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

Let \({\mathbb {N}}\) denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid \(S\subseteq {\mathbb {N}}\). For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid \(S\subseteq {\mathbb {N}}^d\). The cardinality of \({\mathbb {N}}^d \setminus S\) is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let \(N_{g,d}\) denote the number of generalized numerical semigroups \(S\subseteq {\mathbb {N}}^d\) of genus \(g\). We compute \(N_{g,d}\) for small values of \(g,d\) and provide coarse asymptotic bounds on \(N_{g,d}\) for large values of \(g,d\). For a fixed \(g\), we show that \(F_g(d)=N_{g,d}\) is a polynomial function of degree \(g\). We close with several open problems/conjectures related to the asymptotic growth of \(N_{g,d}\) and with suggestions for further avenues of research.

Algorithms and basic asymptotics for generalized numerical semigroups in {\mathbb {N}}^d

Let \(R = \mathbb{C}[x_{1},\ldots,x_{N}]\) and let \(F =\{ f_{1},\ldots,f_{t}\} \subset R\) be a set of generators for an ideal I. Let \(Y =\{ y_{1},\ldots,y_{\ell}\} \subset {\mathbb{C}}^{N}\) be a subset of the set of isolated solutions of the zero locus of F. Let \(\mathfrak{m}_{y_{i}}\) denote the maximal ideal of y i and let \(\mathcal{P}_{y_{i}}\) denote the \(\mathfrak{m}_{y_{i}}\)-primary component of I. Let \(J = \cap _{i=1}^{l}\mathcal{P}_{y_{i}}\) and let \(\mathcal{Z}\) denote the corresponding zero dimensional subscheme supported on Y. This article presents a numerical algorithm for computing the Hilbert function and the regularity of \(\mathcal{Z}\). In addition, the algorithm produces a monomial basis for R∕J. The input for the algorithm is the polynomial system F and a numerical approximation of each element in Y.

Numerical Computation of the Hilbert Function and Regularity of a Zero Dimensional Scheme

We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.

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A method to compute Segre classes of subschemes of projective space

New depth sensors, like the Microsoft Kinect, produce streams of human pose data. These discrete pose streams can be viewed as noisy samples of an underlying continuous ideal curve that describes a trajectory through high-dimensional pose space. This paper introduces a technique for generalized curvature analysis (GCA) that determines features along the trajectory which can be used to characterize change and segment motion. Tools are developed for approximating generalized curvatures at mean points along a curve in terms of the singular values of local mean-centered data balls. The features of the GCA algorithm are illustrated on both synthetic and real examples, including data collected from a Kinect II sensor. We also applied GCA to the Carnegie Mellon University Motion Capture (MoCaP) database. Given that GCA scales linearly with the length of the time series we are able to analyze large data sets without down sampling. It is demonstrated that the generalized curvature approximations can be used to segment pose streams into motions and transitions between motions. The GCA algorithm can identify 94.2 percent of the transitions between motions without knowing the set of possible motions in advance, even though the subjects do not stop or pause between motions.

Motion Segmentation via Generalized Curvatures

We present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the real Grassmann manifold G(k, n) (whose points parameterize the k-dimensional subspaces of Rn) where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information.

Chris Peterson

Papers

Algorithms and basic asymptotics for generalized numerical semigroups in {\mathbb {N}}^d

Numerical Computation of the Hilbert Function and Regularity of a Zero Dimensional Scheme

A method to compute Segre classes of subschemes of projective space

Motion Segmentation via Generalized Curvatures

An application of persistent homology on Grassmann manifolds for the detection of signals in hyperspectral imagery