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.     

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.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

An improved algorithm for solving biobjective integer programs

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.     

A Note on Groups Associated with 4-Arc-Transitive Cubic Graphs

A cubic (trivalent) graph is said to be 4-arc-transitive ifits automorphism group acts transitively on the 4-arcs of (wherea 4-arc is a sequence v₀, v₁, ... v₄ of vertices of such thatv_i–1 is adjacent to v_i for 1 i 4, and v_i–1 v_i+1for 1 i < 4). In his investigations into graphs of thissort, Biggs defined a family of groups 4⁺(a^m), for m = 3,4,5...,each presented in terms of generators and relations under theadditional assumption that the vertices of a circuit of lengthm are cyclically permuted by some automorphism. In this paperit is shown that whenever m is a proper multiple of 6, the group4⁺(a^m) is infinite. The proof is obtained by constructing transitivepermutation representations of arbitrarily large degree.     

Intra- versus intermolecular hydrogen bonds in salicylamide derivatives

The crystal structures, solid-state infrared patterns, and thermal properties of two polymorphs of 4-nitrosalicylanilide are presented. In both polymorphs, intramolecular hydrogen bonds are found between the phenol oxygen and the amide proton, and intermolecular hydrogen bonds are found between the amide carbonyl oxygen and the phenol proton. These hydrogen bond patterns are compared to those found in other known salicylamide derivatives and an analysis is given of the factors contributing to preferences for intra- or intermolecular hydrogen bonds in these structures. Crystal data: polymorph, orthorhombic,Pbca,a=11.003(4),b=27.959(7),c=7.622(5) Å,Z=4,V=2345(3) ų, andR=0.038 (1351 reflections); polymorph, monoclinic,P2₁/a,a=28.36(1),b=11.64(1),c=7.293(8) Å,=90.68(6)°,Z=8,V=2408 ų, andR=0.043 (2425 reflections).     

Chemistry of biureas—I: Cyclisation of ethoxycarbonyl derivatives of thiobiureas and bithioureas

The interaction of ethoxycarbonyl isothiocyanate with semicarbazides or thiosemicarbazides produces 1-ethoxycarbonyl-2-thiobiureas or bithioureas, respectively. The former are cyclisable to 2-ethoxycarbonamido-5-hydroxy-l,3,4-thiadiazole in acid, or to 2(H)-carbamoyl-3-hydroxy-5-alkylthio-1,2,4-triazoles, and thence to 3-hydroxy-5-mercapto-1,2,4-triazole in alkaline media. The ring-closure of the 1-ethoxycarbonylbithioureas proceeds similarly affording, under the influence of acids, 5-amino (or mercapto)-2-ethoxycarbonamido-1,3,4-thiadiazoles. The action of alkali produces compounds derived from 3-hydroxy-5-mercapto-1,2,4-triazole; the isolation, in a selected example, of the 2(H)-phenylthiocarbamoyl-derivative, elucidates the course of this reaction. Hydrazinolysis of 1-ethoxycarbonylbithiourea yields 4-amino-3-hydrazino-5-mercapto-1,2,4-triazole.     

Temperature dependence of development of slip bands and of strain hardening in AgCl crystals

The dependence of the flow stress and the slip band density on the plastic strain has been measured at 201 K, 293 K and 363 K. The growth of deformation concentrated in an average slip band has been stated. The types of obstacles acting against the rise and development of a slip band and the temperature dependence of the strain hardening in AgCl crystals are discussed. An equation stating the dependence of the flow stress on the slip band density is presented. The hardening in AgCl crystals is classified as the stage III — hardening.     

PHOTOREACTIVATING ENZYME FROM NEUROSPORA CRASSA*

Abstract— Extracts of Neurospora crassa contain photoreactivating enzyme by the criteria of ability to split thymine-containing dimers and to increase the transforming ability of u.v.-irradiated Hemophilus influenzae DNA. The latter activity is heat-labile and is destroyed by trypsin. The action spectrum of such in vitro photoreactivation is a simple one (with a single maximum at 405 nm in the range 313 to 436 nm), differing from the more complicated in vitro spectra for yeast and Escherichia coli. However, the in vitro Neurospora spectrum coincides closely with the in vivo spectrum for this organism, suggesting that there is little or no “indirect” photoreactivation in Neurospora. It is concluded that the Neurospora photoreactivating enzyme is probably of a different type than those of yeast and Escherichia coli.