arXiv subjects

Louis Theran

Publications and source records attributed to Louis Theran.

At least 19 recordsLinked to original sources

arXiv2026

On weavings, grillages, tensegrities, and frameworks

We investigate the stability of discrete structures comprising of woven elastic beams in a plane, exposing a connection between admissible over / under patterns and the first-order and static rigidity of an associated tensegrity that is related through a natural polarity transformation. The relationship between the Airy and Whiteley stress functions for the tensegrity and weaving structures is explored. Our results lead to an efficient method for finding such an over / under pattern. The method is illustrated through a worked example modelled on the complete bipartite graph $K_{4,4}$.

math.MG↗

arXiv2026

Combinatorial Admissibility in Control-Affine Networks

We study synchronization of heterogeneous control-affine nonlinear agents interconnected through diffusive (relative-output) measurements. We separate the design into an edge-space step, specifying a stabilizing model evolution for relative outputs, and a lift step, realizing the prescribed edge motion using the agents' allowable input directions, constrained by the control-affine geometry of the agents. We introduce an admissibility notion that characterizes when an edge-driven diffusive design is feasible. We derive checkable combinatorial certificates that connect graph topology and actuation limits directly to admissibility, so that feasible edge dynamics can be verified in a practical and transparent way. The results are illustrated on synchronization of nonlinear oscillators.

math.OC↗

arXiv2025

A geometric view of formation control with application to directed sensing

We propose a geometric approach to distance-based formation control modeled on a minimum-norm lifting of Riemannian gradient descent in edge-space to node-space. This yields a unified family of controllers, including the classical gradient controller and its directed variant. For the directed case, we give a simple numerical test for local convergence that applies to any directed graph and target. We show that persistence is neither necessary nor sufficient for local convergence of our directed controller and propose an alternative that is necessary and more easily checked.

math.OC↗

arXiv2025

Terracini matroids: algebraic matroids of secants and embedded joins

Applications of algebraic geometry have sparked much recent work on algebraic matroids. An algebraic matroid encodes algebraic dependencies among coordinate functions on a variety. We study the behavior of algebraic matroids under joins and secants of varieties. Motivated by Terracini's lemma, we introduce the notion of a Terracini union of matroids, which captures when the algebraic matroid of a join coincides with the matroid union of the algebraic matroids of its summands. We illustrate applications of our results with a discussion of the implications for toric surfaces and threefolds.

math.CO↗

arXiv2025

Uniquely realizable crystalline structures

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in $\mathbb{R}^d$ in the cases when the lattice is either fixed, fully flexible, or flexible with a volume constraint for the fundamental domain. For the fixed and fully flexible lattice variants, we also establish necessary and sufficient conditions for generic infinite periodic bar-joint frameworks to be globally rigid in $\mathbb{R}^d$. These results provide periodic versions of the fundamental results of Connelly, as well as Gortler, Healy and Thurston on the global rigidity of generic finite bar-joint frameworks.

math.MG↗

arXiv2025

Linearizing Algebraic Matroids

Although algebraic matroids were discovered in the 1930s, interest in them was largely dormant until their recent use in applications of algebraic geometry. Because nonlinear algebra is computationally challenging, it is easier to work with an isomorphic linear matroid if one exists. We describe an explicit construction that produces a linear representation over an algebraically closed field of characteristic zero starting with the data used in applications. We will also discuss classical examples of algebraic matroids in the modern language of polynomial ideals, illustrating how the existence of an isomorphic linear matroid depends on properties of the field.

math.AC↗

arXiv2025

Higher Order Rigidity and Energy

In this paper, we revisit the notion of higher-order rigidity of a bar-and-joint framework. In particular, we provide a link between the rigidity properties of a framework, and the growth order of an energy function defined on that framework. Using our approach, we propose a general definition for the rigidity order of a framework, and we show that this definition does not depend on the details of the chosen energy function. Then we show how this order can be studied using higher order derivative tests. Doing so, we obtain a new proof that the lack of a second order flex implies rigidity. Our proof relies on our construction of a fourth derivative test, which may be applied to a critical point when the second derivative test fails. We also obtain a new proof that when the dimension of non-trivial first-order flex coefficients $\p'$ equals $1$, then the lack of a $k$th order flex for some $k$ implies a framework is rigid. The higher order derivative tests that we study here may have applications in more general optimization problems.

math.MG↗

arXiv2025

Equilibrium stresses in frameworks via symmetric averaging

For a bar-joint framework $(G,p)$, a subgroup $\Gamma$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $\Gamma$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.

math.MG↗

arXiv2024

Fibers of point cloud persistence

Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a barcode--a multiset of intervals. The barcode is stable and interpretable, leading to applications within mathematics and data science. We study the spaces of point clouds with the same barcode by connecting persistence with real algebraic geometry and rigidity theory. Utilizing a semi-algebraic setup of point cloud persistence, we give lower and upper bounds on its dimension and provide combinatorial conditions in terms of the local and global rigidity properties of graphs associated with point clouds and filtrations. We prove that for generic point clouds in $\mathbb{R}^d$ ($d \geq 2$), a point cloud is identifiable up to isometry from its VR persistence if the associated graph is globally rigid, and locally identifiable up to isometry from its \v{C}ech persistence if the associated hypergraph is rigid.

math.AT↗

arXiv2024

Reconstruction in one dimension from unlabeled Euclidean lengths

Let $G$ be a $3$-connected ordered graph with $n$ vertices and $m$ edges. Let $\mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $\{1, 2,3, \ldots, 2^b\}$ for $b\ge m^2$. Let $\ell$ be the vector of $m$ Euclidean lengths of $G$'s edges under $\mathbf{p}$. In this paper, we show that, with high probability over $\mathbf{p}$, we can efficiently reconstruct both $G$ and $\mathbf{p}$ from $\ell$. This reconstruction problem is NP-HARD in the worst case, even if both $G$ and $\ell$ are given. We also show that our results stand in the presence of small amounts of error in $\ell$, and in the real setting, with sufficiently accurate length measurements. Our method combines lattice reduction, which has previously been used to solve random subset sum problems, with an algorithm of Seymour that can efficiently reconstruct an ordered graph given an independence oracle for its matroid.

math.MG↗

arXiv2024

The Stress-Flex Conjecture

Recently, it has been proven that a tensegrity framework that arises from coning the one-skeleton of a convex polytope is rigid. Since such frameworks are not always infinitesimally rigid, this leaves open the question as to whether they are at least prestress stable. We prove here that this holds subject to an intriguing new conjecture about coned polytope frameworks, that we call the stress-flex conjecture. Multiple numerical experiments suggest that this conjecture is true, and most surprisingly, seems to hold even beyond convexity and also for higher genus~polytopes.

math.CO↗

arXiv2023

Maximum likelihood thresholds via graph rigidity

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of $G$ and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erd{ö}s-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erd{ö}s-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most $4$ and describe the threshold probability for the MLT to switch from $3$ to $4$. We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.

math.CO↗

arXiv2023

Trilateration using Unlabeled Path or Loop Lengths

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclideanlength in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing $\mathbf{p}$ given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that $\mathbf{p}$ is generic.

math.MG↗

arXiv2023

Proving the Existence of a GOR Without Probability

In this note, we provide a new proof that a $D$-connected graph $G$ on $n$ vertices has a general position orthogonal representation in $\RR^{n-D}$. Our argument, while based on many of the concepts from the original proof due to Lovász, Saks and Schrijver, does not use the probabilistic method.

math.CO↗

arXiv2023

General position stresses

Let $G$ be a graph with $n$ vertices, and $d$ be a target dimension. In this paper we study the set of rank $n-d-1$ matrices that are equilibrium stress matrices for at least one (unspecified) $d$-dimensional framework of $G$ in general position. In particular, we show that this set is algebraically irreducible. Likewise, we show that the set of frameworks with such equilibrium stress matrices is irreducible. As an application, this leads to a new and direct proof that every generically globally rigid graph has a generic framework that is universally rigid.

math.MG↗

arXiv2023

Maximum likelihood thresholds of generic linear concentration models

The maximum likelihood threshold of a statistical model is the minimum number of datapoints required to fit the model via maximum likelihood estimation. In this paper we determine the maximum likelihood thresholds of generic linear concentration models. This turns out to be the number that one might expect from a naive dimension count, which is nontrivial to prove given that the maximum likelihood threshold is a semi-algebraic concept. We also describe geometrically how a linear concentration model can fail to exhibit this generic behavior.

math.ST↗

arXiv2022

Computing maximum likelihood thresholds using graph rigidity

math.CO↗

arXiv2022

Universal Rigidity of Ladders on the line

In "Universal rigidity on the line, point orde" it is shown, answering a question of Jordán and Nguyen, that universal rigidity of a generic bar-joint framework in R^1 depends on more than the ordering of the vertices. The graph G that was used in that paper is a ladder with three rungs. Here we provide a general answer when that ladder with three rungs in the line is universally rigid and when it is not.

math.MG↗