Geometric Approach Research Articles

Blank predetermination in the Iberian Acheulean. Insight from the cleaver on flake assemblage of Casal do Azemel site (Leiria, Portugal) by a Geometric Morphometric approach

Over the last decades, the increase of data available for the study of the archaeological topic in the Iberian Peninsula during the Middle Pleistocene has favoured the understanding of the technological trends of the Iberian Acheulean assemblages. These have features of a Large Flake Acheulean (LFA), displaying, among other traits, a significant presence of cleavers on flake, a specific tool type that is of great cultural and technological value. Particularly, these artefacts are privileged to discuss the importance of blank predetermination in the Acheulean techno-complex. Following this reason, in the present work we aimed to explore this topic through the 2D Geometric Morphometric Analysis of the cleaver on flake assemblage from Casal do Azemel (Leiria, Portugal), an example of a paradigmatic Iberian Acheulean site that has one of the largest collections of this type of tools in Western Europe.
 The results obtained revealed that no significant morphological differences were found according to the technological solutions applied to the acquisition of the blank and its secondary transformation. Considering that in most of the cases these tools display a low degree of secondary transformation, these data suggest that underlying the production of Casal do Azemel’s cleaver on flake assemblage was not only a technological and cognitive flexibility (given its typological composition), but also a conceptual, structural, and morphological standardisation. These observations allowed us to discuss the significance of blank predetermination in the Acheulean, implying the existence of greatly structured technical and cognitive prerequisites.

A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group

As service to the community, we provide – for Euclidean space – a basic treatment of locally rectifiable chains and of the complex of locally integral chains. In this setting, we may beneficially develop the idea of a complete normed commutative group bundle over the Grassmann manifold whose fibre is the coefficient group of the chains. Our exposition also sheds new light on some algebraic aspects of the theory. Finally, we indicate an extension to a geometric approach to locally flat chains centring on locally rectifiable chains rather than completion procedures.

Quantifying the temporal pattern of shape change in trigoniids through the Middle Jurassic–Early Cretaceous of Kutch: A geometric morphometric approach

Quantifying the temporal pattern of shape change in trigoniids through the Middle Jurassic–Early Cretaceous of Kutch: A geometric morphometric approach

Geometric morphometric approach to dental health in an 18th‐ to 19th‐century English infirmary

AbstractThis study evaluated correlations between poor dental health and craniomandibular morphology for the Radcliffe Infirmary population buried between 1770 and 1855. The Radcliffe Infirmary is situated in Oxford, 90 km northwest of London, in what used to be a rural area. The original report for this population observed high prevalence rates of dental caries, antemortem tooth loss (AMTL), and periodontal disease. Industrial societies are acknowledged to have highly processed foods that not only are detrimental to dental health but also require less mastication. This softening of diet is suggested to be the leading influence on the modern morphology of the skull. By using two‐dimensional geometric morphometrics (GM) and linear measurements of the skull, the Radcliffe population was evaluated for the effects of poor dental health and decreased wear on the masticatory structures. High frequencies of AMTL, periodontal disease, and caries and low amounts of dental wear were observed in this sample population. Caries and AMTL showed weak correlations with skull shape and craniomandibular dimensions according to the results of the morphometric analysis and Kendall's tau. Dental wear correlated with a wide bizygomatic breadth and a long narrow skull shape for the males only. Periodontal disease was the only dental health variable not correlated with shape or size. This study confirmed that the Radcliffe population ate a cariogenic diet and suggests that dental wear and pathology weakly correlates with craniomandibular shape and size. This unpredicted finding may be due to the limitations of two‐dimensional shape analysis, a small sample size, and the sampling strategy. Future research is highly encouraged to better understand the consequences of poor dental health for skull morphology.

Analysis and Visualization of Inter-/Intra-surgeon Variability Using Surgical Simulation for Partial Anomalous Pulmonary Venous Connection Repair.

There may be inter-/intra-surgeon variability in the repair of congenital heart defects. The objective was to analyze and visualize inter-/intra-surgeon variability in the level of patch suture lines and in the shape and size of patches developed through surgical simulation for partial anomalous pulmonary venous connection (PAPVC) repair using a patient-specific 3D-printed heart model. A patient with PAPVC and preoperative computed tomography data were selected, and a patient-specific heart model was obtained. Two different exposures on the model were tested. Two surgeons were enrolled. Both surgeons performed a single-patch repair on four heart models with one exposure and four models with the other. On the potential suture line, 20 points common in each model were allocated. The level of patch suture lines was represented as a deviation from the 20 points. The shape and size of the patches were analyzed and visualized using geometric morphometrics approaches, using the 20 points as landmarks to represent the patches. There was inter-surgeon variability in the level of patch suture lines, and the variability was higher in particular locations. There was inter-surgeon variability in the shape and size of patches, which was not affected by the exposures. The inter-surgeon variability in the shape of patches was higher in particular locations. There was intra-surgeon variability in the shape of patches. Inter-/intra-surgeon variability was analyzed and visualized by the surgical simulation using geometric morphometrics approaches. Further studies are needed to scrutinize whether the variability affects postoperative hemodynamics in actual surgery.

Unifying domain gap in Federated Learning: A geometric approach

Federated Learning (FL) is a framework designed to enable distributed and privacy-preserving learning. It typically involves a central server coordinating multiple clients to collaboratively train a global model without the need for sharing private data. Recent studies have revealed that this parameter-only communication strategy can be hindered by non-independent and identically distributed (non-iid) data across clients, leading to statistical heterogeneity. This issue necessitates the development of Personalized Federated Learning (PFL), which aims to enhance the performance of individual clients while still allowing for the learning of a global model. Most existing work on PFL has focused on label shift in supervised tasks, with relatively little attention given to feature shift, another common problem in real-world applications of federated learning. For instance, client data may vary in terms of illumination, angles, and quality for natural images, or in imaging protocols for medical images. To address this issue, we propose a general yet powerful framework that leverages the geometric interpretation of deep neural networks. Geometrically, a deep neural network partitions the feature space into a Power Diagram (PD) or Voronoi Diagram (VD). Regularized losses are introduced to mitigate feature shift from the perspective of PD and VD. Additionally, a personalized generator network is formulated for each client to rectify feature distribution and adapt the input data from its own domain to an ”averaged” or unified domain that generalizes well across all clients. Our method demonstrates competitive performance compared to current state-of-the-art methods.

On the Kostant-Souriau prequantization of scalar fields with polysymplectic structures

In this paper, I present a novel, purely differential geometric approach to the quantization of scalar fields, with a special focus on the familiar case of Minkowski spacetimes. This approach is based on using the natural geometric structures of polysymplectic Hamiltonian field theory to produce an analog of the Kostant-Souriau prequantization map familiar from geometric quantization. I show that while the resulting operators are quite different from those of canonical quantum field theory, the approach is nonetheless able to reproduce a few of canonical quantum field theory's most fundamental results. I finish by elaborating the current limitations of this approach and briefly discussing future prospects.

The geometry of multi-curve interest rate models

We study the problems of consistency and the existence of finite-dimensional realizations for multi-curve interest rate models of Heath–Jarrow–Morton type, generalizing the geometric approach developed by T. Björk and co-authors for the classical single-curve setting. We characterize when a multi-curve interest rate model is consistent with a given parameterized family of forward curves and spreads and when a model can be realized by a finite-dimensional state process. We illustrate the general theory in a number of model classes and examples, providing explicit constructions of finite-dimensional realizations. Based on these theoretical results, we perform the calibration of a three-curve Hull–White model to market data and analyse the stability of the estimated parameters.

Bound-preserving OEDG schemes for Aw–Rascle–Zhang traffic models on networks

Physical solutions to the widely used Aw–Rascle–Zhang (ARZ) traffic model and the adapted pressure ARZ model should satisfy the positivity of density, the minimum and maximum principles with respect to the velocity v and other Riemann invariants. Many numerical schemes suffer from instabilities caused by violating these bounds, and the only existing bound-preserving (BP) numerical scheme (for ARZ model) is random, only first-order accurate, and not strictly conservative. This paper introduces arbitrarily high-order provably BP discontinuous Galerkin (DG) schemes for these two models, preserving all the aforementioned bounds except the maximum principle of v, which has been rigorously proven to conflict with the consistency and conservation of numerical schemes. Although the maximum principle of v is not directly enforced, we find that the strictly preserved maximum principle of another Riemann invariant w actually enforces an alternative upper bound on v. At the core of this work, analyzing and rigorously proving the BP property is a particularly nontrivial task: the Lax–Friedrichs (LF) splitting property, usually expected for hyperbolic conservation laws and employed to construct BP schemes, does not hold for these two models. To overcome this challenge, we formulate a generalized version of the LF splitting property, and prove it via the geometric quasilinearization approach (Wu and Shu, 2023 [47]). To suppress spurious oscillations in the DG solutions, we incorporate the oscillation-eliminating technique, recently proposed in (Peng et al., 2024 [34]), which is based on the solution operator of a novel damping equation. Several numerical examples are included to demonstrate the effectiveness, accuracy, and BP properties of our schemes, with applications to traffic simulations on road networks.

The Bayesian central limit theorem for exponential family distributions: a geometric approach

The Bayesian central limit theorem for exponential family distributions: a geometric approach

Studying the accuracy of geometrized models of ribbon electron beams

Using a set of standard exact solutions described by ordinary differential equations and elementary functions, geometrized models of plane electron beams inl-, andW-representations were studied. A comparison is made of the capabilities of the geometrized approach and the paraxial theory.

Circle-circle intersection. A universal method for solving typical surveying problems

AbstractThe research problem of the article is to devise a universal mathematical procedure for calculating point coordinates from typical planar surveying measurements. The proposed solution involves calculating the intersection points of two circles with radii equal to the measured distances (the distance-distance intersection problem). The author demonstrates a straightforward method for reducing every typical surveying problem to the distance-distance intersection form. The procedure also verifies the accuracy of the calculated coordinates. The derived equations were tested numerically using practical examples. The devised procedure will be integrated into an exhaustive numerical algorithm for diverse surveying problems regardless of the geometric approach during measurements.

Knot data analysis using multiscale Gauss link integral

In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein-ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area-knot deep learning-in data science.

LMVSegRNN and Poseidon3D: Addressing Challenging Teeth Segmentation Cases in 3D Dental Surface Orthodontic Scans.

The segmentation of teeth in 3D dental scans is difficult due to variations in teeth shapes, misalignments, occlusions, or the present dental appliances. Existing methods consistently adhere to geometric representations, omitting the perceptual aspects of the inputs. In addition, current works often lack evaluation on anatomically complex cases due to the unavailability of such datasets. We present a projection-based approach towards accurate teeth segmentation that operates in a detect-and-segment manner locally on each tooth in a multi-view fashion. Information is spatially correlated via recurrent units. We show that a projection-based framework can precisely segment teeth in cases with anatomical anomalies with negligible information loss. It outperforms point-based, edge-based, and Graph Cut-based geometric approaches, achieving an average weighted IoU score of 0.97122±0.038 and a Hausdorff distance at 95 percentile of 0.49012±0.571 mm. We also release Poseidon's Teeth 3D (Poseidon3D), a novel dataset of real orthodontic cases with various dental anomalies like teeth crowding and missing teeth.

Higher-dimensional geometrical approach for the characterization of two-dimensional square-triangle-rhombus tilings

Higher-dimensional geometrical approach for the characterization of two-dimensional square-triangle-rhombus tilings

A Geometric Approach to Triangular-Square Numbers

Summary A triangular-square number is a number that is both triangular and square. We demonstrate how to generate all triangular-square numbers using only elementary geometric ideas. We also show how this method relates to the Pell equations x 2 − 2 y 2 = ± 1 .

Quantum geometric approach: Nature and characteristics of emerged quantum-conditioned spacetime curvatures on three-sphere

General relativity (GR) in its current classical formulation is fundamentally different from quantum mechanics (QM). We argue that the everlasting battle for exploring and understanding the Universe and then privileging a consistent perception of reality is therefore awaiting a consolidator rather than a conqueror. This should be capable of either unifying these two different benchmarks or at least bringing one closer to another! The latter conservatively describes the consolidating quantum geometric approach, which combines generalization of QM to relativistic energies and gravitational fields and continuous Riemann to discretized Finsler geometry. We found that this type of quantum geometric approach seems to unveil additional quantum-conditioned spacetime curvatures (QCC), sources of gravitation), which obviously could not be disclosed in Einstein’s GR, especially at large scales. This study introduces analytical and numerical analyses of QCC. We conclude that (i) nature and characteristics of QCC are fundamentally distinguishable from the classical GR’s spacetime curvatures, (ii) the QCC are intrinsic, real and essential, i.e. not artifacts that would be removed in a certain coordinate transformation and (iii) the magnitude of QCC are nonnegligible to be underestimated. We also conclude that the spacetime at quantum scales seems to be no longer smooth or continuous so that the proposed quantum geometric approach would be regarded as a novel mathematical framework for the emergence of quantum gravity. Last but not least, a maximal proper force is predicted as a new physical constant, which is responsible for the maximal and gravitational acceleration of a quantum particle that would live in the emerged spacetime curvatures. Thereby, the quantum geometric nature of QCC can be accessed and assessed.

On the non-uniqueness of linear Markov perfect equilibria in linear-quadratic differential games: a geometric approach

On the non-uniqueness of linear Markov perfect equilibria in linear-quadratic differential games: a geometric approach

Image enhancement for dichromatic vision by geometric approach in RGB color space

AbstractThe goal of this study is to process digital images to facilitate color discrimination in dichromatic vision without changing their appearances in trichromatic vision as much as possible. Therefore, we propose a simple and effective image enhancement method using the geometric properties of the RGB color space. Concretely, a novel image representation that clarifies the relationship between dichromatic and trichromatic visions in the RGB color space, and an image enhancement method based on this representation are presented. The effectiveness of the proposed method is verified by experiments using several digital images.

Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.