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 Z denote a finite collection of points in projective n-space and let I denote the homogeneous ideal of Z The points in Z are said to be in (i, j)-uniform position if every cardinality i subset of Z imposes the same number of conditions on forms of degree j The points are in uniform position if they are in (i, j)-uniform position for all values of i and j We present a symbolic algorithm that, given I, can be used to determine if the points in Z are in (i, j)-uniform position In addition it can be used to determine if the points in Z are in uniform position, in linearly general position and in general position The algorithm uses the Chow Form of various dUple embeddings of Z and derivatives of these forms The existence of the algorithm provides an answer to a question of Kreuzer

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A symbolic test for (i,j)-Uniformity in reduced zero-schemes

Let &#x1d4a2; = {g1,…,gn} be a set of elements in the polynomial ring R = ℂ[z1,…,zN], let I ⊂ R denote the ideal generated by the elements of &#x1d4a2;, and let I denote the radical of I. There is a unique decomposition I = P1 ∩⋯ ∩ Pk with each Pi a prime ideal corresponding to a minimal associated prime of I over R. Let V (&#x1d4a2;) = V (I) denote the reduced algebraic set corresponding to the common zeroes of the elements of &#x1d4a2;. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of V (&#x1d4a2;) over ℂ. This corresponds to producing a witness set for V (Pi) for each i = 1,…,k together with the degree and dimension of V (Pi) (a point in a witness set for V (Pi) can be considered as a numerical approximation for a general point on V (Pi)). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let F be a number field (i.e. a finite field extension of ℚ) and let &#x1d4a2; = {g1,…,gn} be a set ...

Numerical irreducible decomposition over a number field

. A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with ﬁxed multiplicity m in terms of their m -Ap´ery sets, giving a representation called Kunz coordinates which obey a collection of inequalities deﬁning the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with n sporadic elements and Frobenius number f is polynomial up to periodicity, or quasi-polynomial, as a function of f for ﬁxed n . We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of aﬃne semigroups with a ﬁxed number of elements, and all gaps, contained in an integer dilation of a ﬁxed polytope.

Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.

ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog

Given a collection of M experimentally measured subspaces, and a model-based subspace, this paper addresses the problem of finding a subspace that approximates the collection, under the constraint that it intersects the model-based subspace in a predetermined number of dimensions. This constrained subspace estimation (CSE) problem arises in applications such as beamforming, where the model-based subspace encodes prior information about the direction-of-arrival of some sources impinging on the array. In this paper, we formulate the constrained subspace estimation (CSE) problem, and present an approximation based on a semidefinite relaxation (SDR) of this non-convex problem. The performance of the proposed CSE algorithm is demonstrated via numerical simulation, and its application to beamforming is also discussed.

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Chris Peterson

Papers

A symbolic test for (i,j)-Uniformity in reduced zero-schemes

Numerical irreducible decomposition over a number field

Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog

Constrained subspace estimation via convex optimization