Testing Monotonicity

at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). The complexity of the test is O(n/ε). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions. Received March 29, 1999     

Quasi‐random graphs with given degree sequences

It is now known that many properties of the objects in certain combinatorial structures are equivalent, in the sense that any object possessing any of the properties must of necessity possess them all. These properties, termed quasirandom, have been described for a variety of structures such as graphs, hypergraphs, tournaments, Boolean functions, and subsets of Z _n, and most recently, sparse graphs. In this article, we extend these ideas to the more complex case of graphs which have a given degree sequence. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Multi-linear measure theory and multiple stochastic integration

Summary Stochastic integration in a Lebesgue Stieltjes sense is developed in a general context of multi-linear measure theory based on the Fundamental Grothendieck Inequality and Factorization Theorem.Research partially supported by NSF Grant DMS 8601485     

Electron capture dissociation of peptide hormone changes upon opening of the tocin ring and complexation with transition metal cations

Electron capture dissociation (ECD) is an analytical technique in mass spectrometry (MS) that allows detailed structural study of biomolecules to gain insight in their function. In this work the ECD behavior of two peptide hormones oxytocin (OT1) and vasopressin (VP1) was studied. The results of OT1 and VP1 were compared to structural analogues OT2 and VP2, which have similar amino acid sequences but lack the tocin ring. The ECD results showed that both the fragment type (c/z versus b/y) and the cleavage sites (ring versus tail) changed upon opening of the tocin ring. All four peptides were complexed with three different transition metal cations (Zn²⁺, Ni²⁺ and Cu²⁺) and the ECD results were compared to those obtained from the doubly protonated species. The use of various metal ions yielded different cleavages sites within the same peptide. This can be an effect of the metal ion itself, or a consequence of a change in conformation as was suggested earlier. In addition, the type of fragment ion varied for each metal‐complexed peptide, which is in agreement with previous observations. Copyright © 2008 John Wiley & Sons, Ltd.     

A Review of the Integration of Science and Mathematics: Implications for Further Research

Building on the earlier analysis by Berlin (1991) , this paper reviews various studies on integrating mathematics and science in the 1990s and provides some implications for further research. The areas identified for further exploration include comparison of the nature of mathematics and science, epistemological debates in mathematics and in science education, the bases used to emphasize science over mathematics or vice versa, empirical evidence of effectiveness of integration, connections between teacher education programs for integration and teachers' subsequent classroom teaching practices, perceptions of integration on the part of teacher educators, contextual difficulties in implementing integrated approaches and possible solutions, and rationales of integrating mathematics and science through technology. In order to help all students become scientifically literate, which most reform documents call for, more focused attention on integration of curriculum and instruction is necessary.     

Ryser's Conjecture for Tripartite 3-Graphs

We prove that in a tripartite 3-graph . Received March 17, 1999     

The conjugacy character ofS n tends to be regular

Consider the regular and the conjugacy characters ofS _n as vectors in Euclidean space, with the standard inner product. Asn grows, the angle between them tends to zero and the ratio of their lengths tends to one. The two characters have therefore asymptotically similar decompositions into irreducible components.