Geometric Algebra Research Articles

Geometric Product of Two Oriented Points in Conformal Geometric Algebra

Geometric Product of Two Oriented Points in Conformal Geometric Algebra

Conics, Their Pencils and Intersections in Geometric Algebra

Conics, Their Pencils and Intersections in Geometric Algebra

Highly Productive Laser Annealing Manufacturing Method Using Continuous Blue WBC (Wavelength Beam Combining) Technique

Blue laser annealing can be used to obtain a high-mobility thin-film transistor (TFT) through a laser annealing (i.e., LTPS: low-temperature Poly-Si) process. However, the laser annealing process’s low productivity (as well as high cost) is an issue because the high output power of blue lasers still needs to be addressed. Therefore, productivity can be improved if blue laser energy is efficiently supplied during the laser annealing process using a continuous wave laser instead of a conventional pulsed excimer laser. We developed a blue laser light source (440 ± 10 nm) using the wavelength beam combining (WBC) method, which can achieve a laser power density of 73.7 kW/cm2. In this semiconductor laser, when the power was increased s by 2.9 times, the laser scanning speed was increased by 5.0 times, achieving twice the productivity of conventional lasers. After laser annealing, the size of the crystal grains varied between 2 and 15 μm, resulting in a crystallization rate of 100% by Raman scattering rsult and low resistivity of 0.04 Ωcm. This increase in production capacity is not an arithmetic increase with increased power but a geometric production progression.

Decoupled Edge Physics Algorithms for Collaborative XR Simulations

ABSTRACTThis work proposes a novel approach to transform any modern game engine pipeline, for optimized performance and enhanced user experiences in extended reality (XR) environments decoupling the physics engine from the game engine pipeline and using a client‐server architecture creates a scalable solution, efficiently serving multiple graphics clients on head‐mounted displays (HMDs) with a single physics engine on edge‐cloud infrastructure. This approach ensures better synchronization in multiplayer scenarios without introducing overhead in single‐player experiences, maintaining session continuity despite changes in user participation. Relocating the Physics Engine to an edge or cloud node reduces strain on local hardware, dedicating more resources to high‐quality rendering and unlocking the full potential of untethered HMDs. We present four algorithms that decouple the physics engine, increasing frame rates and Quality of Experience (QoE) in VR simulations, supporting advanced interactions, numerous physics objects, and multiuser sessions with over 100 concurrent users. Incorporating a Geometric Algebra interpolator reduces inter‐calls between dissected parts, maintaining QoE and easing network stress. Experimental validation, with more than 100 concurrent users, 10,000 physics objects, and softbody simulations, confirms the technical viability of the proposed architecture, showcasing transformative capabilities for more immersive and collaborative XR applications without compromising performance.

An Observer-Based View of Euclidean Geometry

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

GASSF-Net: Geometric Algebra Based Spectral-Spatial Hierarchical Fusion Network for Hyperspectral and LiDAR Image Classification

The field of multi-source remote sensing observation is becoming increasingly dynamic through the integration of various remote sensing data sources. However, existing deep learning methods face challenges in differentiating between internal and external relationships and capturing fine spatial features. These models often struggle to effectively capture comprehensive information across remote sensing data bands, and they have inherent differences in the size, structure, and physical properties of different remote sensing datasets. To address these challenges, this paper proposes a novel geometric-algebra-based spectral–spatial hierarchical fusion network (GASSF-Net), which uses geometric algebra for the first time to process multi-source remote sensing images, enabling a more holistic approach to handling these images by simultaneously leveraging the real and imaginary components of geometric algebra to express structural information. This method captures the internal and external relationships between remote sensing image features and spatial information, effectively fusing the features of different remote sensing data to improve classification accuracy. GASSF-Net uses geometric algebra (GA) to represent pixels from different bands as multivectors, thus capturing the intrinsic relationships between spectral bands while preserving spatial information. The network begins by deeply mining the spectral–spatial features of a hyperspectral image (HSI) using pairwise covariance operators. These features are then extracted through two branches: a geometric-algebra-based branch and a real-valued network branch. Additionally, the geometric-algebra-based network extracts spatial information from light detection and ranging (LiDAR) to complement the elevation data lacking in the HSI. Finally, a genetic-algorithm-based cross-fusion module is introduced to fuse the HSI and LiDAR data for improved classification. Experiments conducted on three well-known datasets, Trento, MUUFL, and Houston, demonstrate that GASSF-Net significantly outperforms traditional methods in terms of classification accuracy and model efficiency.

Screening of drought tolerance indices for selection of resistance wheat (Triticum aestivum L.) landraces under drought stress conditions

Abstract During the cropping season 2016-2017, an experiment was conducted in a split plot design to study the response of 25 wheat genotypes to drought stress under irrigated and minimal irrigation conditions. Different drought tolerance indices for 25 bread wheat (Triticum aestivum L.) genotypes were evaluated. The calculation drought tolerance indices was based on the grain yield of wheat under irrigated and water stress conditions. The result indicated a significant effect of drought stress on the yield of different wheat genotypes based on mean values of different drought tolerance indices. Among the studied different indices, mean productivity (MP), stress tolerance index (STI), and geometric mean productivity (GMP) were more efficient to discriminate between drought sensitive and drought tolerant genotypes based on the calculation of correlation coefficient and multivariate analysis. Twenty-five genotypes were classified into tolerant and susceptible groups through drought tolerance indices using cluster analysis. Among the genotypes, C-306, WH 1236, WH 1235, and GW 477 having higher yields under drought conditions which showing their stability under stress conditions. The most yield stable genotypes under adverse environment were identified by multivariate analysis of drought tolerance indices individually or in combination. From the result, we concluded that GMP, MP and STI are more efficient drought tolerance indices among other indices to identify stable genotypes under stress conditions.

Exploring the usefulness of drought tolerance indices in screening of maize inter-racial derivatives

Abstract Among all the abiotic factors affecting maize yield, drought stress is the most destructive one. Since drought tolerance is a complicated trait that is difficult to quantify, screening for stress involves measuring the effects of stress on several traits, most notably plant yield. Selection using the drought tolerance indices is an easy way to identify the stable and tolerant genotypes based on yield performances alone. The current study aims to identify suitable maize hybrids with high drought tolerance. A total of 30 maize-teosinte hybrids and 3 checks were raised under optimal and water-stressed environments and were screened using twelve different drought tolerance indices. It was observed that the inter-racial hybrids viz., G24, G27, and G29 were better than the drought tolerant Zea mays ssp. mays check C3 for yield and most of the indices. The indices STI, TOL, GMP, MP, YI, DI, SSPI, K1STI, and K2STI were observed to be significantly correlated with yield under stress and optimal environments. Based on the yield both environments, the genotypes were classified under four groups, with Group A consisting of genotypes with high yields under both environments. The indices Stress Tolerance Index (STI), Geometric Mean Productivity (GMP), Yield Index (YI), and Modified Stress Tolerance Index (K2STI)were identified for their ability to discriminate Group A from the other genotypes and these indices could be utilized to screen for drought tolerance with high yield.Component Analysis revealed two principal components contributing up to 98.15 percent of the cumulative variation. The PCA biplot identified the hybrids viz., G24, G27, G14, G19, and G25 as tolerant ones. Based on the index scores, the hybrids viz., G27, G29, G17, G20,and G24were identified as the top-ranking genotypes, with high and more stable yields under both stress and optimal environments. Keywords: Drought index, maize, 3D plot, inter-racial hybrids, teosinte

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.

Form Deviation Uncertainty and Conformity Assessment on a Coordinate Measuring Machine

Coordinate measuring machines are widely used in the industrial field due to their ease of automation. However, estimating the measurement uncertainty is a delicate task, especially when controlling for deviation, given the large number of factors that influence the measurement. A precise estimate of the uncertainty is crucial to avoid incorrect conformity assessments. The purpose of this study is to control geometrical-form tolerance specifications, taking into consideration their associated uncertainty. A surface fitting model based on the least squares criterion is proposed, allowing one to obtain the variance–covariance matrix by iterative calculation according to the Levenberg–Marquard optimization method. The form deviation is then evaluated following the Geometrical Product Specifications (GPS) Standard, and its associated uncertainty is estimated using the guide to the expression of uncertainty in measurement (GUM) propagation of the uncertainty law. Finally, the conformity assessment is performed based on the measured deviation and its associated uncertainty. Different results for the measurement of straightness, flatness, circularity, roundness, and cylindricity are presented and detailed. This model is thereafter validated by a Monte Carlo simulation, and interlaboratory comparisons of the obtained results were performed, which showed satisfactory outcome. This contribution is of great use to manufacturing companies and metrology laboratories, allowing them to meet the normative guidelines, which stipulates that each measurement result must be accompanied by its associated uncertainty.

Genome-wide association mapping and genomic prediction of water deficit stress tolerance indices in spring bread wheat

Genome-wide association mapping (GWAM) is crucial for identifying the genetic architecture of quantitative traits, such as drought tolerance indices in bread wheat. This study aims to identify Marker-Trait Associations (MTAs) and genes related to drought tolerance indices in wheat. Seven drought tolerance indices were calculated based on grain weight per spike under field drought stress (FDS) and field non-stress (FNS) conditions. These indices included mean productivity (MP), geometric mean productivity (GMP), relative drought index (RDI), stress tolerance index (STI), tolerance index (TOL), stress susceptibility index (SSI), and yield stability index (YSI). Genotyping of the samples was performed using single nucleotide polymorphism (SNP) markers. A total of 96 MTAs were identified for the studied indices and conditions FNS and FDS in this experiment, with a threshold of -log10p ≥ 3.0. These included FNS, FDS, GMP, MP, STI, SSI, RDI, and YSI, with 15, 11, 16, 16, 20, 6, 6, and 6 MTAs, respectively. The MTAs identified for the drought tolerance indices GMP, MP, and STI were located on chromosomes 2 A, 3B, and 6 A, respectively. Moreover, this study identified four genes related to the indices, namely “TraesCS2B02G000100,” “TraesCS3A02G000100,” “TraesCS2B02G000200,” and “TraesCS4B02G000100”. These genes play a crucial role in drought tolerance and can be utilized for marker-assisted selection to enhance drought tolerance wheat genotypes.

Exactly solvable Stuart-Landau models in arbitrary dimensions.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equationsin the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere S^{D-1} and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus T^{N} on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.

Multi-locus genome-wide association study for grain yield and drought tolerance indices in sorghum accessions.

Drought is a significant factor that causes yield loss in essential cereal crops such as sorghum [Sorghum bicolor (L.) Moench], necessitating the development of drought-tolerant varieties adaptable to various water conditions. This study aimed to pinpoint drought-tolerant sorghum lines and genomic regions for tolerance by utilizing 216 sorghum accessions in stressed and non-stressed environments at two locations. Genetic diversity was evident among accessions in terms of grain yield under different watering regimes. Drought stress indices such as the stress tolerance index, mean productivity, geometric mean productivity, harmonic mean productivity, yield stability index, and yield index were identified as effective measures for selecting drought-tolerant sorghum. Cluster analysis classified genotypes into four groups based on their association with grain yield, highlighting Acc. #28546 and Acc. #216739 as highly drought tolerant across environments. This study identified 32 and 22 quantitative trait nucleotides (QTNs) for drought indices and grain yield under stress and non-stress conditions, respectively, at two locations, with five common QTNs linked to multiple drought indices. Colocation analysis revealed that these QTNs were associated with known stay-green-related quantitative trait loci (QTLs), and 47 putative genes near these QTNs potentially influenced drought tolerance traits. It is suggested that accession selection considers multiple indices for robust evaluation. Understanding the identified genes and their functions provides insights into the genetic mechanisms governing plant responses to drought stress, offering prospects for developing improved drought-resistant sorghum varieties through further genetic research.

Differentials of the basis in Clifford Geometric Algebra

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity

Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA).

Effect of forming strategies on the microstructure and mechanical properties of thin-walled CuCrZr alloy fabricated by selective laser melting

Thin-walled structures are widely used in industrial fields such as radiators and critical aerospace components due to their advantages in lightweight design. However, traditional manufacturing techniques struggle to meet the high-precision requirements for complex components. Selective Laser Melting (SLM) technology, characterized by high design flexibility and high geometric production freedom, is a promising alternative. This study employs SLM technology to fabricate CuCrZr alloy thin walls and systematically investigates the effects of interlayer rotation angles and placement strategies on the forming quality and mechanical properties of the thin walls produced by SLM. The results indicate that different interlayer rotation angles and placement strategies lead to variations in defect types and shapes. When the interlayer rotation angle is 0° and the specimens are placed perpendicular to the recoater blade, numerous lack-of-fusion (LOF) defects are observed on the specimens. These lack-of-fusion defects are caused by factors such as low scanning line flatness and uneven energy distribution when the interlayer rotation angle is 0°. In contrast to the 0° interlayer rotation angle, the specimens with a 67° rotation angle exhibit significantly more curved columnar grains in the growth direction. The thin-walled specimens with a 67° interlayer rotation angle and placed parallel to the recoater blade show the best mechanical performance, with a maximum tensile strength of 254.19 MPa and an elongation of 48.81 %.

Evaluation of barley genotypes for drought adaptability: based on stress indices and comprehensive evaluation as criteria.

The prevalence of drought events worldwide emphasizes the importance of screening and cultivating drought-adapted crops. In this study, 206 germplasm resources were used as materials, dry weight as target trait, and two genotyping methods as criteria to evaluate drought adaptability at the seedling establishment stage. The results showed a significant decrease in average dry weight of the tested germplasm resources (from 746.90 mg to 285.40 mg) and rich variation in the responses of dry weight among each genotype to drought (CV=61.14%). In traditional evaluation method, drought resistance coefficient (DC), geometric mean productivity index (GMP), mean productivity index (MP), stress susceptibility index (SSI), stress tolerance index (STI), and tolerance index (TOL) also exhibited diversity in tested genotypes (CV>30%). However, these indices showed varying degrees of explanation for dry weight under stress and non-stress environments and failed to differentiate drought adaptability among genotypes clearly. In new evaluation method, four stress indices were developed to quantify barley seedling production and stability capacities. Compared to traditional stress indices, the stress production index (SI) explained dry weight more comprehensively under stress conditions (R2 = 0.98), while the ideal production index (II) explained dry weight better under non-stress conditions (R2 = 0.89). Furthermore, the potential index (PI) and elasticity index (EI) eliminated disparities in traditional stress indices and comprehensively clarified the contribution of elasticity and potential to production capacity under drought stress. Ultimately, through grading evaluation and cluster analysis, the tested germplasm resources were effectively categorized, and 11 genotypes were identified as suitable for cultivation in arid areas. Overall, the comprehensive evaluation method based on the newly developed stress indices surpasses the traditional method in screening drought adaptability of crops and serves as a vital tool for identifying high-stability and high-production capacities genotypes in various environments, which is expected to provide practical guidance for barley planting and breeding in arid areas.

Developing GA-FuL: A Generic Wide-Purpose Library for Computing with Geometric Algebra

The Geometric Algebra Fulcrum Library (GA-FuL) version 1.0 is introduced in this paper as a comprehensive computational library for geometric algebra (GA) and Clifford algebra (CA), in addition to other classical algebras. As a sophisticated software system, GA-FuL is useful for practical applications requiring numerical or symbolic prototyping, optimized code generation, and geometric visualization. A comprehensive overview of the GA-FuL design is provided, including its core design intentions, data-driven programming characteristics, and extensible layered design. The library is capable of representing and manipulating sparse multivectors of any dimension, scalar kind, or metric signature, including conformal and projective geometric algebras. Several practical and illustrative use cases of the library are provided to highlight its potential for mathematical, scientific, and engineering applications. The metaprogramming code optimization capabilities of GA-FuL are found to be unique among other software systems. This allows for the automated production of highly efficient code, based on powerful geometric modeling formulations provided by geometric algebra.

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

Geometric algebra and cosine-function based variable step-size adaptive filtering algorithms

Geometric algebra and cosine-function based variable step-size adaptive filtering algorithms