomer is opposite.     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

A note on the Gelfand-Mazur Theorem

Three Gelfand-Mazur type theorems are proved. One of these provides a -property analogue of Zalar's recent generalizations of the Froelich-Ingelstam-Smiley Theorems concerning unital multiplication in Hilbert spaces; the second illustrates that the assumption in Kaplansky's version of the Gelfand-Mazur Theorem can be weakened in the presence of a -norm; whereas the third provides a real analogue of a result due to Srinivasan.

     

On a theorem of Cantor-Bernstein type for algebras

Freytes proved a theorem of Cantor-Bernstein type for algbras; he applied certain sequences of central elements of bounded lattices. The aim of the present paper is to extend the mentioned result to the case when the lattices under consideration need not be bounded; instead of sequences of central elements we deal with sequences of internal direct factors of lattices. This work was supported by Science and Technology Assistance Agency under the contract No APVT-51-032002.