OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

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.     

Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere.     

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.     

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.     

Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems

We present theoretical and numerical comparisons between Arnoldi and nonsymmetric Lanczos procedures for computing eigenvalues of nonsymmetric matrices. In exact arithmetic we prove that any type of eigenvalue convergence behavior obtained using a nonsymmetric Lanczos procedure may also be obtained using an Arnoldi procedure but on a different matrix and with a different starting vector. In exact arithmetic we derive relationships between these types of procedures and normal matrices which suggest some interesting questions regarding the roles of nonnormality and of the choice of starting vectors in any characterizations of the convergence behavior of these procedures. Then, through a set of numerical experiments on a complex Arnoldi and on a complex nonsymmetric Lanczos procedure, we consider the more practical question of the behavior of these procedures when they are applied to the same matrices.This work was supported by NSF grant GER-9450081 while the author was visiting the University of Maryland.     

Attomole protein analysis by CIEF with LIF detection

We have coupled CIEF with an LIF detector that is based on a post‐column sheath flow cuvette. We employed Chromeo P503 as a fluorogenic reagent to label proteins before analysis. This reagent reacts with the ε‐amine of lysine residues, preserving the cationic nature of the residue; labeled proteins generate extremely sharp peaks in CIEF. A set of four standard proteins generated a linear relationship between migration time and pI. A protein homogenate prepared from a Barrett's esophagus cell line resolved over 100 components in a 40 min separation. Detection limits for Chromeo P503‐labeled β‐lactoglobulin were 5 amol injected into the capillary. Fluorescent impurities present in the ampholytes generated a large background signal that degraded the detection limit by four orders of magnitude compared with other forms of capillary electrophoresis with this detector.