80 research outputs found
Stable Complete Intersections
A complete intersection of n polynomials in n indeterminates has only a
finite number of zeros. In this paper we address the following question: how do
the zeros change when the coefficients of the polynomials are perturbed? In the
first part we show how to construct semi-algebraic sets in the parameter space
over which all the complete intersection ideals share the same number of
isolated real zeros. In the second part we show how to modify the complete
intersection and get a new one which generates the same ideal but whose real
zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur
Corrigendum for "Almost vanishing polynomials and an application to the Hough transform"
In this note we correct a technical error occurred in [M. Torrente and M.C.
Beltrametti, "Almost vanishing polynomials and an application to the Hough
transform", J. Algebra Appl. 13(8), (2014)]. This affects the bounds given in
that paper, even though the structure and the logic of all proofs remain fully
unchanged.Comment: 30 page
Thinning out redundant empirical data
Given a set of "empirical" points, whose coordinates are perturbed by
errors, we analyze whether it contains redundant information, that is whether
some of its elements could be represented by a single equivalent point. If this
is the case, the empirical information associated to could be described by
fewer points, chosen in a suitable way. We present two different methods to
reduce the cardinality of which compute a new set of points equivalent to
the original one, that is representing the same empirical information. Though
our algorithms use some basic notions of Cluster Analysis they are specifically
designed for "thinning out" redundant data. We include some experimental
results which illustrate the practical effectiveness of our methods.Comment: 14 pages; 3 figure
Stable Border Bases for Ideals of Points
Let be a set of points whose coordinates are known with limited accuracy;
our aim is to give a characterization of the vanishing ideal independent
of the data uncertainty. We present a method to compute a polynomial basis
of which exhibits structural stability, that is, if is
any set of points differing only slightly from , there exists a polynomial
set structurally similar to , which is a basis of the
perturbed ideal .Comment: This is an update version of "Notes on stable Border Bases" and it is
submitted to JSC. 16 pages, 0 figure
Recognition of feature curves on 3D shapes using an algebraic approach to Hough transforms
Feature curves are largely adopted to highlight shape features, such as sharp lines, or to divide surfaces into meaningful segments, like convex or concave regions. Extracting these curves is not sufficient to convey prominent and meaningful information about a shape. We have first to separate the curves belonging to features from those caused by noise and then to select the lines, which describe non-trivial portions of a surface. The automatic detection of such features is crucial for the identification and/or annotation of relevant parts of a given shape. To do this, the Hough transform (HT) is a feature extraction technique widely used in image analysis, computer vision and digital image processing, while, for 3D shapes, the extraction of salient feature curves is still an open problem. Thanks to algebraic geometry concepts, the HT technique has been recently extended to include a vast class of algebraic curves, thus proving to be a competitive tool for yielding an explicit representation of the diverse feature lines equations. In the paper, for the first time we apply this novel extension of the HT technique to the realm of 3D shapes in order to identify and localize semantic features like patterns, decorations or anatomical details on 3D objects (both complete and fragments), even in the case of features partially damaged or incomplete. The method recognizes various features, possibly compound, and it selects the most suitable feature profiles among families of algebraic curves
Proper measures of connectedness
The concept of connectedness has been widely used in financial applications, in particular for systemic risk detection. Despite its popularity, at the state of the art, a rigorous definition of connectedness is still missing. In this paper we propose a general definition of connectedness introducing the notion of proper measures of connectedness (PMCs). Based on the classical concept of mean introduced by Chisini, we define a family of PMCs and prove some useful properties. Further, we investigate whether the most popular measures of connectedness available in the literature are consistent with the proposed theoretical framework. We also compare different measures in terms of forecasting performances on real financial data. The empirical evidence shows the forecasting superiority of the PMCs compared to the measures that do not satisfy the theoretical properties. Moreover, the empirical results support the evidence that the PMCs can be useful to detect in advance financial bubbles, crises, and, in general, for systemic risk detection
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