arXiv subjects
Publications and source records attributed to Lisbeth Fajstrup.
Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times.
The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The directed Gromov-Hausdorff distance measures the distance between two extended metric spaces, where the new metric, defined on the same underlying space, is induced by the length of zigzag paths. This distance is then computed by isometrically embedding the directed metric spaces into an ambient directed space equipped with the zigzag distance. Analogously to the classical Gromov-Hausdorff distance, we also propose alternative formulations based on the distortion of d-maps and d-correspondences. However, unlike the classical case, these directed distances are not equivalent.
We introduce a max-flow framework for data with periodic boundary conditions, motivated by the analysis of transport in atomistic materials. Starting from a space X equipped with a map into the circle encoding a chosen periodic direction, we use the associated Reeb graph to reduce the geometry of X to a directed one-dimensional tunnel network. We then augment this graph with capacity constraints derived from cross-sectional integrals with respect to Hausdorff measure, so that edge capacities represent bottlenecks in the corresponding level-set components. To obtain a scalar transport descriptor from this capacity-augmented directed Reeb graph, we define circular max-flow for directed graphs mapped to the circle. Unlike classical source-target max-flow, this formulation does not require choosing an inlet and an outlet, and is therefore intrinsic to the periodic setting. We show that circular max-flow can be computed through a linear optimization problem related to minimum-cost circulations, and we prove that its value agrees with the flow obtained on the periodically unrolled graph. We also prove the continuity results needed to justify the capacity construction and verify that the assumptions cover void spaces arising from finite thickened backbones in the torus. The appendix illustrates the framework on simulated periodic point clouds and reports results from a separate materials-science application to self-diffusion in glasses.
Contribution: We demonstrate that it is feasible to include field specific problems in introductory mathematics courses to motivate engineering students. This is done in a way that still allows large parts of the course to be common to all students, ensuring economic viability. Background: Many first-year engineering students perceive mathematics courses as being too abstract and far from their chosen study programme. This may lead to a lack of motivation and effort, thus decreasing the learning outcomes. Intended outcomes: That engineering students recognize that the calculus and linear algebra courses are relevant for their future work within their specific field. This is intended to improve their learning outcomes. Application design: The courses have been restructured into smaller subunits, each of which has a corresponding workshop treating a real-world problem from the specific field of a given group of students. These workshops are developed in consultation with the relevant fields of study, and they are given a prominent role in the course for instance by forming the basis for the oral exams. Findings: Based on the feedback from students, we find that inclusion of field specific problems does help to highlight the applicability and importance of mathematics in engineering. When implementing such a solution, however, there are a number of challenges to keep in mind.
We prove cut-off results for deadlocks and serializability of a $PV$-thread $T$ run in parallel with itself: For a $PV$ thread $T$ which accesses a set $\mathcal{R}$ of resources, each with a maximal capacity $κ:\mathcal{R}\to\mathbb{N}$, the PV-program $T^n$, where $n$ copies of $T$ are run in parallel, is deadlock free for all $n$ if and only if $T^M$ is deadlock free where $M=Σ_{r\in\mathcal{R}}κ(r)$. This is a sharp bound: For all $κ:\mathcal{R}\to\mathbb{N}$ and finite $\mathcal{R}$ there is a thread $T$ using these resources such that $T^M$ has a deadlock, but $T^n$ does not for $n<M$. Moreover, we prove a more general theorem: There are no deadlocks in $p=T1|T2|\cdots |Tn$ if and only if there are no deadlocks in $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$ for any subset $\{i_1,\ldots,i_M\}\subset [1:n]$. For $κ(r)\equiv 1$, $T^n$ is serializable for all $n$ if and only if $T^2$ is serializable. For general capacities, we define a local obstruction to serializability. There is no local obstruction to serializability in $T^n$ for all $n$ if and only if there is no local obstruction to serializability in $T^M$ for $M=Σ_{r\in\mathcal{R}}κ(r)+1$. The obstructions may be found using a deadlock algorithm in $T^{M+1}$. These serializability results also have a generalization: If there are no local obstructions to serializability in any of the $M$-dimensional sub programs, $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$, then $p$ is serializable.
The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.
State-space reduction techniques, used primarily in model-checkers, all rely on the idea that some actions are independent, hence could be taken in any (respective) order while put in parallel, without changing the semantics. It is thus not necessary to consider all execution paths in the interleaving semantics of a concurrent program, but rather some equivalence classes. The purpose of this paper is to describe a new algorithm to compute such equivalence classes, and a representative per class, which is based on ideas originating in algebraic topology. We introduce a geometric semantics of concurrent languages, where programs are interpreted as directed topological spaces, and study its properties in order to devise an algorithm for computing dihomotopy classes of execution paths. In particular, our algorithm is able to compute a control-flow graph for concurrent programs, possibly containing loops, which is "as reduced as possible" in the sense that it generates traces modulo equivalence. A preliminary implementation was achieved, showing promising results towards efficient methods to analyze concurrent programs, with very promising results compared to partial-order reduction techniques.