Geometric Algebra Research Articles

Primal quantization postulate for quantum fields

We formulate a theory of quantum fields underpinned solely by the full geometric algebra structure generated by the vector space of spacetime. A modified dynamical equation of a field with associated background field potentials, distinct from Dirac’s equation, is developed and corrects the inertial potential that acts on the field. A discrete wavenumber spectrum with a possible correlation to the mass spectrum of elementary particles is obtained, and explicit expressions are derived. The formulation is mathematically driven and constitutes what we refer to as the primal field quantization postulate. While manifestly a relativistic quantum field theory, our formulation may provide a natural account of mass hierarchy and family generation. A fourth family of elementary particles is predicted, indicating that lepton and quark oscillations across generation members, as well as quantum decay processes and fluctuations, may need modifying.

An Ordered Tuple Construction of Geometric Algebras

An Ordered Tuple Construction of Geometric Algebras

Analysis of genetic diversity and genome-wide association study for drought tolerance related traits in Iranian bread wheat

BackgroundDrought is most likely the most significant abiotic stress affecting wheat yield. The discovery of drought-tolerant genotypes is a promising strategy for dealing with the world’s rapidly diminishing water resources and growing population. A genome-wide association study (GWAS) was conducted on 298 Iranian bread wheat landraces and cultivars to investigate the genetic basis of yield, yield components, and drought tolerance indices in two cropping seasons (2018–2019 and 2019–2020) under rainfed and well-watered environments.ResultsA heatmap display of hierarchical clustering divided cultivars and landraces into four categories, with high-yielding and drought-tolerant genotypes clustering in the same group. The results of the principal component analysis (PCA) demonstrated that selecting genotypes based on the mean productivity (MP), geometric mean productivity (GMP), harmonic mean (HM), and stress tolerance index (STI) can help achieve high-yield genotypes in the environment. Genome B had the highest number of significant marker pairs in linkage disequilibrium (LD) for both landraces (427,017) and cultivars (370,359). Similar to cultivars, marker pairs on chromosome 4A represented the strongest LD (r2 = 0.32). However, the genomes D, A, and B have the highest LD, respectively. The single-locus mixed linear model (MLM) and multi-locus random-SNP-effect mixed linear model (mrMLM) identified 1711 and 1254 significant marker-trait association (MTAs) (-log10 P > 3) for all traits, respectively. A total of 874 common quantitative trait nucleotides (QTNs) were simultaneously discovered by both MLM and mrMLM methods. Gene ontology revealed that 11, 18, 6, and 11 MTAs were found in protein-coding regions (PCRs) for spike weight (SW), thousand kernel weight (TKW), grain number per spike (GN), and grain yield (GY), respectively.ConclusionThe results identified rich regions of quantitative trait loci (QTL) on Ch. 4A and 5A suggest that these chromosomes are important for drought tolerance and could be used in wheat breeding programs. Furthermore, the findings indicated that landraces studied in Iranian bread wheat germplasm possess valuable alleles, that are responsive to water-limited conditions. This GWAS experiment is one of the few types of research conducted on drought tolerance that can be exploited in the genome-mediated development of novel varieties of wheat.

Evaluation of Four Maize (Zea mays L.) Genotypes Using Drought Tolerance Indices

Drought tolerance is not often considered as an independent trait by breeders. The objective of this study was to evaluate and identify drought tolerant genotypes using eight drought tolerance indices namely the Stress Susceptibility Index (SSI), the YSI, the YR (Yr), yield index (YI), tolerance index (TOL), average productivity (MP), mean geometric productivity (GMP) and stress tolerance index (STI) of maize genotypes (Zea mays L.). A field trial was conducted to evaluate four genotypes during the hot dry season of 2016 and 2017 at the irrigated perimeter of Djirataoua. Drought tolerance indices were calculated on the basis of yield under optimal and stressed conditions. The comparison of the means of drought tolerance demonstrated the effects of drought on yield and showed significant differences between genotypes. The correlation coefficient and principal component analysis showed that the GMP, MP and STI indices were able to discriminate drought-sensitive and tolerant genotypes. Two genotypes CZH131001 and CZH142013 produced high grain yield under both optimal and stressed conditions. Overall, GMP, MP, and STI indices can be used as effectively drought tolerance screening indices and able to identify better genotypes, suitable for both optimal and stress conditions.

An Analytical Method for Sensitivity Analysis of Rigid Multibody System Dynamics Using Projective Geometric Algebra

Abstract The analytical sensitivity analysis, i.e., the analytical first-order partial derivatives of dynamical equations, is one key to improving descent-based optimization methods for motion planning and control of robots. This paper proposes an efficient algorithm that recursively evaluates the analytic gradient of the dynamical equations of a multibody system. The theory of projective geometric algebra (PGA) is used to generate the algorithm. It provides a systemic and geometrically intuitive interpretation for the multibody system dynamics, and the resulting algorithm is highly efficient, with concise formula. The algorithm is first applied to the open-chain system and extended for the cases when kinematic loops are contained. The runtime varying with respect to the degree-of-freedom (DOF) of the system is analyzed. The results are compared with that obtained from the algorithm based on spatial vector algebra (SVA) using open-source matlab codes. A 2DOF serial robot, a 3DOF robot with a kinematic loop and the PUMA560 robot are used for the validation of the minimum-effort motion planning, and it is verified that the proposed algorithm improves the efficiency.

Threshold secret sharing with geometric algebras

In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only “authorized” sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness (“unauthorized” sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras.

Embedding of octonion Fourier transform in geometric algebra of ℝ3 and polar representations of octonion analytic signals in detail

We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three‐dimensional Euclidean space . We apply a new form of dimensionally minimal embedding of octonions in geometric algebra that expresses octonion multiplication nonassociativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas.

Index‐based selection of potato (Solanum tuberosum L.) genotypes for tolerance to soil moisture and ambient heat stress

AbstractDeveloping potato varieties that have enhanced tolerance to soil moisture and high‐temperature stress could improve household food and nutrition security in drought and heat prone areas. A field experiment was conducted at two locations in Ethiopia to evaluate the effectiveness of some stress tolerance indices for the identification of drought and heat tolerant potato genotypes. Treatments consisted of two irrigation regimes and 23 potato genotypes. The experiment was laid out as a split‐plot design in a factorial arrangement with three replications. Irrigation regimes were the main plot factor and genotypes were the sub‐plot factor. Stress tolerance indices were calculated based on the tuber yield under stress (Ys) and non‐stress (Yp) conditions of each genotype. To determine the most desirable stress tolerant genotypes, all indices mean rank (R) and standard deviation of the R were computed. Genotypes Cl‐12(CIP‐304371.67), Cl‐13(CIP‐304383.80), Cl‐2(CIP‐302499.30), Cl‐34(CIP‐397079.6), Cl‐15(CIP‐304394.56), and Cl‐14(CIP‐304387.39) were found to have had top mean ranks and higher tuber yield under drought and heat stress. Ranking methods using all indices best determined desirable drought, heat, and combined drought‐heat tolerant genotypes that consistently coincided with the yield rank of genotypes under a given stress condition. This implies that these indices (stress tolerance index, geometric mean productivity, mean productivity, yield stability index, and tolerance index) could be used in combination to differentiate stress tolerances and susceptible potato genotypes. It is recommended that genotypes that exhibited higher drought and/or heat stress tolerance in this study could be further evaluated in drought and heat prone areas for the development of tolerant potato varieties.

Field theoretic formulation of fluid mechanics according to the geometric algebra

Field theoretic formulation of fluid mechanics according to the geometric algebra

Development of the Method of Averaging in Clifford Geometric Algebras

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli’s theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications.

Индексная оценка засухоустойчивости и адаптивности перспективных сортов диплоидной озимой ржи в контрастных условиях выращивания

Abstract. Winter rye is grown in a variety of soil and climatic conditions, which indicates the evolutionary resistance to the action of abiotic stressors. Due to periodic spring-summer droughts in the conditions of the North-West region of the Russian Federation, the problem of drought tolerance in rye is of particular importance. The purpose of this study was to investigate 13 winter diploid rye collection varieties for drought tolerance and adaptability in the Northwestern region of the Russian Federation and to identify the most promising ones. Methods. Experiments were laid from 2018 to 2022 in a field plot located in the Northwestern region. To determine the drought tolerance of varieties we used indices: average productivity (AP), endurance index (TOL), yield stability index (YSI), drought tolerance index (DI), tolerance index (STI), geometric average productivity (GAP), drought tolerance index (DSI), agronomic drought tolerance (Adt), degree of yield reduction (X3). To calculate adaptability, the following indicators were used: d% - yield range in %, V% - coefficient of variation, Er – response effect, Ca – adaptability coefficient, Ci – intensity coefficient, d t/ha – yield range in t/ha. Results. The study years can be classified as drought years 2018, 2019, and 2022. The average yield in these years ranged from 3.6 t/ha for the variety Niva, to 4.99 t/ha for the variety Lota. Years 2020 and 2021 are favorable, with average yields ranging from 6.3 t/ha for the variety Niva, to 9.9 t/ha for the variety Ofeliya. The most drought-resistant varieties are Lota, Zubrovka, Talisman, Niva, Yaselʼda, the most adaptive are Zubrovka, Era, Ofeliya, Talisman, Yaselʼda. Of the listed methods of determining drought tolerance, we should pay attention to the Yield Stability Index (YSI) and Drought Tolerance Index (DI). To identify adaptability: the coefficient of adaptability (Ca) and the coefficient of intensity (Ci). Scientific novelty. Consists in a comparative assessment of yield drought tolerance and adaptability of winter rye varieties and identifying among them possessing a set of economically useful features and resistant to drought conditions of the North-West region.

Modeling orientational features via geometric algebra for 3D protein coordinates prediction

Protein structure prediction (PSP) is the prediction of the three‐dimensional (3D) folding of a protein (its tertiary structure) starting from its amino acid sequence (its primary structure). The state of the art in PSP is achieved by deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids positively impact the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of geometric algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA‐based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as additional input features to a graph transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle‐based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative and can be (i) closely associated to protein secondary structures and (ii) more easily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.

Indicators for evaluating the resistance of tomato genotypes to low light

ABSTRACT Low light is one of the most important factors limiting tomato (Solanum lycopersicum L.) production. In this experiment, tomato genotypes under low light (winter planting), and adequate light (summer planting) were evaluated in the same greenhouse condition for other environmental conditions where 11 tolerance and susceptibility indices were evaluated in terms of fruit weight per plant produced. Using the rank of each genotype for the calculated indices, the mean rank index was calculated. The highest yield under adequate light was in genotypes “9318,” “9314,” “2216,” “324,” and “5239.” Under low light, genotypes “321,” “9318,” “9314,” “523,” and “324” had the highest fruit yield per plant, respectively. Evaluation of tolerance and sensitivity indices indicated that tolerance indices, geometric mean productivity index, mean relative performance index and relative efficiency index had a positive, significant, correlation with fruit weight per plant in low light and adequate light are the best indicators for selecting genotypes tolerant to low light. Based on these indices, genotypes “9318,” “9314,” “321,” “324,” and “523” had the highest ranks and more tolerance to low light, and might be successfully cultivated in low light areas.

Survey of new applications of geometric algebra

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.

Another Geometric Interpretation of Cramer’s Rule

Summary We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.

On Some Lie Groups in Degenerate Clifford Geometric Algebras

On Some Lie Groups in Degenerate Clifford Geometric Algebras

Wigner–Ville Distribution Associated with Clifford Geometric Algebra Cln,0, n=3(mod 4) Based on Clifford–Fourier Transform

In this study, the Wigner–Ville distribution is associated with the one sided Clifford–Fourier transform over Rn, n = 3(mod 4). Accordingly, several fundamental properties of the WVD-CFT have been established, including non-linearity, the shift property, dilation, the vector differential, the vector derivative, and the powers of τ∈Rn. Moreover, powerful results on the WVD-CFT have been derived such as Parseval’s theorem, convolution theorem, Moyal’s formula, and reconstruction formula. Eventually, we deduce a directional uncertainty principle associated with WVD-CFT. These types of results, as well as methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

Unification of the four forces in the Spin(11,1) geometric algebra

SO(10), or equivalently its covering group Spin(10), is a well-known promising grand unified group that contains the standard-model group. The spinors of the group Spin(N) of rotations in N spacetime dimensions are indexed by a bitcode with [N/2] bits. Fermions in Spin(10) are described by five bits yzrgb, consisting of two weak bits y and z, and three colour bits r, g, b. If a sixth bit t is added, necessary to accommodate a time dimension, then the enlarged Spin(11, 1) algebra contains the standard-model and Dirac algebras as commuting subalgebras, unifying the four forces. The minimal symmetry breaking chain that breaks Spin(11, 1) to the standard model is unique, proceeding via the Pati-Salam group. The minimal Higgs sector is similarly unique, consisting of the dimension 66 adjoint representation of Spin(11, 1); in effect, the scalar Higgs sector matches the vector gauge sector. Although the unified algebra is that of Spin(11, 1), the persistence of the electroweak Higgs field after grand symmetry breaking suggests that the gauge group before grand symmetry breaking is Spin(10, 1), not the full group Spin(11, 1). The running of coupling parameters predicts that the standard model should unify to the Pati-Salam group Spin(4) w × Spin(6) c at 1012 GeV, and thence to Spin(10, 1) at 1015 GeV. The grand Higgs field breaks t-symmetry, can drive cosmological inflation, and generates a large Majorana mass for the right-handed neutrino by flipping its t-bit. The electroweak Higgs field breaks y-symmetry, and generates masses for fermions by flipping their y-bit.

The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a basis‐free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalized to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems ofanalysis of evolution of the entangled quantum states.

Vector properties of geometric calculus

After its introduction by Grossman and Katz, the geometry part of geometric calculus has been mostly left unattended. In this paper, we give the elementary theorems to build the geometry on geometric space. We apply the well‐known theorems for inner product in geometric space. The terms geometric determinant and geometric vector product are introduced, and their properties are studied. We give the definitions of the circle, the plane, and the sphere and visualize them along with a line in geometric space. Finally, we discuss the Gram–Schmidt process in geometric space.