Let H g denote the coarse moduli space of smooth hyperelliptic curves of genus g in characteristic p ≥ 3 , and let H g f denote the p -rank f stratum of H g for 0 ≤ f ≤ g . Achter and Pries note that determining the number of irreducible components of H g f would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various p -rank strata. Our strategy involves sampling curves over finite fields and calculating their p -ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera g > 1 . The data also leads us to conjecture that the moduli space H g g − 2 is irreducible and suggests that H g f is irreducible for all 1 ≤ f ≤ g . We conclude with a brief discussion on H g 0 .

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various parameters of random graphs. In this paper, we investigate whether this normal limit behavior extends to the topological invariants of geometric spaces. We show that the Betti numbers of the moduli space of rational curves with [Formula: see text] marked points [Formula: see text] and the Fulton-MacPherson configuration space [Formula: see text] are asymptotically normally distributed. Based on numerical evidence and established log-concavity, we conjecture that the Betti numbers of the quotients of these spaces by the symmetric group [Formula: see text] are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.

We prove that the geodesic flow on a geometrically finite locally symmetric space of negative curvature is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The approach is based on constructing a suitable anisotropic Banach space on which the infinitesimal generator of the flow admits an essential spectral gap. A key step in the proof involves estimating certain oscillatory integrals against the Patterson-Sullivan measure. For this purpose, we prove a general result of independent interest asserting that Fourier transforms of measures on R d \mathbb {R}^d , which do not concentrate near proper affine subspaces, enjoy polynomial decay outside of a sparse set of frequencies. As an intermediate step, we show that the L q L^q -dimension ( 1 > q ≤ ∞ 1>q\leq \infty ) of iterated self-convolutions of such measures tend towards that of the ambient space. Our analysis also yields polynomial bounds on the Patterson-Sullivan mass of neighborhoods of certain proper subvarieties of the boundary at infinity which are saturated along the vertical foliation.

The real locus of the moduli space of stable genus zero curves with marked points, \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) , is known to be a smooth manifold and is the Eilenberg–MacLane spaces for the so-called pure cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) . In particular, we show that the operad \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) is not formal. As an application of these operadic constructions, we prove that for each n , the cohomology ring H^{\bullet}(\overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}});{\mathbb{Q}}) is a Koszul algebra, and that the manifold \overline{{\mathcal{M}}}_{0,{n+1}}({\mathbb{R}}) is not formal for n\geq 6 but is a rational K(\pi,1) -space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure cactus groups.

We prove that the existence of a divisor of degree 3 and Baker-Norine rank at least 1 on a 3 -edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree 3 from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of 3 -edge connected tropical trigonal covers and of 3 -edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of 3 -edge connected genus g tropical trigonal curves has the same dimension as the moduli space of genus g algebraic trigonal curves.

Abstract We investigate(pseudo-)Riemannian spaces whose curvature tensor possesses a specific structure called split curvature. First, we provide a direct and simplified proof that every space of split curvature is semisymmetric. This result refines previous approaches by eliminating unnecessary assumptions and clarifying inaccuracies in earlier classifications. We then propose a natural classification of these spaces according to the rank of the Ricci tensor, distinguishing three geometric types. The results of Sinyukov and Mikeš on geodesic mappings of semisymmetric spaces and on equidistant spaces are employed here. Furthermore, we analyze geodesic mappings of spaces of split curvature and demonstrate the existence of spaces that admit nontrivial geodesic mappings while having non-constant scalar curvature. This disproves a previously published theorem by Kiosak asserting that the scalar curvature must be constant in this context. By constructing explicit examples, we provide a more precise and transparent description of the geometric structure of spaces of split curvature, including their classification, mapping properties, and curvature characteristics. Special attention is given to the role of concircular vector fields in equidistant spaces, which yields new insights into their structure and mapping behavior.

Purpose of research . The widespread adoption of dynamic signatures in various biometric technology applicationssupported by clearly defined legal procedures in many countries-drives significant attention toward the reliability of corresponding biometric authentication algorithms. While dynamic signatures are partially free from the drawbacks inherent in static signatures, the problem of authentication reliability remains critical due to the complex interplay of heterogeneous factors. Therefore, the aim of this study is to improve the reliability of user authentication based on the dynamic signature using experimental structural and parametric synthesis of problem-oriented neural networks and comparison with classical detection-discrimination algorithms for multidimensional signals. Methods . The proposed method involves comprehensive identification of the user's dynamic signature in the sample space of multidimensional curves by means of parallel recognition of curve fragments using multiple detectors/classifiers, followed by integration and analysis of the results. Results . Neural network algorithms for identifying the user’s dynamic signature in the sample space of multidimensional curves were experimentally studied and compared with optimal detection-discrimination algorithms for multidimensional signals. The experiments demonstrated that 3–5 key parameters-including two stylus coordinates on the tablet plane, screen pressure, and stylus velocity vectors-ensure acceptable identification reliability in the range of 0,8 to 0,95 for a small number of users, and maintain a reliability level of about 0,7 with unlimited user scaling. The average gain in accuracy from using the developed models and algorithms, compared to statistical methods, amounted to 25– 35%, and compared to metric methods, 5–15%. Conclusion . To achieve the required reliability of user authentication, hardware-software identification models for dynamic signatures should be decomposed into groups with a limited number of users. There exists an optimal combination of algorithms that delivers maximum accuracy in result integration: Euclidean metric, correlation-based, and neural network classifiers.

Abstract We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves with rational components in terms of the combinatorics of the corresponding graphs.

Local Rigidity of Convex Hypersurfaces in Spaces of Constant Curvature

We construct, for all g\ge 2 and n\ge 0 , a spectral sequence of rational S_{n} -representations which computes the S_{n} -equivariant rational cohomology of the tropical moduli spaces of curves \Delta_{g,n} in terms of compactly supported cohomology groups of configuration spaces of n points on graphs of genus g . Using the canonical S_{n} -equivariant isomorphisms \widetilde{H}^{i-1}(\Delta_{g,n};\mathbb{Q}) \cong W_{0} H^{i}_{c}(\mathcal{M}_{g,n};\mathbb{Q}) , we calculate the weight 0 , compactly supported rational cohomology of the moduli spaces \mathcal{M}_{g,n} in the range g=3 and n\le 9 , with partial computations available for n\le 13 .

In this paper, we examine geometric relationships between metric spaces with curvature bounded below and their corresponding model spaces of constant curvature. Let \(\gamma\) be a closed spherical curve in a metric space whose curvature is bounded below by \(K\), lying at a distance \(r < \tfrac{\pi}{2\sqrt{K}}\) from a point. Let \(\gamma^{\prime}\) denote the circle of radius \(r\) centered at the corresponding point in the model space of constant curvature \(K\). Under suitable geometric equivalence conditions—namely, the preservation of pairwise distances between corresponding points of \(\gamma\) and \(\gamma^{\prime}\), the isometry of convex hulls of corresponding geodesic triangles, and the equality of arc lengths or total curvature—we show that the geodesic surface enclosed by \(\gamma\) is isometric to the region bounded by \(\gamma^{\prime}\). This result offers a foundational geometric characterization of metric spaces with curvature bounded below through their model counterparts and provides a framework for further study of total curvature, convexity, and isometric embeddings in such spaces.

In this article, we derive a unified inversion formula for the k-dimensional totally geodesic Radon transform on simply connected spaces of constant curvature, namely R n , H n , and S n . The unification of the formulae is motivated from the observation that the geometries of constant curvature spaces are interlinked. To obtain the unified inversion formula, we use wavelet-like transforms.

Abstract In this paper we prove the unirationality of the locus of bielliptic curves in the Hilbert scheme of canonical curves of genus $$g \ge 11$$ . As a consequence, we obtain another proof for the unirationality of the bielliptic locus in the moduli space of curves of genus $$g \ge 11$$ .

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a conjectural characterization of weighted projective spaces as Fano orbifolds in terms of orbifold rational curves, which would imply that the equality holds only for smooth points.

The online version contains supplementary material available at 10.1007/s00332-026-10266-8.

In their homonymous article, Sam Payne and Thomas Willwacher construct a combinatorial graph complex to compute the weight 11 part of the compactly supported cohomology of the moduli space of curves $\cM_{g,n}$ and compute explicitly the cohomology of the introduced graph complexes in cases of complexes of excess zero, one, two, and three. In this paper, we extend the computation the cohomology to excess four graph complexes. Along the way, we give more details on generators, the computation process, and the definition of the graph complex.

Scissor mechanisms for non-constant curvature space reflectors: Design and geometric compatibility

This study aims to evaluate the effects of percutaneous large-channel spinal endoscopic decompression on stress response, lumbar stability, and disability index in elderly patients with single-segment degenerative lumbar spinal stenosis (LSS). A retrospective analysis was performed on 120 elderly patients with single-segment degenerative LSS treated from January 2020 to January 2024. Fifty-seven underwent percutaneous transforaminal endoscopic discectomy group, and 63 underwent percutaneous large-channel spinal endoscopic decompression (large-channel group). Surgical indicators, complications, stress response, lumbar and leg pain, lumbar function [Japanese Orthopaedic Association (JOA) score, Oswestry Disability Index (ODI) score], and lumbar stability were compared. The large-channel group had shorter operative time and fewer intraoperative fluoroscopy sessions than the percutaneous transforaminal endoscopic discectomy group (P<.05), while intraoperative blood loss and incision length were greater (P<.05). Hospital stay and complication rates showed no difference (P>.05). At 3 days postoperatively, serum NE, DA, and 5-HT levels were elevated in both groups (P<.05) but were lower in the large-channel group (P<.05). Lumbar and leg visual analogue scale scores at 1 week, 3 months, and 6 months were lower than preoperative values in both groups (P<.05), with greater improvement in the large-channel group at 3 and 6 months (P<.05). At 1, 3, and 6 months, JOA scores increased and ODI scores decreased in both groups (P<.05). At 3 and 6 months, JOA scores were higher and ODI scores lower in the large-channel group (P<.05). At 6 months, lumbar curvature, lordosis angle, pelvic tilt, and intervertebral space height improved in both groups (P<.05), with greater gains in the large-channel group (P<.05). Percutaneous large-channel spinal endoscopic decompression for elderly patients with single-segment degenerative LSS can shorten operative time, alleviate stenosis and stress response, improve function, and enhance lumbar stability. Its definite efficacy supports clinical application.

Context. The article addresses the problem of image similarity assessment based on the Fréchet distance metric and its modifications. In this context, images are approximated by polygonal curves. The problem arises from the need to quantitatively evaluate image similarity for tasks such as image generation, clustering, and recognition. Quantitative assessment of the proximity of biomedical images supports decision-making in automated diagnostic systems. The object of the study is the process of image similarity evaluation. The subject of the study is the Fréchet distance metric and its modifications.Objective. To develop a method for determining the fuzzy discrete Fréchet distance, to evaluate the computational complexity of the proposed method, to implement the algorithm for determining the fuzzy discrete Fréchet distance in software, and to conduct computational experiments to evaluate the fuzzy discrete Fréchet distance between polygons.Method. The article presents a method for determining the fuzzy discrete Fréchet distance based on the fuzzy Fréchet metric between polygonal curves. The fuzzy Fréchet metric is grounded in the classical Fréchet distance defined on the space of parameterized curves. The required approximation for practical applications is achieved through the discretization of the fuzzy Fréchet metric. The developed method estimates the fuzzy discrete Fréchet distance between polygonal curves by adapting the algorithm for computing the classical discrete Fréchet distance.Results. The computer experiments were conducted on a set of predefined regions approximated by polygonal curves. Based on the proposed method, an algorithm was developed to evaluate the discrete fuzzy Fréchet distance. The developed algorithm exhibits low computational complexity, equal to the product of the discretized segments of the polygonal curves: O(Cm·n). This enables the estimation of the discrete Fréchet distance with a specified similarity threshold. The software implementation of the method is intended to be integrated into an automatic medical diagnostic system.Conclusions. The results obtained in the study allow recommending the developed method for evaluating image similarity based on the fuzzy discrete Fréchet distance for broad application in computer vision systems, including image generation, clustering, and recognition.

We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape spaces, such as the Kendall shape space, as well as elastic spaces of discrete curves and surfaces. The shape module further implements the abstract mathematical structures of group actions, fiber bundles, quotient spaces and associated Riemannian metrics which allow users to build their own shape spaces. The Riemannian geometry tools enable users to compare, average, interpolate between shapes inside a given shape space. These essential operations can then be leveraged to perform statistics and machine learning on shape data. We present the object-oriented implementation of the shape module along with illustrative examples and show how it can be used to perform statistics and machine learning on shape spaces.