We introduce the lcm-filtration and stepwise filtration, comparing their performance across various scenarios in terms of computational complexity, efficiency, and redundancy. The lcm-filtration often involves identical steps or ideals, leading to unnecessary computations. To address this, we analyse how stepwise filtration can effectively compute only the non-identical steps, offering a more efficient approach. We compare these filtrations in applications to networks, system signatures, and sensitivity analysis.

Abstract It has been shown by Pronzato et al. (Bernoulli 23(4A):2617–2642, 2017; J Multivar Anal 168:276–289, 2018) that simplicial volumes formed by independent copies of random variables can be used to extend the definition of generalised variances. It is shown in this paper that exterior algebra is a natural environment in which to study these constructions. This is used to extend the formulation to covariances and correlations. The theory leads naturally to dispersion ordering, that is partial orderings in which one random variable is more disperse than another if one squared simplicial volume stochastically dominates the other.

We use a monomial ideal I to model a discrete preference function on a set of n factors. We can measure the sensitivity of each point represented by a monomial m by calculating its formal partial derivatives with respect to each variable. These derivatives can be used to define the Koszul simplicial complex of the ideal I at m. We refer to points at which the homology of their Koszul complex is not null as sensitive corners. In the context of preference analysis, the ranks of the homology groups are not precise enough to distinguish between sensitive corners that have the same homology but correspond to different sensitivity behaviors. To address this issue, we propose using a filtration on the Koszul complexes of the sensitive corners based on the lcm-lattice of the ideal I. This filtration induces a persistent homology at each corner m. We then use unsupervised Machine Learning methods to classify the corners based on the distance between their persistence diagrams.

We apply the methods of algebraic reliability to the study of percolation on trees. To a complete $k$-ary tree $T_{k,n}$ of depth $n$ we assign a monomial ideal $I_{k,n}$ on $\sum_{i=1}^n k^i$ variables and $k^n$ minimal monomial generators. We give explicit recursive formulae for the Betti numbers of $I_{k,n}$ and their Hilbert series, which allow us to study explicitly percolation on $T_{k,n}$. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.

The algebraic approach to the analysis of system reliability associates an algebraic object, a monomial ideal, to a coherent system (CS), and studies the reliability of the system using the Hilbert series of the monomial ideal. New capabilities of the algebraic method in system design are shown, in particular related to enumeration of working states. The algebraic method should be a useful tool in reliability, both for performing different computations on system features, and to study the structure of systems.

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multicriteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

There is a close mathematical relationship between integer grids of a particular echelon form and coherent systems in reliability in the case of states coded as integer grid points. This paper shows that such an integer representation is the link between abstract tube theory, which gives improved inclusion-exclusion bounds, and an algebraic method, Grobner bases, based on the polynomial ideal of the failure event.

The growth function for a class of subsets C of a set X is defined by mC(N)max{Δc(F): F⫅X, |F| =N}, N=1,2,…, whereΔc(F)|{F∩C: CϵC}| the number of possible sets obtained by intersecting an element of C with the set F. Sauer (1972) showed that if C forms a Vapnik-Chervonenkis class with dimension V(C), then mc(N)⩽∑j=0V(C)−1Njfor N⩾ V(C) −1. The collection C of Euclidean balls in Rd has been shown by Dudley (1979) to have VC dimension equal to d + 2. It is well known, by using a standard geometric transformation, that Sauer's bound gives the exact number of subsets in this case. We give a more direct construction of the subsets picked out by balls, and as a corollary we obtain the number of such subsets.

Recent work on characterizing the covariance functions of stationary binary sequences has been geometric in flavor. Extreme distributions induced in windows of a given length are obtained from special periodic sequences. The work is extended to sequences with a given mean m, revealing some finer structure.>