The topological structure of 2D disordered cellular systems

We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These cellular structures are studied as topological trees rooted on a central cell or as closed shells arranged concentrically around a germ cell. We single out the most significant parameters that characterize statistically the organization of these patterns. Universality and specificity in disordered cellular structures are discussed. Received: 23 September 1997 / Received in final form: 14 January 1998 / Accepted: 29 January 1998     

Correlations and aggregate statistics in granular packs

We study how the aggregate statistical properties for density fluctuations in granular aggregates scale with the sample size and how such a scaling is associated with the correlations between grains. Correlations are studied both between grain positions and between Vorono? cell volumes, showing distinct behaviors and properties. A non-linear scaling in the aggregate volume fluctuations as function of the sample size is discovered and the connection between such anomalous scaling and correlations is explained. It emerges that volume fluctuations might be described by means of a single universal equation for all samples at all cluster sizes.     

Foreword

Steve Smale set the agenda for FoCM in his call for the 1995 conference in Park City, Utah. No stranger he to ambitious agendas and extraordinary accomplishments. He is one of the dominant figures in the mathematics of the second half of the twentieth century. Smale’s theory of immersions, the generalized Poincare conjecture, and H-cobordism theorems with their far-reaching consequences are the bedrock of differential topology. His horseshoe is the hallmark of chaos, and his hyperbolic dynamics the rejuvenation of the geometric theory of dynamical systems. He is one of the pioneers in the introduction of infinite-dimensional manifolds for the study of nonlinear problems in the calculus of variations and partial differential equations. The list goes on: the systematic use of differential techniques in microeconomics, electrical circuit theory, chaos in predator–prey equations and, finally, for the twentieth century, the foundations of computational mathematics and complexity theory, and now, in the twenty-first century, the theory of learning. It has been our privilege to be among his collaborators and students in the broadest sense of the word. With these issues (Volume 5 Number 4 and Volume 6 Number 1, as well as an earlier article appearing in Volume 5 Number 2, are dedicated to Steve Smale’s 75th Birthday) we celebrate Steve’s 75th birthday and continuing vitality. He sets the bar high. We do our best.     

Regularization Networks and Support Vector Machines

Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular, the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik's theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case. This revised version was published online in June 2006 with corrections to the Cover Date.     

On discrete rational least squares approximation

Summary The paper deals with the finite rational least squares approximation to a discrete function. An approximation without poles and depending on a parameter is defined which tends to the least squares approximation for 0. It gives an acceptable approximation when the least squares approximation does not exist. Further it is shown that, if the discrete function to be fitted is sufficiently close to a rational function, then the least squares approximation exists.     

Stratifications of cellular patterns: hysteresis and convergence

A foam is a space-filling cellular pattern, that can be decomposed into successive layers or strata. Each layer contains all cells at the same topological distance to an origin (cell, cluster of cells, or basal layer). The disorder of the underlying structure imposes a characteristic roughening of the layers. In this paper, stratifications are described as the results a deterministic invasion" process started from different origins in the same, given foam. We compare different stratifications of the same foam. Our main results are 1) hysteresis and 2) convergence in the sequence of layers. 1) If the progression direction is reversed, the layers in the up and down sequences differ (irreversibility of the invasion process); nevertheless, going back up, the layers return exactly to the top profile. This hysteresis phenomenon is established rigorously from elementary properties of graphs and processes. 2) Layer sequences based on different origins (e.g. different starting cells) converge, in cylindrical geometry. Jogs in layers may be represented as pairs of opposite dislocations, that move erratically because the underlying structure is disordered, and end up annihilating when colliding. Convergence is demonstrated and quantified by numerical simulations on a two dimensional columnar model.     

Onset of mechanical stability in random packings of frictional spheres

Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction phiRLP of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure p and increasing interparticle friction coefficient mu. Using x-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit p-->0 a new lower bound, phiRLP=0.550+/-0.001.     

The use of predicted boiling points for the identification of halobenzenes and substituted phenols in capillary gas chromatography

Quantitative correlations between physicochemical parameters and structure of various solutes and their gas chromatographic behavior were investigated in order to predict the retention values. The identification of unknown samples in gas chromatographic routine analysis of environmental samples is enhanced by the use of these correlations in conjuction with other chromatographic methods and with mass spectrometry. The boiling points of many compounds are easily found in literature and therefore their correlation with retention values obtained in isothermal or programmed temperature analysis permit restriction of the choice of names of unknown compound to within a narrow range. The correlation between the boiling points and the retention times of chloro- and bromo-benzenes and of some chloro- and nitro- substituted phenols was investigated on non-polar capillary columns and permitted the tentative identification of many compounds belonging to these homologous series.