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11
Multigraded commutative algebra of graph decompositions
, 2012
"... The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe ..."
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Cited by 8 (3 self)
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The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in nonzero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K4minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.
Toric algebra of hypergraphs
, 2014
"... The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings o ..."
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Cited by 4 (1 self)
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The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are wellknown in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.
Computing algebraic matroids
, 2014
"... An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. Decorated algebraic mat ..."
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Cited by 2 (2 self)
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An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. Decorated algebraic matroids can be computed via symbolic computation using Gröbner bases, or through linear algebra in the space of differentials (with decorations calculated using numerical algebraic geometry). Both algorithms are developed here. Failure of the second algorithm occurs on a subvariety called the nonmatroidal or NMlocus. Decorated algebraic matroids have widespread relevance anywhere that coordinates have combinatorial significance. Examples are computed from applied algebra, in algebraic statistics and chemical reaction network theory, as well as more theoretical examples from algebraic geometry and matroid theory.
Sizebiased permutation of a finite sequence with independent and identically distributed terms. arXiv preprint arXiv:1206.2081v1
, 2012
"... Abstract. This paper focuses on the sizebiased permutation of n independent and identically distributed (i.i.d) positive random variables. Our setting is a finite dimensional analogue of the sizebiased permutation of ranked jumps of a subordinator studied in PermanPitmanYor (PPY) [27], as well ..."
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Abstract. This paper focuses on the sizebiased permutation of n independent and identically distributed (i.i.d) positive random variables. Our setting is a finite dimensional analogue of the sizebiased permutation of ranked jumps of a subordinator studied in PermanPitmanYor (PPY) [27], as well as a special form of induced order statistics [3, 8]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simpler proofs of existing ones. Our main contribution, Theorem 19 in Section 5, describes the asymptotic distribution of the last few terms in a finite i.i.d sizebiased permutation via a Poisson coupling with its few smallest order statistics. 1.
Markov degree of the Birkhoff model
 Journal of Algebraic Combinatorics
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Olteanu, Algebraic properties of classes of path ideals
"... Abstract. We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially CohenMacaulay in terms of the underlying poset. Moreover, monomial ideals, which arise from the Lucedecomposable model in algebraic statistics, ..."
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Abstract. We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially CohenMacaulay in terms of the underlying poset. Moreover, monomial ideals, which arise from the Lucedecomposable model in algebraic statistics, can be viewed as path ideals of certain posets. We study invariants of these socalled Lucedecomposable monomial ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their regularity and their Betti numbers.
Goodnessoffit for loglinear network models: Dynamic Markov bases using hypergraphs
, 2014
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RESEARCH STATEMENT  ALGEBRAIC MATROIDS: STRUCTURE AND APPLICATIONS
"... Algebraic matroids are combinatorial objects that can be extracted from geometric problems, describing the independence structure on the coordinates. In this proposal, algebraic matroids are used to analyze applied problems, and their structure is explored. We give historical background to this topi ..."
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Algebraic matroids are combinatorial objects that can be extracted from geometric problems, describing the independence structure on the coordinates. In this proposal, algebraic matroids are used to analyze applied problems, and their structure is explored. We give historical background to this topic, then set forth four projects.