Related papers: Abstract simplicial complexes in {\tt Macaulay2}
The famous asynchronous computability theorem (ACT) relates the existence of an asynchronous wait-free shared memory protocol for solving a task with the existence of a simplicial map from a subdivision of the simplicial complex…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
We design various logics for proving hyper properties of iterative programs by application of abstract interpretation principles. In part I, we design a generic, structural, fixpoint abstract interpreter parameterized by an algebraic…
Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental…
Static program analysis is a valuable tool for any programming language that people write programs in. The prevalence of scripting languages in the world suggests programming language interpreters are relatively easy to write. Users of…
Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices…
We introduce a new Macaulay2 package, Nauty, which gives access to powerful methods on graphs provided by the software nauty by Brendan McKay. The primary motivation for accessing nauty is to determine if two graphs are isomorphic. We also…
For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and…
The Python package ComCH is a lightweight specialized computer algebra system that provides models for well known objects, the surjection and Barratt-Eccles operads, parameterizing the product structure of algebras that are commutative in a…
We describe a new software package for computing multiplier ideals in certain cases, including monomial ideals, monomial curves, generic determinantal ideals, and hyperplane arrangements. In these cases we take advantage of combinatorial…
This paper presents a novel set of algorithms for heap abstraction, identifying logically related regions of the heap. The targeted regions include objects that are part of the same component structure (recursive data structure). The result…
We introduce the package Posets for Macaulay2. This package provides a data structure and the necessary methods for working with partially ordered sets, also called posets. In particular, the package implements methods to enumerate many…
partycls is a Python framework for cluster analysis of systems of interacting particles. By grouping particles that share similar structural or dynamical properties, partycls enables rapid and unsupervised exploration of the system's…
We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for…
In this paper we inititate the study of abstract simplicial complexes which are initial segments of qualitative probability orders. This is a natural class that contains the threshold complexes and is contained in the shifted complexes, but…
The "simplicial complexes" and "join" (*) today used within combinatorics aren't the classical concepts, cf. Spanier (1966) p. 108-9, but, exept for \emptyset, complexes having {\emptyset} as a subcomplex resp. \Sigma1 * \Sigma2 := {\sigma1…
This note describes a Macaulay2 package for handling divisors. Group operations for divisors are included. There are methods for converting divisors to reflexive or invertible sheaves. Additionally, there are methods for checking whether…
A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological…