931 research outputs found
On Border Basis and Groebner Basis Schemes
Hilbert schemes of zero-dimensional ideals in a polynomial ring can be
covered with suitable affine open subschemes whose construction is achieved
using border bases. Moreover, border bases have proved to be an excellent tool
for describing zero-dimensional ideals when the coefficients are inexact. And
in this situation they show a clear advantage with respect to Groebner bases
which, nevertheless, can also be used in the study of Hilbert schemes, since
they provide tools for constructing suitable stratifications.
In this paper we compare Groebner basis schemes with border basis schemes. It
is shown that Groebner basis schemes and their associated universal families
can be viewed as weighted projective schemes. A first consequence of our
approach is the proof that all the ideals which define a Groebner basis scheme
and are obtained using Buchberger's Algorithm, are equal. Another result is
that if the origin (i.e. the point corresponding to the unique monomial ideal)
in the Groebner basis scheme is smooth, then the scheme itself is isomorphic to
an affine space. This fact represents a remarkable difference between border
basis and Groebner basis schemes. Since it is natural to look for situations
where a Groebner basis scheme and the corresponding border basis scheme are
equal, we address the issue, provide an answer, and exhibit some consequences.
Open problems are discussed at the end of the paper.Comment: Some typos fixed, some small corrections done. The final version of
the paper will be published on "Collectanea Mathematica
Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves
In this paper, we shall prove a Carleman estimate for the so-called Zaremba
problem. Using some techniques of interpolation and spectral estimates, we
deduce a result of stabilization for the wave equation by means of a linear
Neumann feedback on the boundary. This extends previous results from the
literature: indeed, our logarithmic decay result is obtained while the part
where the feedback is applied contacts the boundary zone driven by an
homogeneous Dirichlet condition. We also derive a controllability result for
the heat equation with the Zaremba boundary condition.Comment: 37 pages, 3 figures. Final version to be published in Amer. J. Mat
PAC-Bayesian High Dimensional Bipartite Ranking
This paper is devoted to the bipartite ranking problem, a classical
statistical learning task, in a high dimensional setting. We propose a scoring
and ranking strategy based on the PAC-Bayesian approach. We consider nonlinear
additive scoring functions, and we derive non-asymptotic risk bounds under a
sparsity assumption. In particular, oracle inequalities in probability holding
under a margin condition assess the performance of our procedure, and prove its
minimax optimality. An MCMC-flavored algorithm is proposed to implement our
method, along with its behavior on synthetic and real-life datasets
The Geometry of Border Bases
The main topic of the paper is the construction of various explicit flat
families of border bases. To begin with, we cover the punctual Hilbert scheme
Hilb^\mu(A^n) by border basis schemes and work out the base changes. This
enables us to control flat families obtained by linear changes of coordinates.
Next we provide an explicit construction of the principal component of the
border basis scheme, and we use it to find flat families of maximal dimension
at each radical point. Finally, we connect radical points to each other and to
the monomial point via explicit flat families on the principal component
Deformations of Border Bases
Here we study the problem of generalizing one of the main tools of Groebner
basis theory, namely the flat deformation to the leading term ideal, to the
border basis setting. After showing that the straightforward approach based on
the deformation to the degree form ideal works only under additional
hypotheses, we introduce border basis schemes and universal border basis
families. With their help the problem can be rephrased as the search for a
certain rational curve on a border basis scheme. We construct the system of
generators of the vanishing ideal of the border basis scheme in different ways
and study the question of how to minimalize it. For homogeneous ideals, we also
introduce a homogeneous border basis scheme and prove that it is an affine
space in certain cases. In these cases it is then easy to write down the
desired deformations explicitly.Comment: 21 page
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