Moments of Two-Variable Functions and the Uniqueness of Graph Limits

For a symmetric bounded measurable function W on [0, 1]² and a simple graph F, the homomorphism density $t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G _n ) of dense graphs is said to be convergent if the probability, t(F, G _n ), that a random map from V(F) into V(G _n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]². Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.     

Bayesian analysis of constant elasticity of variance models

This paper focuses on the estimation of some models in finance and in particular, in interest rates. We analyse discretized versions of the constant elasticity of variance (CEV) models where the normal law showing up in the usual discretization of the diffusion part is replaced by a range of heavy‐tailed distributions. A further extension of the model is to allow the elasticity of variance to be a parameter itself. This generalized model allows great flexibility in modelling and simplifies the model implementation considerably using the scale mixtures representation. The mixing parameters provide a means to identify possible outliers and protect inference by down‐weighting the distorting effects of these outliers. For parameter estimation, Bayesian approach is adopted and implemented using the software WinBUGS (Bayesian inference using Gibbs sampler). Results from a real data analysis show that an exponential power distribution with a random shape parameter, which is highly leptokurtic compared with the normal distribution, forms the best CEV model for the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Frame theory in directional statistics

Distinguishing between uniform and non-uniform sample distributions is a common problem in directional data analysis; however for many tests, non-uniform distributions exist that fail uniformity rejection. To predict these distributions, we merge directional statistics with frame theory and find that probabilistic tight frames yield non-uniform distributions that minimize directional potentials, leading to failure of uniformity rejection for the Bingham test. Finally, we apply our results to model patterns found in granular rod experiments.     

Weighing Matrices and String Sorting

In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion, that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find new weighing matrices of order 2n and weights 2n – 13, 2n – 17 constructed from two circulants.     

Complex variables in junior high school: the role and potential impact of an outreach mathematician

Outreach mathematicians are college faculty who are trainedin mathematics but who undertake an active role in improvingprimary and secondary education. This role is examined througha study where an outreach mathematician introduced the conceptof complex variables to junior high school students in the UnitedStates with the goal of stimulating their interest in mathematicsand improving their algebra skills. Comparison of pre- and post-testresults showed that ninth-grade students displayed a significantchange in algebraic skills while the eighth-grade students madelittle progress. The outreach mathematician lacked some awarenessof the eighth-grade students’ foundational backgroundand motivation. This illustrates the importance of working moreclosely with the participating teacher, who understands betterthe curriculum and the students’ background knowledge,levels of maturity and levels of motivation.     

On the number of generators of Cohen-Macaulay ideals

Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.

     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].