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In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented… (More)

Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very complicated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structures which… (More)

- HARM DERKSEN
- 2001

Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends… (More)

- Harm Derksen
- 2005

The vanishing ideal I of a subspace arrangement V 1 ∪ V 2 ∪ · · · ∪ V m ⊆ V is an intersection I 1 ∩ I 2 ∩ · · · ∩ I m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J = I 1 I 2 · · · I m without any assumptions on the subspace… (More)

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices. Résumé Nous… (More)

Let ρ: G → GL(V) be a representation of a group G on a vector space V of dimension n < ∞. For simplicity, we assume that the base field k is algebraically closed and of characteristic zero. As usual, the group G acts linearly on the k-algebra O(V) of polynomial functions on V , the coordinate ring of V. Of special interest is the subalgebra of invariant… (More)

- HARM DERKSEN
- 2008

To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T [X].… (More)

- HARM DERKSEN
- 2005

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem-Mahler-Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological… (More)

- HARM DERKSEN
- 2002

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in [2]. Here we study the minimal free resolution of the invariant ring. Recently it was shown that if G is a finite linearly reductive group, then the ring of invariants is generated in degree ≤ |G| (see [5, 6, 3]). This extends the classical… (More)

To a directed graph without loops or 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph Γ is the set of all isomorphism classes of graphs… (More)