We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series of rank one, and discuss connections to other concepts in tropical algebra and combinatorial algebraic geometry.

We prove quadratic generation for the ideal of the Cox ring of the blow-up of P 3 \mathbb {P}^3 at 7 7 points, solving a conjecture of Lesieutre and Park. To do this we compute Khovanskii bases, implementing techniques which proved successful in the case of del Pezzo surfaces. Such bases give us degenerations to toric varieties whose associated polytopes encode toric degenerations with respect to all projective embeddings. We study the edge-graphs of these polytopes and we introduce the Mukai edge graph .

This paper introduces novel subfamilies of analytic and bi-univalent functions in Ω=ς∈C:|ς|<1, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate Fibonacci polynomials. The proposed framework provides a unified treatment that generalizes numerous earlier studies by incorporating parameters controlling both the operator’s fractional calculus features and the domain’s combinatorial geometry. For these subfamilies, we establish initial coefficient bounds (d2, d3) and solve the Fekete–Szegö problem (d3−ξd22). The derived inequalities are interesting, and their proofs leverage the intricate interplay between the series expansions of the Mittag–Leffler function and the generating function of the Fibonacci polynomials. By specializing the parameters governing the operator and the polynomial domain, we show how our main theorems systematically recover and extend a wide range of known results from the literature, thereby demonstrating the generality and unifying power of our approach.

PurposeCurrent methods for evaluating scleral staphyloma morphology fail to provide curvature data of global scleral deformation. This study aimed to develop a quantitative method based on discrete differential geometry for analyzing scleral deformation caused by staphyloma.MethodsThis study retrospectively analyzed 128 eyes of 73 patients with pathological myopia. All patients underwent orbital magnetic resonance imaging and three-dimensional (3D) reconstruction. The discrete Gaussian curvature measures (DGCMs) and discrete mean curvature measures (DMCMs) of all vertices on the ocular 3D model were calculated. We further established a computational method for the degree of scleral staphyloma expansion (E/U ratio).ResultsThe posterior scleral DGCM (pDGCM) and DMCM (pDMCM) standard deviations (SDs) were significantly greater in the staphyloma group than in the non-staphyloma group (0.069 ± 0.026 vs. 0.025 ± 0.005 and 0.335 ± 0.096 vs. 0.154 ± 0.027, respectively; both P < 0.0001). The E/U ratio was strongly linearly correlated with both the pDGCM and pDMCM SDs (both P < 0.01). For staphyloma diagnosis, the areas under the receiver operating characteristic (ROC) curves for pDGCM and pDMCM SDs were 0.994 (95% confidence interval [CI], 0.984–1.000) and 0.993 (95% CI, 0.982–1.000), respectively (both P < 0.001).ConclusionsThe variation in the curvature of the posterior sclera is significantly greater in eyes with staphyloma than in those without. This variation is highly specific and sensitive for staphyloma diagnosis.Translational RelevanceThis method, based on discrete differential geometry, enables the direct quantification of scleral deformation, potentially providing a quantitative basis for the diagnosis and evaluation of staphyloma.

Discrete differential geometry for simulating nonlinear behaviors of flexible systems: A survey

Combinatorial geometry of rational quasi-independence models as toric fiber products

Abstract Thin plates and shells are central to emerging technologies such as deployable space structures, wearable devices, and flexible electronics, where large geometric nonlinearities are not only unavoidable but often exploited for functionality. While the widely used bar-and-hinge model in the discrete differential geometry approach offers computational simplicity, it lacks physical consistency and suffers from mesh-dependent artifacts, limiting its predictive capability. In this technical brief, we show that the mid-edge-based formulation can provide an accurate and consistent simulation for thin plates and shells. By constructing discrete analogues of the first and second fundamental forms from a mesh and its edge-adjacent neighbors, the method naturally recovers in-plane and bending strain tensors and their associated strain energy. Benchmark comparisons against finite element simulations demonstrate that the mid-edge model achieves superior accuracy, stronger consistency, and faster convergence than the bar-and-hinge formulation. Crucially, the method delivers mesh-shape-independent convergence, enabling robust modeling of geometrically nonlinear responses on arbitrary meshes. These advantages make the framework highly suitable for rapid simulation, optimization, and inverse design of morphable and programmable plate structures, with potential applications in metasurfaces, kirigami, and origami-inspired systems.

Recent advances in connectomics have been led by high-resolution reconstruction of large volumes of neural tissues using electron microscopy (EM), providing unprecedented insights into brain structure and function. Dendritic spines—dynamic protrusions on neuronal dendrites—play crucial roles in synaptic plasticity, influencing learning, memory, and various neurological disorders. However, current spine analysis methods often rely on manual annotation of subcellular features, limiting their ability to handle the complexity of spines in dense dendritic networks. This paper introduces a novel automated computational framework that integrates discrete differential geometry, machine learning, and 3D image processing to analyze dendritic spines in these intricate environments. By generating distributions of spine morphology from high resolution images including many thousands of spines, our approach captures subtle variations in spine shapes, offering a nuanced understanding of their roles in synaptic function. This framework is tested on multiple EM datasets, with the aim of enhancing our understanding of synaptic plasticity and its alterations in disease states. The proposed method is poised to accelerate neuroscience research by providing a scalable, objective, and comprehensive solution for spine analysis, uncovering insights into the role of spine geometry for neural function.

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies n i of orbitals &amp;#x03C6; i according to 0 &amp;#x2264; n i &amp;#x2264; 2 . In this work, we first refine the underlying one-body N -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness w of the N -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope &amp;#x03A3; N , S ( w ) &amp;#x2282; [ 0 , 2 ] d . These constraints are independent of M and the number d of orbitals, while their dependence on N , S is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

Abstract A major research area in discrete geometry is to consider the best way to partition the d -dimensional Euclidean space $$\mathbb {R}^d$$ R d under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space $$\mathbb {R}^d$$ R d to a discrete subset of representative values. Specifically, we study partitions of $$\mathbb {R}^d$$ R d into bounded-size tiles colored by one of k colors, such that tiles of the same color have a distance of at least t from each other. Such tilings allow for error-resilient rounding, as two points of the same color and distance less than t from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors k and the distance t , for various dimensions d . On the qualitative side, we show that in $$\mathbb {R}^d$$ R d , using $$k=d+1$$ k = d + 1 colors is both sufficient and necessary to achieve $$t&gt;0$$ t &gt; 0 . On the quantitative side, we achieve numerous upper and lower bounds on t as a function of k . In particular, for $$d=3,4,8,24$$ d = 3 , 4 , 8 , 24 , we obtain sharp asymptotic bounds on t , as $$k \rightarrow \infty $$ k → ∞ . We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat’s connector-free lemma, and Čech cohomology.

Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width . The foremost open problem in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $$P\in \{0,1\}^{k\times l}$$ P ∈ { 0 , 1 } k × l is acyclic , meaning it is the bipartite incidence matrix of a forest, then $$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ Ex ( P , n ) = O ( n log C P n ) , where $$\operatorname {Ex}(P,n)$$ Ex ( P , n ) is the maximum number of 1s in a P -free $$n\times n$$ n × n 0–1 matrix and $$C_P$$ C P is a constant depending only on P . This conjecture has been confirmed on many small patterns, specifically all P with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ Ex ( S 0 , n ) , Ex ( S 1 , n ) ≥ n 2 Ω ( log n ) , where $$S_0,S_1$$ S 0 , S 1 are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $$(P_t)$$ ( P t ) , specifically that for every $$t\ge 2$$ t ≥ 2 , $$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ Ex ( P t , n ) = Θ ( n ( log n / log log n ) t ) . This is the first proof of an asymptotically sharp bound that is $$\omega (n\log n)$$ ω ( n log n ) .

Abstract Quantitative versions of Helly's and Steinitz' theorems were first introduced by Bárány, Katchalski and Pach in 1982, and have grown into a well-studied field within discrete and convex geometry in the last decade. This note is an invitation to the field in the form of an incomplete collection of open problems.

Abstract Factorization structures occur in toric differential and discrete geometry and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such or as an apex generalization of cyclic polytopes featuring a generalized Gale’s evenness condition. This article presents a comprehensive study of this new concept called factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains a construction of polytopes and cones compatible with a given factorization structure and exemplifies it for the product Segre–Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalized Gale’s evenness condition for compatible cones, polytopes, and their duals and explicitly describes faces of these. Factorization structures naturally provide generalized Vandermonde identities, which relate normals of any compatible polytope, and which are used to find examples of Delzant and rational Delzant polytopes compatible with the Veronese factorization structure. The article offers a myriad of factorization structure examples, which are later characterized to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.

Impossible objects, geometric constructions that humans can perceive but that cannot exist in real life, have been a topic of intrigue in visual arts, perception, and graphics, yet no satisfying computer representation of such objects exists. Previous work embeds impossible objects in 3D, cutting them or twisting/bending them in the depth axis. Cutting an impossible object changes its local geometry at the cut, which can hamper downstream graphics applications, such as smoothing, while bending makes it difficult to relight the object. Both of these can invalidate geometry operations, such as distance computation. As an alternative, we introduce Meschers, meshes capable of representing impossible constructions akin to those found in M.C. Escher's woodcuts. Our representation has a theoretical foundation in discrete exterior calculus and supports the use-cases above, as we demonstrate in a number of example applications. Moreover, because we can do discrete geometry processing on our representation, we can inverse-render impossible objects. We also compare our representation to cut and bend representations of impossible objects.

Catalan numbers, a classical sequence in many combinactorics problems, are defined asCn=1n+1(2nn)and have diverse applications, such as interpreted as Dyck paths. This paper mainly explores the higher-dimensional generalizations of Catalan numbers, including the higher-dimensional Catalan numbersCd(n)=(nd)!i=0d-1i!(n+i)!and Fuss-Catalan numbersCn(s)=1(s-1)n+1(snn). We revisit Zeilberger's reflection principle proof for higher-dimensional Catalan numbers and present a novel combinatorial proof using ordered Catalan sequences. Additionally, we combine Fuss-Catalan numbers with higher-dimensional paths, deriving a product formula for d-dimensional cases. The paper also generalizes the coefficient of Fuss-Catalan numbers to more real numbers and establishes bijections between m-ary trees and polygon division problems, providing their enumeration via Fuss-Catalan numbers. Our results extend the understanding of Catalan-type numbers and their combinatorial interpretations, highlighting connections across multiple mathematical domains.These results enhance our understanding of Catalan-type numbers and construct deep connections across multiple mathematical domains, including algebraic combinatorics, probability, and discrete geometry. The generalizations presented in the article offer new tools for solving complex enumeration problems and provide theoretical foundations for future research in combinatorial mathematics.\\

Soft robots have garnered significant attention due to their promising applications across various domains. A hallmark of these systems is their bilayer structure, where strain mismatch caused by differential expansion between layers induces complex deformations. Despite progress in theoretical modeling and numerical simulation, accurately capturing their dynamic behavior, especially during environmental interactions, remains challenging. This study presents a novel simulation environment based on the discrete elastic rod (DER) model to address the challenge. By leveraging discrete differential geometry, the DER approach offers superior convergence compared to conventional methods like finite element method, particularly in handling contact interactions—an essential aspect of soft robot dynamics in real‐world scenarios. The simulation framework incorporates key features of bilayer structures, including stretching, bending, twisting, and interlayer coupling. This enables the exploration of a wide range of dynamic behaviors for bilayer soft robots, such as gripping, crawling, jumping, and swimming. The insights gained from this work provide a robust foundation for the design and control of advanced bilayer soft robotic systems.

Abstract Brualdi and Li introduced tournament interchange graphs. In such a graph, each vertex represents a tournament. Traversing an edge corresponds to reversing a cyclically directed triangle. Such a triangle is neutral, in that its reversal does not affect the score sequence. An interchange graph encodes the combinatorics of the set of tournaments with a given score sequence, or equivalently, of a given fiber of the classical permutahedron from discrete geometry. Coxeter tournaments were introduced by the first author and Sanchez, in relation to the Coxeter permutahedra in Ardila, Castillo, Eur and Postnikov. Coxeter tournaments have collaborative and solitaire games, in addition to the usual competitive games in classical tournaments. We introduce Coxeter interchange graphs. These graphs are more intricate, as there are multiple neutral structures at play, which interact with one another. Our main result shows that the Coxeter interchange graphs are regular, and we describe the degree geometrically, in terms of distances in the Coxeter permutahedra. We also characterize the set of score sequences of Coxeter tournaments, generalizing a classical result of Landau.

Abstract Flexible elastic structures, such as beams, rods, ribbons, plates, and shells, exhibit complex nonlinear dynamical behaviors that are central to a wide range of engineering and scientific applications, including soft robotics, deployable structures, and biomedical devices. While various numerical methods have been developed to simulate these behaviors, many conventional approaches struggle to simultaneously capture geometric and material nonlinearities, as well as nonlinear external interactions, particularly in highly deformable and dynamically evolving systems. The Discrete Differential Geometry (DDG) method has emerged as a robust and efficient numerical framework that intrinsically preserves geometric properties, accommodates material nonlinearity, and accurately models interactions with external environments and fields. By directly discretizing geometric and mechanical quantities, DDG provides an accurate, stable, and efficient approach to modeling flexible structures, addressing key limitations of traditional numerical methods. This tutorial provides a systematic introduction to the DDG method for simulating nonlinear behaviors in flexible structures. It covers DDG theory, numerical framework, and simulation implementation, with examples spanning dynamic systems, geometric and material nonlinearities, and external interactions like magnetics, fluids and contact, culminating in practical insights and future directions. By offering a comprehensive and practical guide–together with open-source MATLAB code–this tutorial aims to facilitate the broader adoption of DDG-based numerical tools among researchers and engineers in computational mechanics, applied mathematics, and structural design. We seek to enhance the accessibility and applicability of DDG methods, fostering further advancements in the simulation and analysis of highly flexible structures across diverse scientific and engineering domains.

Abstract We obtain many objects of discrete differential geometry as reductions of skew parallelogram nets, a system of lattice equations that may be formulated for any unital associative algebra. The Lax representation is linear in the spectral parameter, and paths in the lattice give rise to polynomial dependencies. We prove that generic polynomials in complex 2 × 2 matrices factorize, implying that skew parallelogram nets encompass all systems with such a polynomial representation. We demonstrate factorization in the context of discrete curves by constructing pairs of Bäcklund transformations that induce Euclidean motions on discrete elastic rods. More generally, we define a hierarchy of discrete curves by requiring such an invariance after an integer number of Bäcklund transformations. Moreover, we provide the factorization explicitly for discrete constant curvature surfaces and reveal that they are slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method, this interpretation constructs such surfaces from given discrete holomorphic maps.

Abstract We derive the one-loop partition function for three-dimensional quantum gravity (QG) in a finite-radius thermal twisted flat space with a conical defect, reproducing the massive BMS3 character. We perform the computation in both discrete and continuum geometry formulations, showing consistency between them. In the discrete case, we integrate out bulk degrees of freedom in a Regge gravity framework, while in the continuum, we construct a dual non-local boundary field theory encoding geodesic length fluctuations. Our study shows that the additional modes of the massive character, compared to the vacuum case, originate from the explicit breaking of radial diffeomorphism symmetry by the defect. This provides a concrete geometric mechanism in Regge gravity, tracing the appearance of massive BMS3 particles to diffeomorphism breaking by conical defects, and highlights the broader relevance of discrete geometry approaches to QG with matter.