Geometric Context Research Articles

A Problem-Solving Approach Aimed at Helping Calculus I Students Solve Related Rates Problems

After analyzing our Calculus I students’ performance on a related rates problem from a final exam, we designed a teaching experiment aimed at improving our students’ performance at problem solving. For our analysis, we used two frameworks – one specific to related rates problems and Polya’s general framework for problem solving. The analysis of student work on a final exam problem (N = 57) and interviews with 10 students revealed difficulties understanding the problem and making connections between data and unknowns. The results informed a teaching experiment the following semester when a different group of students (N = 13) used Polya’s approach to problem solving. The students solved and discussed related rates problems in geometric contexts, wrote their algorithms, and were assessed by a related rates problem on the final exam. All 13 students understood the problem, used diagrams, and all but one, established a meaningful relation with quantities from the problem. Having students create their algorithms seems to be a promising strategy in the teaching and learning of related rates problems.

Perform assessment of COLREGs onboard a maritime autonomous surface ship: Narrow Channels and Traffic Separation Schemes

Abstract The usage of Maritime Autonomous Surface Ships in coastal and heavily used maritime environments necessitates advanced navigation systems capable of fully complying with the International Regulations for Preventing Collisions at Sea 1972 and not only open sea encounters. A significant challenge in this domain is enabling autonomous vessels to understand and navigate complex collision avoidance scenarios dictated by Rule 9 Narrow Channels and Rule 10 Traffic Separation Schemes. These rules are not well covered by research and only partial solutions have been proposed. To address this, we propose a ruleset for a system that can blend the traffic situation and geometric constraints extracted from Electronic Navigational Charts in S-57 format. The proposed ruleset extends our previous work on operationalization of COLREGs by introducing spatial features describing the maritime geometries relevant for traffic situations in Narrow Channels and Traffic Separation Schemes. We apply the ruleset to various traffic situations, showing how the inclusion of geometric context improves decision-making by reducing ambiguity and therefore enables the assessment of traffic situations and ship encounters in the vicinity of Narrow Channels and Traffic Separation Schemes. Our work outlines how the integration of spatial features into a rule-based system enables autonomous navigation capabilities of MASS for safe operations and compliance with maritime regulations.

Upper Bounds of the Third Hankel Determinant for Bi-Univalent Functions in Crescent-Shaped Domains

This paper investigates the third Hankel determinant, denoted H3(1), for functions within the subclass RS∑*(λ) of bi-univalent functions associated with crescent-shaped regions φ⦅z=z+1+z2. The primary aim of this study is to establish upper bounds for H3(1). By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

A survey of Spanish research in mathematics education

AbstractThis survey paper presents recent relevant research in mathematics education produced in Spain, which allows the identification of different broad lines of research developed by Spanish groups of scholars. First, we present and describe studies whose research objectives are related to student learning of specific curricular contents and process-oriented competencies, namely arithmetic, algebra, geometry, functions and calculus, probability and statistics, and argumentation or proof in geometric contexts. Next, we present characteristics and foci of investigations dealing with different aspects of mathematics teacher education, encompassing a large part of Spanish research in mathematics education. The descriptions of other transversal lines of research complement the previous two big blocks: research on students with special educational needs and the effects of using technology in different curricular contents and educational levels. Finally, we report on the research activities and advances of Spanish research in mathematics education from two main theoretical frameworks created or developed by Spanish researchers. This plurality of research strands also corresponds to a wide range of international collaborations, especially with Latin American colleagues.

3D Point Cloud Semantic Segmentation based on Multi-scale Dense Nested Networks

Aiming at the problem that the relationship between geometric features and semantic features is ignored in the point cloud data downsampling process, which leads to inaccurate segmentation of object boundaries and structural details, this paper proposes a 3D point cloud semantic segmentation network based on multi-scale dense nested type. Firstly, a dense nested network architecture is constructed by nesting multiple multi-scale feature fusion modules to fuse multi-scale features of different directions between encoder-decoder paths, so as to effectively propagate local ge-ometric context information and enhance the ability of cross-scale information interaction. Secondly, a local feature aggregation unit is constructed in the multi-scale feature fusion module, which strengthens the structural awareness within the local point set based on graph convolution and attention mechanism, and promotes the complementarity of local geometric features and abstract semantic information. Then, the cross-layer multi-loss supervision module is combined to further optimize the multi-scale feature propagation, which makes the network training more stable and improves the point cloud segmentation accuracy. Finally, this paper verified the proposed network on the S3DIS dataset. The experimental results show that the proposed network has the mean intersection over Union of 71.2% and the overall accuracy of 88.7%, which is 1.2 and 0.7 percentage points higher than RandLA-Net, respectively, which proves that the proposed network can effectively improve the accuracy of 3D point cloud semantic segmentation.

Adaptive Surface Normal Constraint for Geometric Estimation From Monocular Images.

We introduce a novel approach to learn geometries such as depth and surface normal from images while incorporating geometric context. The difficulty of reliably capturing geometric context in existing methods impedes their ability to accurately enforce the consistency between the different geometric properties, thereby leading to a bottleneck of geometric estimation quality. We therefore propose the Adaptive Surface Normal (ASN) constraint, a simple yet efficient method. Our approach extracts geometric context that encodes the geometric variations present in the input image and correlates depth estimation with geometric constraints. By dynamically determining reliable local geometry from randomly sampled candidates, we establish a surface normal constraint, where the validity of these candidates is evaluated using the geometric context. Furthermore, our normal estimation leverages the geometric context to prioritize regions that exhibit significant geometric variations, which makes the predicted normals accurately capture intricate and detailed geometric information. Through the integration of geometric context, our method unifies depth and surface normal estimations within a cohesive framework, which enables the generation of high-quality 3D geometry from images. We validate the superiority of our approach over state-of-the-art methods through extensive evaluations and comparisons on diverse indoor and outdoor datasets, showcasing its efficiency and robustness.

Reduced resonance schemes and Chen ranks

Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

Learning Effective Geometry Representation from Videos for Self-Supervised Monocular Depth Estimation

Recent studies on self-supervised monocular depth estimation have achieved promising results, which are mainly based on the joint optimization of depth and pose estimation via high-level photometric loss. However, how to learn the latent and beneficial task-specific geometry representation from videos is still far from being explored. To tackle this issue, we propose two novel schemes to learn more effective representation from monocular videos: (i) an Inter-task Attention Model (IAM) to learn the geometric correlation representation between the depth and pose learning networks to make structure and motion information mutually beneficial; (ii) a Spatial-Temporal Memory Module (STMM) to exploit long-range geometric context representation among consecutive frames both spatially and temporally. Systematic ablation studies are conducted to demonstrate the effectiveness of each component. Evaluations on KITTI show that our method outperforms current state-of-the-art techniques.

A note on the Hill–Ogden generalised strains

This brief contribution provides an overview of the Hill–Ogden generalised strain tensors, and some considerations on their representation in generalised (curvilinear) coordinates, in a fully covariant formalism that is adaptable to a more general theory on Riemannian manifolds. These strains may be naturally defined with covariant components or naturally defined with contravariant components. Each of these two macro-families is best suited to a specific geometrical context.

RIGA: Rotation-Invariant and Globally-Aware Descriptors for Point Cloud Registration.

Successful point cloud registration relies on accurate correspondences established upon powerful descriptors. However, existing neural descriptors either leverage a rotation-variant backbone whose performance declines under large rotations, or encode local geometry that is less distinctive. To address this issue, we introduce RIGA to learn descriptors that are Rotation-Invariant by design and Globally-Aware. From the Point Pair Features (PPFs) of sparse local regions, rotation-invariant local geometry is encoded into geometric descriptors. Global awareness of 3D structures and geometric context is subsequently incorporated, both in a rotation-invariant fashion. More specifically, 3D structures of the whole frame are first represented by our global PPF signatures, from which structural descriptors are learned to help geometric descriptors sense the 3D world beyond local regions. Geometric context from the whole scene is then globally aggregated into descriptors. Finally, the description of sparse regions is interpolated to dense point descriptors, from which correspondences are extracted for registration. To validate our approach, we conduct extensive experiments on both object- and scene-level data. With large rotations, RIGA surpasses the state-of-the-art methods by a margin of 8 ° in terms of the Relative Rotation Error on ModelNet40 and improves the Feature Matching Recall by at least 5 percentage points on 3DLoMatch.

Genome-scale annotation of protein binding sites via language model and geometric deep learning.

Revealing protein binding sites with other molecules, such as nucleic acids, peptides, or small ligands, sheds light on disease mechanism elucidation and novel drug design. With the explosive growth of proteins in sequence databases, how to accurately and efficiently identify these binding sites from sequences becomes essential. However, current methods mostly rely on expensive multiple sequence alignments or experimental protein structures, limiting their genome-scale applications. Besides, these methods haven't fully explored the geometry of the protein structures. Here, we propose GPSite, a multi-task network for simultaneously predicting binding residues of DNA, RNA, peptide, protein, ATP, HEM, and metal ions on proteins. GPSite was trained on informative sequence embeddings and predicted structures from protein language models, while comprehensively extracting residual and relational geometric contexts in an end-to-end manner. Experiments demonstrate that GPSite substantially surpasses state-of-the-art sequence-based and structure-based approaches on various benchmark datasets, even when the structures are not well-predicted. The low computational cost of GPSite enables rapid genome-scale binding residue annotations for over 568,000 sequences, providing opportunities to unveil unexplored associations of binding sites with molecular functions, biological processes, and genetic variants. The GPSite webserver and annotation database can be freely accessed at https://bio-web1.nscc-gz.cn/app/GPSite.

Genome-scale annotation of protein binding sites via language model and geometric deep learning

Revealing protein binding sites with other molecules, such as nucleic acids, peptides, or small ligands, sheds light on disease mechanism elucidation and novel drug design. With the explosive growth of proteins in sequence databases, how to accurately and efficiently identify these binding sites from sequences becomes essential. However, current methods mostly rely on expensive multiple sequence alignments or experimental protein structures, limiting their genome-scale applications. Besides, these methods haven’t fully explored the geometry of the protein structures. Here, we propose GPSite, a multi-task network for simultaneously predicting binding residues of DNA, RNA, peptide, protein, ATP, HEM, and metal ions on proteins. GPSite was trained on informative sequence embeddings and predicted structures from protein language models, while comprehensively extracting residual and relational geometric contexts in an end-to-end manner. Experiments demonstrate that GPSite substantially surpasses state-of-the-art sequence-based and structure-based approaches on various benchmark datasets, even when the structures are not well-predicted. The low computational cost of GPSite enables rapid genome-scale binding residue annotations for over 568,000 sequences, providing opportunities to unveil unexplored associations of binding sites with molecular functions, biological processes, and genetic variants. The GPSite webserver and annotation database can be freely accessed at https://bio-web1.nscc-gz.cn/app/GPSite.

Characterizing sub-glacial hydrology using radar simulations

Abstract. The structure and distribution of sub-glacial water directly influences Antarctic ice mass loss by reducing or enhancing basal shear stress and accelerating grounding line retreat. A common technique for detecting sub-glacial water involves analyzing the spatial variation in reflectivity from an airborne radar echo sounding (RES) survey. Basic RES analysis exploits the high dielectric contrast between water and most other substrate materials, where a reflectivity increase ≥ 15 dB is frequently correlated with the presence of sub-glacial water. There are surprisingly few additional tools to further characterize the size, shape, or extent of hydrological systems beneath large ice masses. We adapted an existing radar backscattering simulator to model RES reflections from sub-glacial water structures using the University of Texas Institute for Geophysics (UTIG) Multifrequency Airborne Radar Sounder with Full-phase Assessment (MARFA) instrument. Our series of hypothetical simulation cases modeled water structures from 5 to 50 m wide, surrounded by bed materials of varying roughness. We compared the relative reflectivity from rounded Röthlisberger channels and specular flat canals, showing both types of channels exhibit a positive correlation between size and reflectivity. Large (> 20 m), flat canals can increase reflectivity by more than 20 dB, while equivalent Röthlisberger channels show only modest reflectivity gains of 8–13 dB. Changes in substrate roughness may also alter observed reflectivity by 3–6 dB. All of these results indicate that a sophisticated approach to RES interpretation can be useful in constraining the size and shape of sub-glacial water features. However, a highly nuanced treatment of the geometric context is necessary. Finally, we compared simulated outputs to actual reflectivity from a single RES flight line collected over Thwaites Glacier in 2022. The flight line crosses a previously proposed Röthlisberger channel route, with an obvious bright bed reflection in the radargram. Through multiple simulations comparing various water system geometries, such as canals and sub-glacial lakes, we demonstrated the important role that topography and water geometry can play in observed RES reflectivity. From the scenarios that we tested, we concluded the bright reflector from our RES flight line cannot be a Röthlisberger channel but could be consistent with a series of flat canals or a sub-glacial lake. However, we note our simulations were not exhaustive of all possible sub-glacial water configurations. The approach outlined here has broad applicability for studying the basal environment of large glaciers. We expect to apply this technique when constraining the geometry and extent of many sub-glacial hydrologic structures in the future. Further research may also include comprehensive investigations of the impact of sub-glacial roughness, substrate heterogeneity, and computational efficiencies enabling more complex and complete simulations.

Quantitative rates of convergence to equilibrium for the degenerate linear Boltzmann equation on the torus

AbstractWe study the linear relaxation Boltzmann equation on the torus with a spatially varying jump rate which can be zero on large sections of the domain. In Bernard and Salvarani (Arch. Ration. Mech. Anal. 208 (2013), no. 3, 977–984), Bernard and Salvarani showed that this equation converges exponentially fast to equilibrium if and only if the jump rate satisfies the geometric control condition of Bardos, Lebeau and Rauch. Han‐Kwan and Léautaud showed a more general result for linear Boltzmann equations under the action of potentials in different geometric contexts, including the case of unbounded velocities. In this paper, we obtain quantitative rates of convergence to equilibrium when the geometric control condition is satisfied, using a probabilistic approach based on Doeblin's theorem from Markov chains.

PU-Flow: a Point Cloud Upsampling Network with Normalizing Flows.

Point cloud upsampling aims to generate dense point clouds from given sparse ones, which is a challenging task due to the irregular and unordered nature of point sets. To address this issue, we present a novel deep learning-based model, called PU-Flow, which incorporates normalizing flows and weight prediction techniques to produce dense points uniformly distributed on the underlying surface. Specifically, we exploit the invertible characteristics of normalizing flows to transform points between Euclidean and latent spaces and formulate the upsampling process as ensemble of neighbouring points in a latent space, where the ensemble weights are adaptively learned from local geometric context. Extensive experiments show that our method is competitive and, in most test cases, it outperforms state-of-the-art methods in terms of reconstruction quality, proximity-to-surface accuracy, and computation efficiency. The source code will be publicly available at https://github.com/unknownue/puflow.

Flag matroids with coefficients

This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebro-geometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian.

Nuclear Shape-Phase Transitions and the Sextic Oscillator

This review delves into the utilization of a sextic oscillator within the β degree of freedom of the Bohr Hamiltonian to elucidate critical-point solutions in nuclei, with a specific emphasis on the critical point associated with the β shape variable, governing transitions from spherical to deformed nuclei. To commence, an overview is presented for critical-point solutions E(5), X(5), X(3), Z(5), and Z(4). These symmetries, encapsulated in simple models, all model the β degree of freedom using an infinite square-well (ISW) potential. They are particularly useful for dissecting phase transitions from spherical to deformed nuclear shapes. The distinguishing factor among these models lies in their treatment of the γ degree of freedom. These models are rooted in a geometrical context, employing the Bohr Hamiltonian. The review then continues with the analysis of the same critical solutions but with the adoption of a sextic potential in place of the ISW potential within the β degree of freedom. The sextic oscillator, being quasi-exactly solvable (QES), allows for the derivation of exact solutions for the lower part of the energy spectrum. The outcomes of this analysis are examined in detail. Additionally, various versions of the sextic potential, while not exactly solvable, can still be tackled numerically, offering a means to establish benchmarks for criticality in the transitional path from spherical to deformed shapes. This review extends its scope to encompass related papers published in the field in the past 20 years, contributing to a comprehensive understanding of critical-point symmetries in nuclear physics. To facilitate this understanding, a map depicting the different regions of the nuclide chart where these models have been applied is provided, serving as a concise summary of their applications and implications in the realm of nuclear structure.

Pressure at infinity and strong positive recurrence in negative curvature

In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincaré series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said to be strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show, in particular, that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.

CoupleUNet: Swin Transformer coupling CNNs makes strong contextual encoders for VHR image road extraction

ABSTRACT Accurately segmenting roads is challenging due to substantial intra-class variations, indistinct inter-class distinctions, and occlusions caused by shadows, trees, and buildings. To address these challenges, it is crucial to simultaneously pay attention to important texture details and perceive global geometric context information. Recent research has shown that CNN-Transformer hybrid structures outperform using CNN or Transformer alone. CNN excels at extracting local detail features and Transformer naturally has the ability to perceive global context information. In this paper, we propose a dual-branch network module called ConSwin, which combines the advantages of ResNet and Swin Transformer for better road information extraction. Taking ConSwin as the basic encoding module, we build an hourglass-shaped network with an encoder-decoder structure. In order to better transfer texture and structural detail information between the encoder and decoder, we also design two novel connection blocks. We conduct comparative experiments on the Massachusetts and CHN6-CUG datasets, and the proposed method outperforms state-of-the-art methods in terms of overall accuracy, IOU, and F1 metrics. Other experiments verify the effectiveness of our method, while visualization results demonstrate its ability to obtain better road representations.

Geometric interpretation of risk and prevention, by implementation of the “Level of Preventive Action” risk assessment method

Risk assessment methodologies are determining tools in risk prevention for construction works. However, frequently the evaluation is presented in a linguistic context and may involve interpretation difficulties in multicultural contexts. This research aims to visually show the evaluation, in a geometric context, the risk conditions and the necessary prevention level. The methodology arises from the theoretical-practical model of occupational risk assessment, adapted to construction works called “Level of Preventive Action”. It is based on the preventive environments’ observation: initial, documentary, constructive, and social environments (through technical observation and psychosocial survey). Implementing and developing this new methodology by resourcing a digital tool is essential for integrating BIM approach procedures. The Level of Preventive Action procedures shows the existing risk in the different preventive environments, from its so-called Characteristic Value and its interaction with the assessed risk (in the scope of Safety at Work, Industrial Hygiene, Ergonomics and Psychosociology). Finally, the existing risks and the level of preventive actions required according to the workers’ behaviors and emotional states are shown. The Level of Preventive Action geometric interpretation is qualitative, quantitative, visual, and dynamic. It allows interaction with the worker in real-time, with electronic devices, joining technology to human behavior.