Related papers: Differential cumulants, hierachical models and mon…

We study the normalization of a monomial ideal, and show how to compute its Hilbert function (using Ehrhart polynomials) if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown.

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…

This paper develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale. Perfect dependence is attainable for all marginal distributions. It furthermore proposes a set of dependence…

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.

All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do…

Let $M$ be a model set meeting two simple conditions: (1) the internal space $H$ is a product of $R^n$ and a finite group, and (2) the window $W$ is a finite union of disjoint polyhedra. Then any point pattern with finite local complexity…

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial…

Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical…

Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert…

Following our previous work on copula-based nonsymmetric dependence measures, we introduce similar measures for discrete random variables. The measures cover the range between two extremes: independence and complete dependence, which take…

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and…

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F.…

In a central lemma we characterize "generating functions" of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to "unital, associative universal…

The number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis…

We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation…

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…

We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to…

Given a stochastic structure with a filtration $\mathbb{F}$, the class of all random times whose conditional distribution functions are differentiable with respect to some $\mathbb{F}$ adapted non decreasing processes is considered. The…