arXiv subjects

Lisbeth Fajstrup

Publications and source records attributed to Lisbeth Fajstrup.

9 recordsLinked to original sources

arXiv2026

Periodic Tunnel Systems: Reeb Graph Representations and Circular Max-Flow

Inspired by the study of transport properties in materials science, we investigate the Reeb graph of a map $f:X\to\Ss$, interpreting $X$ as a periodic tunnel system in which transport takes place. The map $f$ encodes a chosen periodic direction and induces a directed structure on the graph. Under constructibility, we establish a lifting result for directed paths that makes precise how the graph represents tunnel segments in $X$. To turn this representation into a numerical transport descriptor, we equip the graph with cross-sectional capacities defined, in a Riemannian setting, by weighted Hausdorff-measure integrals. Under boundary regularity and Morse assumptions in dimension $d\ge2$, we establish continuity over regular Reeb values and continuity of the total capacity across fibers. Taking edgewise infima gives the bottleneck capacities for a circular max-flow formulation, which measures flow through fibers of the map to the circle without choosing an arbitrary inlet or outlet. The resulting descriptor is computable by linear optimization and is related to minimum-cost circulations. For nonnegative real capacities, we prove that the max-flow value is invariant under periodic unrolling: the same descriptor is obtained whether the medium is modeled by a toroidal cell or by its periodic replication along the chosen direction. Simulations on periodic point clouds illustrate the construction and the effects of imposing source--target cuts.

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arXiv2026

A Directed Cubical Framework for k-critical Bipersistence Modules

In topological data analysis, multiparameter persistence modules (MPMs) are used to analyze data with multiple parameters. Summarizing topological information for MPMs is a non-trivial task; one useful summary of bipersistence modules is the fibered barcode \cite{lesnick2015interactive,lesnick2025fast}. In this paper, we use a directed topological construction of the parameter space of a bifiltration to extend the fibered barcode. Specifically, given a $k$-critical bifiltration $F$, we build an associated directed Euclidean cubical complex $C$ whose directed paths each correspond to a one-parameter subfiltration of $F$. The path space of $C$, $\di{C}$, is used to define the \emph{extended fibered barcode}, which contains all possible monotonically increasing paths in $C$. We show that there can be paths in $\di{C}$ which generate filtrations not obtainable along a line with positive slope, and that all lines with positive slope have a path in $\di{C}$ that generate an equivalent barcode. We introduce two types of equivalence for these restricted modules: combinatorial equivalence of diagrams, and simplicial equivalence of the filtrations generating the modules. We show that removing certain cubes from $C$ yields a subcomplex on which dihomotopic paths induce simplicially equivalent filtrations, and hence combinatorially equivalent diagrams.

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arXiv2025

Gromov-Hausdorff Distance for Directed Spaces

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.

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arXiv2025

Identifying cobordisms using kernel persistence

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.

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arXiv2023

Transforming First-Year Calculus Teaching for Engineering Students -- Blocks with Field Specific Examples, Problems, and Exams

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.

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arXiv2022

Cut-off Theorems for the PV-model

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.

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arXiv2022

Combinatorial Conditions for Directed Collapsing

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.

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arXiv2019

Towards Directed Collapsibility

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.

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arXiv2012

Trace Spaces: an Efficient New Technique for State-Space Reduction

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.

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