Data with Non-Euclidean Geometry and Its Characterization

Recently,dealing with the non-Euclidean data and its characterization is considered as one of the major issues by researchers.The first problem arises while defining the distinction among Euclidean and non-Euclidean geometry with its examples.The second problem arises while dealing with the non-Euclidean geometry in true,false,and uncertain regions.The third problem arises while investigating some patterns in non-Euclidean data sets.This paper focused on tackling these issues with some real-life examples in data processing,data visualization,knowledge representation,and quantum computing.

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Unveiling the geometric site dependent activity of spinel Co_(3)O_(4)for electrocatalytic chlorine evolution reaction

Spinel cobalt oxide(Co_(3)O_(4)),consisting of tetrahedral Co^(2+)(CoTd)and octahedral Co^(3+)(CoOh),is considered as promising earth-abundant electrocatalyst for chlorine evolution reaction(CER).Identifying the catalytic contribution of geometric Co site in the electrocatalytic CER plays a pivotal role to precisely modulate electronic configuration of active Co sites to boost CER.Herein,combining density functional theory calculations and experiment results assisted with operando analysis,we found that the Co_(Oh) site acts as the main active site for CER in spinel Co_(3)O_(4),which shows better Cl^(-)adsorption and more moderate intermediate adsorption toward CER than CoTd site,and does not undergo redox transition under CER condition at applied potentials.Guided by above findings,the oxygen vacancies were further introduced into the Co_(3)O_(4) to precisely manipulate the electronic configuration of Co_(Oh) to boost Cl^(-)adsorption and optimize the reaction path of CER and thus to enhance the intrinsic CER activity significantly.Our work figures out the importance of geometric configuration dependent CER activity,shedding light on the rational design of advanced electrocatalysts from geometric configuration optimization at the atomic level.

Modeling and Performance Analysis of UAV-Aided Millimeter Wave Cellular Networks with Stochastic Geometry

UAV-aided cellular networks,millimeter wave(mm-wave)communications and multi-antenna techniques are viewed as promising components of the solution for beyond-5G(B5G)and even 6G communications.By leveraging the power of stochastic geometry,this paper aims at providing an effective framework for modeling and analyzing a UAV-aided heterogeneous cellular network,where the terrestrial base stations(TBSs)and the UAV base stations(UBSs)coexist,and the UBSs are provided with mm-wave and multi-antenna techniques.By modeling the TBSs as a PPP and the UBSs as a Matern hard-core point pro-´cess of type II(MPH-II),approximated but accurate analytical results for the average rate of the typical user of both tiers are derived through an approximation method based on the mean interference-to-signal ratio(MISR)gain.The influence of some relevant parameters is discussed in detail,and some insights into the network deployment and optimization are revealed.Numerical results show that some trade-offs are worthy of being considered,such as the antenna array size,the altitude of the UAVs and the power control factor of the UBSs.

Estimation of surface geometry on combustion characteristics of AP/HTPB propellant under rapid depressurization

The 2D sandwich model serves as a potent tool in exploring the influence of surface geometry on the combustion attributes of Ammonium perchlorate/Hydroxyl-terminated polybutadiene(AP/HTPB)propellant under rapid pressure decay.The thickness of the sandwich propellant is derived from slicing the 3D random particle packing,an approach that enables a more effective examination of the micro-flame structure.Comparative analysis of the predicted burning characteristics has been performed with experimental studies.The findings demonstrate a reasonable agreement,thereby validating the precision and soundness of the model.Based on the typical rapid depressurization environment of solid rocket motor(initial combustion pressure is 3 MPa and the maximum depressurization rate is 1000 MPa/s).A-type(a flatter surface),B-type(AP recesses from the combustion surface),and C-type(AP protrudes from the combustion surface)propellant combustion processes are numerically simulated.Upon comparison of the evolution of gas-phase flame between 0.1 and 1 ms,it is discerned that the flame strength and form created by the three sandwich models differ significantly at the beginning stage of depressurization,with the flame structures gradually becoming harmonized over time.Conclusions are drawn by comparison extinction times:the surface geometry plays a pivotal role in the combustion process,with AP protrusion favoring combustion the most.

The Effects of the Geometry of a Current Collector with an Equal Open Ratio on Output Power of a Direct Methanol Fuel Cell

The open ratio of a current collector has a great impact on direct methanol fuel cell(DMFC)performance.Although a number of studies have investigated the influence of the open ratio of DMFC current collectors,far too little attention has been given to how geometry(including the shape and feature size of the flow field)affects a current collector with an equal open ratio.In this paper,perforated and parallel current collectors with an equal open ratio of 50%and different feature sizes are designed,and the corresponding experimental results are shown to explain the geometry effects on the output power of the DMFC.The results indicate that the optimal feature sizes are between 2 and 2.5 mm for both perforated and parallel flow field in the current collectors with an equal open ratio of 50%.This means that for passive methanol fuel cells,to achieve the highest output power,the optimal feature size of the flow field in both anode and cathode current collectors is between 2 and 2.5 mm under the operating mode of this experiment.The effects of rib and channel position are also investigated,and the results indicate that the optimum pattern depends on the feature sizes of the flow field.

Geometries and electronic structures of Zr_(n)Cu(n=2–12) clusters: A joint machine-learning potential density functional theory investigation

Zr-based amorphous alloys have attracted extensive attention because of their large glassy formation ability, wide supercooled liquid region, high elasticity, and unique mechanical strength induced by their icosahedral local structures.To determine the microstructures of Zr–Cu clusters, the stable and metastable geometry of Zr_(n)Cu(n=2–12) clusters are screened out via the CALYPSO method using machine-learning potentials, and then the electronic structures are investigated using density functional theory. The results show that the Zr_(n)Cu(n ≥ 3) clusters possess three-dimensional geometries, Zr_(n)Cu(n≥9) possess cage-like geometries, and the Zr_(12)Cu cluster has icosahedral geometry. The binding energy per atom gradually gets enlarged with the increase in the size of the clusters, and Zr_(n)Cu(n=5,7,9,12) have relatively better stability than their neighbors. The magnetic moment of most Zr_(n)Cu clusters is just 1μB, and the main components of the highest occupied molecular orbitals(HOMOs) in the Zr_(12)Cu cluster come from the Zr-d state. There are hardly any localized two-center bonds, and there are about 20 σ-type delocalized three-center bonds.

Galerkin-based quasi-smooth manifold element(QSME)method for anisotropic heat conduction problems in composites with complex geometry

The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method(FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element(QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional(2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity.The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom(DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

Outage Probability Analysis for D2D-Enabled Heterogeneous Cellular Networks with Exclusion Zone:A Stochastic Geometry Approach

Interference management is one of the most important issues in the device-to-device(D2D)-enabled heterogeneous cellular networks(HetCNets)due to the coexistence of massive cellular and D2D devices in which D2D devices reuse the cellular spectrum.To alleviate the interference,an efficient interference management way is to set exclusion zones around the cellular receivers.In this paper,we adopt a stochastic geometry approach to analyze the outage probabilities of cellular and D2D users in the D2D-enabled HetCNets.The main difficulties contain three aspects:1)how to model the location randomness of base stations,cellular and D2D users in practical networks;2)how to capture the randomness and interrelation of cellular and D2D transmissions due to the existence of random exclusion zones;3)how to characterize the different types of interference and their impacts on the outage probabilities of cellular and D2D users.We then run extensive Monte-Carlo simulations which manifest that our theoretical model is very accurate.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Noncommutative-Geometry Wormholes Based on the Casimir Effect

While wormholes are as good a prediction of Einstein’s theory as black holes, they are subject to severe restrictions from quantum field theory. In particular, holding a wormhole open requires a violation of the null energy condition, calling for the existence of exotic matter. The Casimir effect has shown that this physical requirement can be met on a small scale, thereby solving a key conceptual problem. The Casimir effect does not, however, guarantee that the small-scale violation is sufficient for supporting a macroscopic wormhole. The purpose of this paper is to connect the Casimir effect to noncommutative geometry, which also aims to accommodate small-scale effects, the difference being that these can now be viewed as intrinsic properties of spacetime. As a result, the noncommutative effects can be implemented by modifying only the energy momentum tensor in the Einstein field equations, while leaving the Einstein tensor unchanged. The wormhole can therefore be macroscopic in spite of the small Casimir effect.

Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry

The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaussian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of transformations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Riemannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of modeling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials.

Fractal Geometry:Axioms,Fractal Derivative and Its Geometrical Meaning

Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geometry.In modem scientific literature often write from a firm that"there is no strict definition of fractals",to the more moderate that"objects in a certain sense,fractal and similar."We show that fractal geometry is a strict mathematical theory,defined by their axioms.This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials.Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension,geometri-cal interpretation of fractal derivative gain and open dual symmetry.

Solving Algebraic Problems with Geometry Diagrams Using Syntax-Semantics Diagram Understanding

Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.

Aging and Biological Oscillation: A Question of Geometry

Previous studies in different ethnic groups show changes in heart rate, respiratory rate, cortisol cycle, and sleep-wake cycle throughout life. Our purpose is to verify such changes by comparing the values of each variable before and after puberty. Puberty is associated with the end of growth and is an important point in our theoretical framework: when growth ends, changes occur in the geometry of the biological system. At the same time, this causes phase changes in the oscillatory variables, which are seen as chronodisruption. The results confirm the changes found by other authors in the evolution of the variables throughout life. Then, we can conclude that the variables studied present phase changes when growth ends, in accordance with the proposed theoretical framework.

A Data-Driven Intersection Geometry Mapping Technique to Enhance the Scalability of Trajectory-Based Traffic Signal Performance Measures

Connected vehicle (CV) trajectory data provides practitioners with opportunities to assess traffic signal performance with no investment in detection or communication infrastructure. With over 500 billion trajectory records generated each month in the United States, operations can be evaluated virtually at any of the over 400,000 traffic signals in the nation. The manual intersection mapping required to generate accurate movement-level trajectory-based performance estimations is the most time-consuming aspect of using CV data to evaluate traffic signal operations. Various studies have utilized vehicle location data to update and create maps;however, most proposed mapping techniques focus on the identification of roadway characteristics that facilitate vehicle navigation and not on the scaling of traffic signal performance measures. This paper presents a technique that uses commercial CV trajectory and open-source OpenStreetMap (OSM) data to automatically map intersection centers and approach areas of interest to estimate signal performance. OSM traffic signal tags are processed to obtain intersection centers. CV data is then used to extract intersection geometry characteristics surrounding the intersection. To demonstrate the proposed technique, intersection geometry is mapped at 500 locations from which trajectory-based traffic signal performance measures are estimated. The results are compared to those obtained from manual geometry definitions. Statistical tests found that at a 99% confidence level, upstream-focused performance estimations are strongly correlated between both methodologies. The presented technique will aid agencies in scaling traffic signal assessment as it significantly reduces the amount of manual labor required.

Solving Geometry Problems via Feature Learning and Contrastive Learning of Multimodal Data

This paper presents an end-to-end deep learning method to solve geometry problems via feature learning and contrastive learning of multimodal data.A key challenge in solving geometry problems using deep learning is to automatically adapt to the task of understanding single-modal and multimodal problems.Existing methods either focus on single-modal ormultimodal problems,and they cannot fit each other.A general geometry problem solver shouldobviouslybe able toprocess variousmodalproblems at the same time.Inthispaper,a shared feature-learning model of multimodal data is adopted to learn the unified feature representation of text and image,which can solve the heterogeneity issue between multimodal geometry problems.A contrastive learning model of multimodal data enhances the semantic relevance betweenmultimodal features and maps them into a unified semantic space,which can effectively adapt to both single-modal and multimodal downstream tasks.Based on the feature extraction and fusion of multimodal data,a proposed geometry problem solver uses relation extraction,theorem reasoning,and problem solving to present solutions in a readable way.Experimental results show the effectiveness of the method.

Geometry Flow-Based Deep Riemannian Metric Learning

Deep metric learning(DML)has achieved great results on visual understanding tasks by seamlessly integrating conventional metric learning with deep neural networks.Existing deep metric learning methods focus on designing pair-based distance loss to decrease intra-class distance while increasing interclass distance.However,these methods fail to preserve the geometric structure of data in the embedding space,which leads to the spatial structure shift across mini-batches and may slow down the convergence of embedding learning.To alleviate these issues,by assuming that the input data is embedded in a lower-dimensional sub-manifold,we propose a novel deep Riemannian metric learning(DRML)framework that exploits the non-Euclidean geometric structural information.Considering that the curvature information of data measures how much the Riemannian(nonEuclidean)metric deviates from the Euclidean metric,we leverage geometry flow,which is called a geometric evolution equation,to characterize the relation between the Riemannian metric and its curvature.Our DRML not only regularizes the local neighborhoods connection of the embeddings at the hidden layer but also adapts the embeddings to preserve the geometric structure of the data.On several benchmark datasets,the proposed DRML outperforms all existing methods and these results demonstrate its effectiveness.

Effect of tool geometry on ultraprecision machining of soft-brittle materials:a comprehensive review

Brittle materials are widely used for producing important components in the industry of optics,optoelectronics,and semiconductors.Ultraprecision machining of brittle materials with high surface quality and surface integrity helps improve the functional performance and lifespan of the components.According to their hardness,brittle materials can be roughly divided into hard-brittle and soft-brittle.Although there have been some literature reviews for ultraprecision machining of hard-brittle materials,up to date,very few review papers are available that focus on the processing of soft-brittle materials.Due to the‘soft’and‘brittle’properties,this group of materials has unique machining characteristics.This paper presents a comprehensive overview of recent advances in ultraprecision machining of soft-brittle materials.Critical aspects of machining mechanisms,such as chip formation,surface topography,and subsurface damage for different machining methods,including diamond turning,micro end milling,ultraprecision grinding,and micro/nano burnishing,are compared in terms of tool-workpiece interaction.The effects of tool geometries on the machining characteristics of soft-brittle materials are systematically analyzed,and dominating factors are sorted out.Problems and challenges in the engineering applications are identified,and solutions/guidelines for future R&D are provided.

Study of residual stresses and distortions from the Ti6Al4V based thinwalled geometries built using LPBF process

The current paper focuses on the prediction of residual stresses and distortions in the Laser Powder Bed Fusion(LPBF)built Ti6Al4V thin-walled geometries using Ansys Additive Print(AAP)software which employs a layer-by-layer accumulation of inherent strain to calculate the deformations.Isotropic and anisotropic strain scaling factors were calibrated initially within the APP software for the Ti6Al4V based single cantilever beam geometry.Subsequently,the numerical simulations were performed in APP software and computed the residual stresses and distortions for the varied process parameters including laser power,scan speed and hatch distance while maintaining the layer thickness constant for all the design iterations.The numerical predictions were compared;they were found to match reasonably well with the XRD measurements within the calibrated regime.