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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)

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come… (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)

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the " Cluster algebras IV " paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family… (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.

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, Jerzy Weyman
- 2007

Contents 1. Introduction 1 1.1. Main results 1 1.2. Horn's conjecture and related problems 2 1.3. The quiver method 4 1.4. Organization 5 2. Preliminaries 6 2.1. Basic notions for quivers 6 2.2. Semi-invariants for quiver representations 7 2.3. Representations in general position 8 2.4. The canonical decomposition 9 2.5. The combinatorics of dimension… (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)