A Six-Vertex Theorem for Bounding Normal Planar Curves

We prove that the number of vertices of a smooth normal planarcurve is at least 6, if it bounds a surface other than the disk.     

Intended Treatments of Arithmetic Average in U.S. and Asian School Mathematics Textbooks

This study examined how selected U.S. and Asian mathematics curricula are designed to facilitate students' understanding of the arithmetic average. There is a consistency regarding the learning goals among these curriculum series, but the focuses are different between the Asian series and the U.S. reform series. The Asian series and the U.S. commercial series focus the arithmetic average more on conceptual and procedural understanding of the concept as a computational algorithm than on understanding the concept as a representative of a data set; however, the two U.S. reform series focus the concept more on the latter. Because of the different focuses, the Asian and the U.S. curriculum series treat the concept differently. In the Asian series, the concept is first introduced in the context of “equal‐sharing” or “per‐unit‐quantity,” and the averaging formula is formally introduced at a very early stage. In the U.S. reform series, the concept is discussed as a measure of central tendency, and after students have some intuitive ideas of the statistical aspect of the concept, the averaging algorithm is briefly introduced.     

Radiotracer studies of the antitumor metallocene molybdocene dichloride with biomolecules

Neutron irradiation of Cp₂MoCl₂ for 24 h afforded the radiotracer Cp₂⁹⁹MoCl₂ which was characterised by UV–Vis spectroscopy and thin layer chromatography. Binding experiments with the thiol containing protein human serum albumin (HSA) or calf thymus DNA, were monitored for ⁹⁹Mo using a gamma counter. Under the conditions investigated, molar ratios of binding of 0.2:1 (Cp₂MoCl₂:DNA) and 9.4:1 (Cp₂MoCl₂:HSA) were calculated. The results are consistent with in vitro coordination studies that have shown strong preferential interaction of Cp₂MoCl₂ with thiols versus other donor sites in biomolecules including DNA.     

Iterative methods for solvingAx=b,GMRES/FOM versus QMR/BiCG

We study the convergence of the GMRES/FOM and QMR/BiCG methods for solving nonsymmetric systems of equationsAx=b. We prove, in exact arithmetic, that any type of residual norm convergence obtained using BiCG can also be obtained using FOM but on a different system of equations. We consider practical comparisons of these procedures when they are applied to the same matrices. We use a unitary invariance shared by both methods, to construct test matrices where we can vary the nonnormality of the test matrix by variations in simplified eigenvector matrices. We used these test problems in two sets of numerical experiments. The first set of experiments was designed to study effects of increasing nonnormality on the convergence of GMRES and QMR. The second set of experiments was designed to track effects of the eigenvalue distribution on the convergence of QMR. In these tests the GMRES residual norms decreased significantly more rapidly than the QMR residual norms but without corresponding decreases in the error norms. Furthermore, as the nonnormality ofA was increased, the GMRES residual norms decreased more rapidly. This led to premature termination of the GMRES procedure on highly nonnormal problems. On the nonnormal test problems the QMR residual norms exhibited less sensitivity to changes in the nonnormality. The convergence of either type of procedure, as measured by the error norms, was delayed by the presence of large or small outliers and affected by the type of eigenvalues, real or complex, in the eigenvalue distribution ofA. For GMRES this effect can be seen only in the error norm plots.In honor of the 70th birthday of Ted RivlinThis work was supported by NSF grant GER-9450081.     

The effect of olefinic terminal chains on the mesomorphic properties of 4,4'-disubstituted diphenyldiacetylenes

New diphenyldiacetylenes of the type with A, B = H and/or F; m = 0, 1; n = 1-4; and X = C n H 2n + 1 , F, CF 3 or CN were synthesized and their mesomorphic properties determined by hot stage polarizing microscopy and DSC. When m = 0, all of these compounds showed only a nematic phase except when X = CF 3 when both nematic and smectic A phases were seen. Both clearing and melting temperatures were higher than those reported for substitution with the corresponding alkyl chains but the much larger increase in clearing temperatures produced considerably wider nematic phases. Eutectic mixtures of a few of these olefins yielded nematic materials also having much wider temperature ranges and higher clearing temperatures than the eutectic mixtures of the alkyl compounds, while retaining their high birefringence and low viscosities. Such materials are of interest for beam-steering devices.

Four of the diacetylenes with m = 1 ( A, B = H) were also prepared ( X = C 6 H 13 , F, n = 2, 3). When X was C 6 H 13 ( n = 2), the nematic range was smaller in the 2- than in the 1-olefin but wider than in the alkyl series. When X = F, either no nematic phase or a monotropic one was observed, whereas the 1-olefins gave a much wider nematic phase. Both transition temperatures were lower than those for the corresponding 1-olefin and alkyl analogues. The compound with X = C 6 H 13 and n = 2 had a melting temperature below room temperature.     

Fast atom bombardment mass spectra of keto acid and keto ester phosphoranes

Positive-ion fast atom bombardment mass spectra of 13 keto acid and keto ester phosphoranes were examined. All compounds gave a protonated molecular ion and characteristic peaks at m/z 303, 279, 262 and 201.     

Anomalous capacitance of quantum well double-barrier diodes

The S-parameters of several different quantum well double barrier diodes have been measured. A technique has been developed for measuring whisker contacted diodes with and HP 8510B automatic network analyzer. Special coaxial mounts using K-connectors were designed to enable measurements up to 20 GHz. The voltage-dependent conductance and capacitance were derived from the measured reflection coefficient of each device. The C/V characteristics were observed to exhibit an anomalous increase at voltages corresponding the the negative differential resistance region (NDR). These are the first reported S-parameter measurements in the negative differential resistance region of quantum well double barrier diodes. A theory is presented that explains, in part, the observed results.     

Synthesis and characterization of a semiflexible liquid crystalline polyester with a broad nematic region

In this paper, we report on the synthesis and detailed characterization of a new semiflexible nematic liquid crystalline polyester which could serve as a 'model' polyester for a variety of physical and physico-chemical investigations. The polymer is a nematic liquid over a wide temperature range-from the glass transition temperature at ∼95°C to the isotropic transition at ∼240°C. We expect this polyester to be particularly useful for studying the effect of flow on the orientation of liquid crystalline polymers, as well as the production and removal of disclinations.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002