Discrepancy in different numbers of colors

In this article, we investigate the interrelation between the discrepancies of a given hypergraph in different numbers of colors. Being an extreme example we determine the multi-color discrepancies of the k-balanced hypergraph on partition classes of (equal) size n. Let . Set k₀ k mod c and b_nkc (n−n/c/k)k/c. For the discrepancy in c colors we show

if k₀≠0, and , if c divides k. This shows that, in general, there is little correlation between the discrepancies of in different numbers of colors. If c divides k though, holds for any hypergraph .     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

The hereditary discrepancy is nearly independent of the number of colors

We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers of colors is the same apart from constant factors, i.e.,

This contrasts the ordinary discrepancy problem, where no correlation exists in many cases.

     

Non-independent randomized rounding and coloring

We propose an advanced randomized coloring algorithm for the problem of balanced colorings of hypergraphs (discrepancy problem). Instead of independently coloring the vertices with a random color, we try to use structural information about the hypergraph in the design of the random experiment by imposing suitable dependencies. This yields colorings having smaller discrepancy. We also obtain more information about the coloring, or, conversely, we may enforce the random coloring to have special properties. There are some algorithmic advantages as well.We apply our approach to hypergraphs of d-dimensional boxes and to finite geometries. Among others results, we gain a factor 2^d/2 decrease in the discrepancy of the boxes, and reduce the number of random bits needed to generate good colorings for the geometries down to (from n). The latter also speeds up the corresponding derandomization by a factor of .     

Matrix rounding with respect to small submatrices

We show that any real valued matrix A can be rounded to an integer one B such that the error in all 2 × 2 (geometric) submatrices is less than 1.5, that is, we have |a_ij ? b_ij| < 1 and for all i,j. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Synthesis and characterization of a fluorinated polyanhydride

A fluorinated aromatic polyanhydride ( B ) was synthesized from the melt condensation of mixed anhydrides of 4,4′‐(hexafluoroisopropylidene)bis benzoic acid. Although the mixed anhydride from acetic anhydride yielded only a mixture of oligomers (weight‐average molecular weight < 2000), higher weight‐average molecular weight materials in the range of 15,000–18,000 were obtained with trifluoroacetic anhydride. Polymer B was soluble in chloroform and tetrahydrofuran, had a relatively high glass‐transition temperature of 176 °C with no melting point detected to 310 °C, and showed excellent thermal stability (5% weight loss observed at 380 °C by thermogravimetric analysis). The hydrolytic degradation of the fluorinated polyanhydride in a 0.1 M phosphate buffer of pH 7.4 at 37 °C was initially zero‐order, with 35% degradation occurring in 10 days. Loss of film integrity following that led to accelerated degradation, and almost complete dissolution was observed by the 16th day. The stability of the fluorinated polyanhydride in the solid state and in the solvent tetrahydrofuran was also evaluated. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 3027–3036, 2002