Discrepancy, chaining and subgaussian processes

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs $\inf_{(\epsilon_i)}{\sup_{f\in F}}|{\sum_{i=1}^k\epsilon_i}f(X_i)|$ is asymptotically smaller than the expectation over signs as a function of the dimension $k$, if the canonical Gaussian process indexed by $F$ is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of $\mathbb {R}^k$ using properties of the canonical Gaussian process the set indexes, and then obtain quantitative structural information on a typical coordinate projection of a subgaussian class.Comment: Published in at http://dx.doi.org/10.1214/10-AOP575 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The use of parabolic variations and the direct determination of stress intensity factors using the BIE method

Two advances in the numerical techniques of utilizing the BIE method are presented. The boundary unknowns are represented by parabolas over each interval which are integrated in closed form. These integrals are listed for easy use. For problems involving crack tip singularities, these singularities are included in the boundary integrals so that the stress intensity factor becomes just one more unknown in the set of boundary unknowns thus avoiding the uncertainties of plotting and extrapolating techniques. The method is applied to the problems of a notched beam in tension and bending, with excellent results

The Composition of Business Establishment in Smaller and Larger Communities in Canada

Between 1993 and 1996 there was a large number of business establishment starts in smaller and larger communities in Canada, indicating significant business establishment development in the post recessionary period of the 1990s. Between 1993 and 1996 smaller communities had a larger ratio of smaller establishments (under 10 employees) to population than larger communities. In today's economy, new establishments tend to have fewer employees (under 10 employees) and are located in the service and construction sectors of the economy.Community/Rural/Urban Development,