An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

Dependence of biodiesel fuel properties on the structure of fatty acid alkyl esters

IEEE Transactions on Pattern Analysis and Machine Intelligence

Production of liquid biofuels from renewable resources

Progress and recent trends in biodiesel fuels

Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

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Persistence images: a stable vector representation of persistent homology

The biodegradability of various biodiesel fuels was examined by the CO2 evolution method (EPA 560/6-82-003),
BOD5 (EPA 405.1), COD (EPA 410), and gas chromatography (GC) analyses in an aquatic system. The fuels examined
included the methyl- and ethyl-esters of rapeseed oil and soybean oil, neat rapeseed oil, neat soybean oil and Phillips 2-D
low sulfur, reference petroleum diesel. Blends of biodiesel/petroleum diesel at different volumetric ratios, including 80/20,
50/50, and 20/80, were also examined. The results demonstrate that all the biodiesel fuels are “readily biodegradable”.
Moreover, in the presence of REE, the degradation rate of petroleum diesel increased to twice that of petroleum diesel alone.
The pattern of biodegradation in the blends and reasons why biodiesel is more readily degradable than petroleum diesel are
discussed. The biodegradation monitoring results from both CO2 evolution and GC methods are compared.

Biodegradability of biodiesel in the aquatic environment

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-secant varieties to Segre varieties for p < 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P1 x P1 x Pn x Pn and P2 x P3 x P3. In the final section we propose a series of conjectures about defective Segre varieties.

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Induction for secant varieties of Segre varieties

Introduction Preliminaries Gaeta's theorem Divisors on an ACM subscheme of projective spaces Gorenstein ideals and Gorenstein liaison CI-liaison invariants Geometric applications of the CI-liaison invariants Glicci curves on arithmetically Cohen-Macaulay surfaces Unobstructedness and dimension of families of subschemes Dimension of families of determinantal subschemes Bibliography.

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

Chris Peterson

Papers

Persistence images: a stable vector representation of persistent homology

Biodegradability of biodiesel in the aquatic environment

Induction for secant varieties of Segre varieties

Gorenstein liaison, complete intersection liaison invariants and unobstructedness

Persistence Images: A Stable Vector Representation of Persistent Homology