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.

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.

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.

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.

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.

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.

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.

Spatio-Temporal Context-Guided Algorithm for Lossless Point Cloud Geometry Compression

Point cloud compression is critical to deploy 3D representation of the physical world such as 3D immersive telepresence,autonomous driving,and cultural heritage preservation.However,point cloud data are distributed irregularly and discontinuously in spatial and temporal domains,where redundant unoccupied voxels and weak correlations in 3D space make achieving efficient compression a challenging problem.In this paper,we propose a spatio-temporal context-guided algorithm for lossless point cloud geometry compression.The proposed scheme starts with dividing the point cloud into sliced layers of unit thickness along the longest axis.Then,it introduces a prediction method where both intraframe and inter-frame point clouds are available,by determining correspondences between adjacent layers and estimating the shortest path using the travelling salesman algorithm.Finally,the few prediction residual is efficiently compressed with optimal context-guided and adaptive fastmode arithmetic coding techniques.Experiments prove that the proposed method can effectively achieve low bit rate lossless compression of point cloud geometric information,and is suitable for 3D point cloud compression applicable to various types of scenes.

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.

Influences of cement asphalt emulsified mortar construction on track slab geometry status

Purpose–The construction of cement asphalt(CA)emulsified mortar can obviously disturb the slab status after the fine adjustment.To decrease or eliminate the influence of CA mortar grouting on track slab geometry status,the effects of grouting funnel,slab pressing method,mortar expansion ratio,seepage ratio and grouting area on China Railway Track System Type(CRTS I)track slab geometry status were discussed in this paper.Design/methodology/approach–Combined with engineering practice,this paper studied the expansion law of filling layer mortar,the liquid level height of the filling funnel,the pressure plate device and the amount of exudation water and systematically analyzed the influence of filling layer mortar construction on the state of track slab.Relevant precautions and countermeasures were put forward.Findings–The results showed that the track slab floating values of four corners were different with the CA mortar grouting and the track slab corner near CA mortar grouting hole had the maximum floating values.The anti-floating effect of“7”shaped slab pressing device was more efficient than fixed-joint angle iron,and the slab floating value could be further decreased by increasing the amount of“7”shaped slab pressing devices.After CA mortar grouting,the track slab floating pattern had a close correlation with the expansion rate and water seepage rate of CA mortar over time and the expansion and water seepage rate of the mortar were faster when the temperature was high.Furthermore,the use of strip CA mortar filling under the rail bearing platform on bothsides could effectively reduce the float under the track slab,and it could also save mortar consumption and reduce costs.Originality/value–This study plays an important role in controlling the floating values,CA mortar dosage and the building cost of projects by grouting CA mortar at two flanks of filling space.The research results have guiding significance for the design and construction of China’s CRTS I,CRTS II and CRTS III track slab.

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.

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.

Geometry Clipmap算法扩展研究

Geometry Clipmap算法是一种基于GPU优化的LOD算法,在研究Geometry Clipmap算法的基础上,对该算法进行了扩展。增加了地形数据调度模块,采用地形块预读取和预卸载策略,实现对海量地形数据的内外存管理。对内存中Clipmap堆栈的数据结构进行了改进,取消了原始算法中Clipmap Pyramid部分。将Geometry Instancing技术引入渲染过程,进一步提升了渲染速度。实验表明,该算法实用有效,能够满足大规模地形可视化的实时渲染要求。

基于Geometry因子计算校正的下行TD-LTE调度优化研究

由于无线频谱资源日益短缺,多层次组网、高复用方式频繁应用造成同频干扰现象时有发生,导致网络传输不畅。为得到较好改善,首先从3GPP已有协议出发,采用两种方案推导获取关键参数Geometry因子;其次针对网络有效性、规避潜在风险性,通过分析对比及实验室验证,确定RSRP作为其计算模式;最后将该因子应用于已有系统,以减小资源碰撞为目的,调整调度时序以协调资源分配。仿真以两种网络标准拓扑进行,并依据网络实际探测能力进行限制,结果表明,可以达到提升系统吞吐量的目的,且密集组网场景下增益提升更明显。

基于Geometry Clipmap的地形绘制技术研究

提出一种改进的Geometry Clipmap算法,增加了地形数据调度模块,重构了节点的组织方式,并引入了渐进纹理更新技术。仿真实验表明,改进的算法显著地提高了飞行模拟系统的工作效率和显示效果。

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

A Study on Computer Consciousness on Intuitive Geometry Based on Mathematics Experiments and Statistical Analysis

In this paper, we present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference. The investigation consists of the following five steps. At first, we select a set of geometric configurations and for each configuration we construct a large amount of geometric data as observation data using dynamic geometry programs together with the pseudo-random number generator. Secondly, we refer to the geometric predicates in the algebraic method of machine proof of geometric theorems to construct statistics suitable for measuring the approximate geometric relationships in the observation data. In the third step, we propose a geometric relationship detection method based on the similarity of data distribution, where the search space has been reduced into small batches of data by pre-searching for efficiency, and the hypothetical test of the possible geometric relationships in the search results has be performed. In the fourth step, we explore the integer relation of the line segment lengths in the geometric configuration in addition. At the final step, we do numerical experiments for the pre-selected geometric configurations to verify the effectiveness of our method. The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can actually attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia.

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.

基于Geometry Clipmap的大规模地形绘制算法

针对地形渲染的实时性与大规模地形数据的海量性之间的矛盾,基于Geometry Clipmap算法的基本思想,对其进行了简化和改进。在原始地形数据处理阶段,精简了原算法的数据结构,并按Clipmap格式进行存储;在绘制阶段,针对相邻层次间的裂缝问题,提出了一种删除边的方法有效地消除了裂缝。实验表明,该方法充分利用了GPU的优势,为大规模地形的实时绘制提供了解决方案。