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We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.
Neural network models capable of storing memory have been extensively studied in computer science and computational neuroscience. The Hopfield network is a prototypical example of a model designed for associative, or content-addressable, memory and has been analyzed in many forms. Further, ideas and methods from complex network theory have been incorporated into artificial neural networks and learning, emphasizing their structural properties. Nevertheless, the temporal dynamics also play a vital role in biological neural networks, whose temporal structure is a crucial feature to examine. Biological neural networks display complex intermittency and, thus, can be studied through the lens of the temporal complexity (TC) theory. The TC approach look at the metastability of self-organized states, characterized by a power-law decay in the inter-event time distribution and in the total activity distribution or a scaling behavior in the corresponding event-driven diffusion processes. In this study, we present a temporal complexity (TC) analysis of a biologically-inspired Hopfield-type neural network model. We conducted a comparative assessment between scale-free and random network topologies, with particular emphasis on their global activation patterns. Our parametric analysis revealed comparable dynamical behaviors across both neural network architectures. Furthermore, our investigation into temporal complexity characteristics uncovered that seemingly distinct dynamical patterns exhibit similar temporal complexity behaviors. In particular, similar power-law decay in the activity distribution and similar complexity levels are observed in both topologies, but with a much reduced noise in the scale-free topology. Notably, most of the complex dynamical profiles were consistently observed in scale-free network configurations, thus confirming the crucial role of hubs in neural network dynamics.
Canonical neural circuit motifs are usually described functionally: divisive normalization rescales population activity by a pooled signal, and winner-take-all competition selects one pattern through recurrent excitation and shared inhibition. We represent them, and their compositions, algebraically as finite transformation systems and analyze the transition monoids generated by their input-conditioned updates, distinguishing structure already present in a generator from structure that appears only through composition, and, on a joint state space, structure inherited from one factor from structure that lives on a joint configuration. Individually aperiodic updates can generate non-aperiodic monoids. In the WTA, every frozen-drive generator collapses to fixed points, yet short input sequences create local cycles of winner-dependent inhibitory gating: globally dissipative dynamics with a reversible action. The strongest result arises in WTA-to-DN composition. The composed monoid then contains a genuinely composite local cycle in which normalization state and the winner's gating state change together, although every primitive generator is aperiodic. Holonomy analysis certifies this as a group component of the Krohn-Rhodes cascade rather than an incidental cycle, and finds most group-carrying image sets on joint configurations, whereas the uncoupled product has none. An exhaustive interface sweep shows that the composite cycle is a property of the coupling rather than of a chosen map. If motifs are building blocks of neural computation, composing them is a form of programming: one chooses primitives and interfaces so that the generated algebra has the intended repertoire. The transition monoid is that repertoire - what a primitive presents to any later construction. Recurrent circuits are compositional transformation systems; their algebra constrains what they can be programmed to compute.
In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.
Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?
For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior and with at most one tangent at every boundary point, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.
In this work we introduce a triangular Delaunay mesh generator that can be trained using reinforcement learning to maximize a given mesh quality metric. Our mesh generator consists of a graph neural network that distributes and modifies vertices, and a standard Delaunay algorithm to triangulate the vertices. We explore various design choices and evaluate our mesh generator on various tasks including mesh generation, mesh improvement, and producing variable resolution meshes. The learned mesh generator outputs meshes that are comparable to those produced by Triangle and DistMesh, two popular Delaunay-based mesh generators.
Effective analysis in neuroscience benefits significantly from robust conceptual frameworks. Traditional metrics of interbrain synchrony in social neuroscience typically depend on fixed, correlation-based approaches, restricting their explanatory capacity to descriptive observations. Inspired by the successful integration of geometric insights in network science, we propose leveraging discrete geometry to examine the dynamic reconfigurations in neural interactions during social exchanges. Unlike conventional synchrony approaches, our method interprets inter-brain connectivity changes through the evolving geometric structures of neural networks. This geometric framework is realized through a pipeline that identifies critical transitions in network connectivity using entropy metrics derived from curvature distributions. By doing so, we significantly enhance the capacity of hyperscanning methodologies to uncover underlying neural mechanisms in interactive social behavior.
Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensional ones due to high computational complexity. A recursive algorithm is proposed to compute certain high-dimensional (reduced) path homologies of stratified digraphs. By recursion on matrix representations of homologies of subgraphs, the algorithm efficiently computes the full-depth path homology of a stratified digraph, i.e. homology with dimension equal to the depth of the graph. The algorithm can be used to compute the maximal path homology of acyclic digraphs, i.e., path homology with dimension equal to the maximum path length of a graph. Numerical exper- iments show that the algorithm has a significant advantage over the general algorithm in computation time as the depth of stratified digraph increases.
This paper presents reproducible experimental evidence on unit-distance graph density. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures, used in the literature for generating unit-distance graphs. When applied to large unit-distance graphs, our method generates large graphs with a scaling exponent of 1.17, that surpasses recent theoretical predictions on graph density growth.
In this work, we revisit the problem of computing minimum enclosing Bregman balls (Bregman MEBs) of finite sets of parameters. First, we show that Bregman MEBs are equivalent to MEBs of corresponding weighted point sets with respect to the power distance. We then report an efficient Frank--Wolfe $(1+ε)$-approximation algorithm for computing power MEBs, for any $ε>0$. This power MEB approximation algorithm coincides with the Bregman MEB approximation algorithm of Nock and Nielsen (2005) when expressed in the dual gradient space. Finally, we show that the Bregman potential lifting transforms used to construct Bregman Voronoi diagrams can be reinterpreted as the classical paraboloid lifting transform applied to corresponding weighted point sets. In particular, Bregman MEB circumcenters lie on the farthest Bregman Voronoi diagrams or equivalently on the corresponding farthest power diagrams.
Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of $O(\log^6 n)$ with $O(n \log^2 n)$ preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of $O(\log^4 n)$ and, via randomization, an expected query time of $O(\log^{5/2} n \log\log n)$ with the same preprocessing bounds.
Most shape reconstruction methods assume measurements defined over planar sensing domains, such as RGB images or depth maps. In this paper, we use a curved capacitive textile as a shape sensor, treating its surface as a non-planar sensing manifold. Each scan is represented as a capacitive proximity field on this manifold, induced by the interaction between the curved electrode layout and nearby object geometry. We introduce a multi-view feedforward reconstruction model that aggregates these fields across known sensor views and recovers the observed object shape. Simulated and physical experiments demonstrate robust reconstruction from capacitive proximity signals acquired on curved sensing surfaces, pointing toward a new route to robotic near-field geometric awareness via embodied sensing.
We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.
Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.
Let $0 < α\leq 1/2$. We say that a finite point set $P$ in $\mathbb{R}^d$ is $α$-split by a hyperplane $h$ if each of the closed half-spaces determined by $h$, contains at least $α|P|$ of the points of $P$. We further say $P$ is $α$-split by a $k$-dimensional flat $τ$ if $P$ is $α$-split by any hyperplane through $τ$. In the standard notation (which coincides with Tukey depth for $k= 0$), the $k$-flat $τ$ has depth $α$ with respect to $P$. We establish interesting Helly-type theorems for splitting families of finite point sets in $\mathbb{R}^d$. Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for point and hyperplanes, our results extend to splitting families of point sets by collections of $k$-flats of arbitrary dimensionality $ 0 \leq k \leq d-1$.
This paper investigates $k$-visibility, where a line of sight can penetrate up to $k$ obstacles. While computing the $k$-visibility polygon from a single query point is well-studied, existing spatial preprocessing approaches rely on full $O(n^2)$ line arrangements through all vertex pairs without characterizing the minimal set of topological boundaries. We present a refined cell decomposition framework that isolates the exact geometric events governing $k$-visibility: primary vertex horizon lines and secondary mutually critical hinge lines. We prove that this minimal set of partition lines yields a spatial decomposition of $Θ(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $δ$-compression scheme across cell boundaries, we achieve an overall storage complexity of $\mathcal{O}(n^4)$ while supporting optimal $\mathcal{O}(\log n + m)$ query time to reconstruct explicit $k$-visibility polygons of size $m$. Our framework naturally extends to polygons containing holes.
The dilation of a geometric graph measures how much longer the path between pairs of points becomes when restricted to graph edges, rather than following the direct path through the ambient space. The minimum dilation tree of a point set is the spanning tree with minimum dilation, where edge lengths in the tree are given by distances in the ambient space. In the Euclidean plane, computing the minimum dilation tree is NP-hard, but no hardness of approximation result is known. On the other hand, the minimum spanning tree is an $(n-1)$-approximation to the minimum dilation tree, but no asymptotically-better approximation algorithm is known for general point sets in the Euclidean plane. We give the first sublinear approximation algorithm for the minimum dilation tree in the Euclidean plane. Our approximation ratio is $\tilde{O}(n^{14/15})$ and our algorithm runs in polynomial time. This resolves an open problem proposed by Eppstein in 1996.