Geometric Representation Research Articles

Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏

Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated. Novelty: From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). Keywords: Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions

Building Change Detection Network Based on Multilevel Geometric Representation Optimization Using Frame Fields

To address the challenges of accurately segmenting irregular building boundaries in complex urban environments faced by existing remote sensing change detection methods, this paper proposes a building change detection network based on multilevel geometric representation optimization using frame fields called BuildingCDNet. The proposed method employs a multi-scale feature aggregation encoder–decoder architecture, leveraging contextual information to capture the characteristics of buildings of varying sizes in the imagery. Cross-attention mechanisms are incorporated to enhance the feature correlations between the change pairs. Additionally, the frame field is introduced into the network to model the complex geometric structure of the building target. By learning the local orientation information of the building structure, the frame field can effectively capture the geometric features of complex building features. During the training process, a multi-task learning strategy is used to align the predicted frame field with the real building outline, while learning the overall segmentation, edge outline, and corner point features of the building. This improves the accuracy of the building polygon representation. Furthermore, a discriminative loss function is constructed through multi-task learning to optimize the polygonal structured information of the building targets. The proposed method achieves state-of-the-art results on two commonly used datasets.

Interaction of complex particles: A framework for the rapid and accurate approximation of pair potentials using neural networks

Motivated by the limitations of conventional coarse-grained molecular dynamics for simulation of large systems of nanoparticles and the challenges in efficiently representing general pair potentials for rigid bodies, we present a method for approximating general rigid body pair potentials based on a specialized type of deep neural network that maintains essential properties, such as conservation of energy and invariance to the chosen origins of the particles. The network uses a specialized geometric abstraction layer to convert the relative coordinates of the rigid bodies to input more suitable to a conventional artificial neural network, which is trained together with the specialized layer. This results in geometric representations of the particles optimized for the specific potential. The network can be trained directly on scalar values to fit a model without explicit gradient and then used to efficiently evaluate the force and torque on the particles resulting from the potential. The concept is demonstrated with an atomistic interaction model for carbon nanotubes and the resulting model is compared with a common type of coarse-grained model optimized for the same potential, with even very small networks comparing favourably and larger networks achieving up to two orders of magnitude lower cost. The sensitivity to noise in the training data is investigated and the model is found to strongly reject noise up to 12.5% given a dataset of 107 samples. The performance of a proof-of-concept implementation is demonstrated on a variety of hardware, showing the models viability for large-scale simulations. Furthermore, generalization to soft bodies and potentials for polydisperse systems are discussed. Published by the American Physical Society 2024

Integration of molecular coarse-grained model into geometric representation learning framework for protein-protein complex property prediction

Structure-based machine learning algorithms have been utilized to predict the properties of protein-protein interaction (PPI) complexes, such as binding affinity, which is critical for understanding biological mechanisms and disease treatments. While most existing algorithms represent PPI complex graph structures at the atom-scale or residue-scale, these representations can be computationally expensive or may not sufficiently integrate finer chemical-plausible interaction details for improving predictions. Here, we introduce MCGLPPI, a geometric representation learning framework that combines graph neural networks (GNNs) with MARTINI molecular coarse-grained (CG) models to predict PPI overall properties accurately and efficiently. Extensive experiments on three types of downstream PPI property prediction tasks demonstrate that at the CG-scale, MCGLPPI achieves competitive performance compared with the counterparts at the atom- and residue-scale, but with only a third of computational resource consumption. Furthermore, CG-scale pre-training on protein domain-domain interaction structures enhances its predictive capabilities for PPI tasks. MCGLPPI offers an effective and efficient solution for PPI overall property predictions, serving as a promising tool for the large-scale analysis of biomolecular interactions.

A non-newtonian approach to geometric phase through optic fiber via multiplicative quaternions

In this paper, we researched magnetic and electromagnetic curves in multiplicative Euclidean 3-space via multiplicative quaternion algebra. Firstly, we examined the geometric phase representation of the polarized light wave in the optic fiber by multiplicative Frenet frame. Using the quaternionic approaches, we were able to derive the magnetic curve equations and theorems. Then, three particular instances have been illustrated with examples of electromagnetic curves and magnetic field equations. Lastly, we provided an interpretation of the findings. With the help of the results, we were able to present an alternative viewpoint on the construction of trajectories (such as circular or spiral-like ones) that do not exist uniquely in the realm of physics.

NeuFG: Neural Fuzzy Geometric Representation for 3-D Reconstruction

NeuFG: Neural Fuzzy Geometric Representation for 3-D Reconstruction

GeoNet enables the accurate prediction of protein-ligand binding sites through interpretable geometric deep learning.

The identification of protein binding residues is essential for understanding their functions invivo. However, it remains a computational challenge to accurately identify binding sites due to the lack of known residue binding patterns. Local residue spatial distribution and its interactive biophysical environment both determine binding patterns. Previous methods could not capture both information simultaneously, resulting in unsatisfactory performance. Here, we present GeoNet, an interpretable geometric deep learning model for predicting DNA, RNA, and protein binding sites by learning the latent residue binding patterns. GeoNet achieves this by introducing a coordinate-free geometric representation to characterize local residue distributions and generating an eigenspace to depict local interactive biophysical environments. Evaluation shows that GeoNet is superior compared to other leading predictors and it shows a strong interpretability of learned representations. We present three test cases, where interaction interfaces were successfully identified with GeoNet.

Research on 3D Model Calculation of Green and low-carbon New Buildings

Due to the widespread use of solid clay bricks in building houses in China, the annual destruction of farmland by burning bricks amounts to about 100000 acres. Faced with these problems, China has an extremely urgent demand for green and low-carbon buildings. Therefore, this article conducts computational research on the 3D model of a new type of green and low-carbon building. Due to the significant differences in the geometric form and texture representation of a large number of green and low-carbonl new buildings, it is difficult to uniformly quantify their level of detail. This article considers the usefulness and accessibility of 3D model calculation data separately. The shadow length is generally the distance from the selected building's facade roof to the end of the shadow, which can avoid the influence of building shape on its length. The determination of these two points and the calculation of the distance between them will directly affect the accuracy of shadow length calculation. Therefore, when users need to add an application subsystem to the system, they must add information such as the input and output data structures of the subsystem to the module. The height information calculation method uses the building and shadow imaging geometric modeling to calculate the building height by establishing the relationship between the shadow length and the corresponding building height.

Application of Clifford’s Algebra to Describe the Early Universe

This article is a shortened review of previous results obtained by the author. The advantages of describing the geometric nature of the physical properties of the early universe using the Clifford algebra approach are demonstrated. A geometric representation of the wave function of the early universe is used, and a new mechanism of spontaneous symmetry breaking with different degrees of freedom is proposed. A possible supersymmetry is revealed, and it is shown that the energy of the initial vacuum can be considered equal to zero. The origin of baryonic asymmetry and the nature of dark matter can be explained using a geometric representation of the wave function of the early universe.

Graph masked self-distillation learning for prediction of mutation impact on protein–protein interactions

Assessing mutation impact on the binding affinity change (ΔΔG) of protein–protein interactions (PPIs) plays a crucial role in unraveling structural-functional intricacies of proteins and developing innovative protein designs. In this study, we present a deep learning framework, PIANO, for improved prediction of ΔΔG in PPIs. The PIANO framework leverages a graph masked self-distillation scheme for protein structural geometric representation pre-training, which effectively captures the structural context representations surrounding mutation sites, and makes predictions using a multi-branch network consisting of multiple encoders for amino acids, atoms, and protein sequences. Extensive experiments demonstrated its superior prediction performance and the capability of pre-trained encoder in capturing meaningful representations. Compared to previous methods, PIANO can be widely applied on both holo complex structures and apo monomer structures. Moreover, we illustrated the practical applicability of PIANO in highlighting pathogenic mutations and crucial proteins, and distinguishing de novo mutations in disease cases and controls in PPI systems. Overall, PIANO offers a powerful deep learning tool, which may provide valuable insights into the study of drug design, therapeutic intervention, and protein engineering.

A Preisach Model Defining Correlation between Monotonic and Cyclic Response of Structural Mild Steel

This article delivers a new Preisach model representing the correlation between the elastoplastic behavior of structural mild steel under axial monotonic and cyclic loading with damage. The newly formed model is based on the experimentally defined correlation between axial monotonic and cyclic behavior of structural mild steel. To examine the monotonic and cyclic behavior of structural mild steel and find fitting material properties for the model, monotonic and cyclic axial tensile tests are performed. Tests are executed on coupons of the commonly used European structural steel S275. The model represents a mathematical description of modified single-crystal material behavior under monotonic loading. Two different approaches were used to describe damage in the multilinear mechanical model. The excellent agreement with experimental results is achieved by infinitely linking many single-crystal elements in parallel, forming the polycrystalline model. This model provides a good solution for everyday engineering practice due to its geometric representation in the form of the Preisach triangle and the lower costs of monotonic tests used to define material properties compared to cyclic tests.

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2,1)-labeling.

Geometric representations of braid and Yang–Baxter gates

Geometric representations of braid and Yang–Baxter gates

Historical Architecture and BIM Modelling: Between Representation of Reality and Conceptual Abstraction

The essay aims to explore the applications of BIM to design regarding historic buildings, which will become mandatory in Italy from 2025 for public works above €1 million. Despite the advantages in coordinated data management and information sharing among the actors involved in contracts, problems related to the geometric representation and semantic definition of the virtual entities that populate the models have been recognized by the scientific community. Critical issues that have emerged from the study of significant experiences can be traced back to the difficulty of adapting tools designed for the projects of new buildings to the characteristics of the cultural heritage. In particular, the complexity of proposing congruent categorisations and parameterisations seems related to only partial correspondence between historical building components, IFC classes and BIM software categories. These limitations often lead to the identification of contingent solutions and expedients, which solve only the problems of geometric representation. In contrast, the ability to adequately share information stored in databases, interoperable through the IFC format, is conveyed by an appropriate semantic description. Formal representation in the BIM environment effectively refers to standardized industrial production, reproducibility of building elements and, more generally, to the goals of globalisation. Thus, the proposed reflections aim to encourage a conscious use of these tools and to outline implementation perspectives useful to bring the digital environment closer to the concrete reality of historical architecture, unique and irreproducible in origin and transformation, often realized through artisanal processes and anchored to specific contexts in multiple aspects.

Semiclassical Analysis, Geometric Representation and Quantum Ergodicity

Semiclassical Analysis, Geometric Representation and Quantum Ergodicity

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.

Loop corrections versus marginal deformations in celestial holography

Four-dimensional all-loop amplitudes in quantum electrodynamics and gravity exhibit universal IR singularities with a factorization structure. This structure is governed by tree amplitudes and a universal IR-divergent factor representing the exchange of soft particles between external lines. This Letter offers a precise dual interpretation of these universal IR-divergent factors within celestial holography. Considering the tree amplitude as the foundation of the celestial conformal field theory (CCFT), these universal factors correspond to marginal deformations with a construction in the CCFT. Remarkably, a novel geometric representation of these deformations through topological gauging provides an exact description of transitions within bulk vacuum moduli spaces which extends the celestial holography beyond the perturbative level. Our findings establish a concrete dictionary for celestial holography and offer a holographic lens to understand loop corrections in scattering amplitudes. Published by the American Physical Society 2024

Nonprojective Bell-state measurements

The Bell-state measurement (BSM) is the projection of two qubits onto four orthogonal maximally entangled states. Here we first propose how to appropriately define more general BSMs, which have more than four possible outcomes, and then study whether they exist in quantum theory. We observe that nonprojective BSMs can be defined in a systematic way in terms of equiangular tight frames of maximally entangled states, i.e., a set of maximally entangled states, where every pair is equally, and in a sense maximally, distinguishable. We show that there exists a five-outcome BSM through an explicit construction and find that it admits a simple geometric representation. Then we prove that there exists no larger BSM on two qubits by showing that no six-outcome BSM is possible. We also determine the most distinguishable set of six equiangular maximally entangled states and show that it falls only somewhat short of forming a valid quantum measurement. Finally, we study the nonprojective BSM in the contexts of both local state discrimination and entanglement-assisted quantum communication. Our results put forward natural forms of nonprojective joint measurements and provide insight into the geometry of entangled quantum states. Published by the American Physical Society 2024

A review of graph neural network applications in mechanics-related domains

Mechanics-related tasks often present unique challenges in achieving accurate geometric and physical representations, particularly for non-uniform structures. Graph neural networks (GNNs) have emerged as a promising tool to tackle these challenges by adeptly learning from graph data with irregular underlying structures. Consequently, recent years have witnessed a surge in complex mechanics-related applications inspired by the advancements of GNNs. Despite this process, there is a notable absence of a systematic review addressing the recent advancement of GNNs in solving mechanics-related tasks. To bridge this gap, this review article aims to provide an in-depth overview of the GNN applications in mechanics-related domains while identifying key challenges and outlining potential future research directions. In this review article, we begin by introducing the fundamental algorithms of GNNs that are widely employed in mechanics-related applications. We provide a concise explanation of their underlying principles to establish a solid understanding that will serve as a basis for exploring the applications of GNNs in mechanics-related domains. The scope of this paper is intended to cover the categorisation of literature into solid mechanics, fluid mechanics, and interdisciplinary mechanics-related domains, providing a comprehensive summary of graph representation methodologies, GNN architectures, and further discussions in their respective subdomains. Additionally, open data and source codes relevant to these applications are summarised for the convenience of future researchers. This article promotes an interdisciplinary integration of GNNs and mechanics and provides a guide for researchers interested in applying GNNs to solve complex mechanics-related tasks.

Building Krylov complexity from circuit complexity

Krylov complexity has emerged as a probe of operator growth in a wide range of nonequilibrium quantum dynamics. However, a fundamental issue remains in such studies: the definition of the distance between basis states in Krylov space is ambiguous. Here we show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Whereas circuit complexity characterizes the geodesic distance in a multidimensional operator space, Krylov complexity measures the height of the final operator in a particular direction. The geometric representation of circuit complexity thus unambiguously designates the distance between basis states in Krylov space. This geometric approach also applies to time-dependent Liouvillian superoperators, where a single Krylov complexity is no longer sufficient. Multiple Krylov complexity may be exploited jointly to fully describe operator dynamics. Published by the American Physical Society 2024