Geometric Algebra Research Articles

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.

Identifying Drought Tolerant Finger Millet Landraces for the Hills of Nepal

Finger millet is in the fourth rank among the major cereals in Nepal after rice, maize and wheat. Since it is cultivated in rainfed conditions, drought is one of the vital production constraints for this crop. With the objective of enhancing the utilization of native landraces conserved ex situ in genebank, 300 finger millet accessions were evaluated under drought stress and normal conditions at Khumaltar (1,360 masl), Lalitpur, Nepal during 2017 and 2018. In both the years, genotypes were significantly different for the yield and other agronomic characters. Eight different drought tolerant indices namely drought resistance index, geometric mean productivity, harmonic mean, mean productivity, stress susceptibility index, tolerance, yield index and yield stability index for each genotype were calculated using grain yield data recorded from the stress (Yds) and non-stress (Yns) experiment. Majority of the landraces were grouped under moderately susceptible group in both the years followed by moderately tolerant and susceptible group based on all the drought tolerant indices (DTIs). Five landraces, namely NGRC4849, NGRC6487, NGRC04852, NGRC03491 and NGRC6490 with average grain yield of 4670, 3624, 3426, 3191 and 3132 kg/ha, respectively, were identified as stable drought tolerant landraces compared to the released varieties. These landraces could be the potential sources of drought tolerance for finger millet improvement program for the mid-hills of Nepal.

Point Pair-Based Expression of Cutter Swept Envelopes in Five-Axis Milling

This paper proposes an efficient method of formulating cutter swept envelopes in five-axis milling based on conformal geometric algebra. The surface of the rotary cutter is represented as a canal surface, as the envelope of a one-parameter family of spheres. The swept envelopes generated by the general rigid motion of the cutter can be regarded as the envelopes of a two-parameter family of spheres. With the help of the dual representations of spheres and planes in conformal space, a point pair-based expression of the swept envelopes is formulated analytically, which represents two surfaces uniformly. By using the present method, the swept surfaces of a conical cutter, a drum-shaped cutter, and a toroidal cutter are illustrated. As an application, two types of collision detection methods for 5-axis milling are presented based on the point pair-based envelope expression. The benchmark examples demonstrate that the use of the proposed envelope expression significantly improves computational efficiency.

A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure

A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure

Connections between matrix products for 3-vectors and geometric algebra

Connections between matrix products for 3-vectors and geometric algebra

A comparison of geometric algebra and harmonic domain for linear circuit analysis

A comparison of geometric algebra and harmonic domain for linear circuit analysis

A Machine Proof System of Point Geometry Based on Coq

An important development in geometric algebra in recent years is the new system known as point geometry, which treats points as direct objects of operations and considerably simplifies the process of geometric reasoning. In this paper, we provide a complete formal description of the point geometry theory architecture and give a rigorous and reliable formal verification of the point geometry theory based on the theorem prover Coq. Simultaneously, a series of tactics are also designed to assist in the proof of geometric propositions. Based on the theoretical architecture and proof tactics, a universal and scalable interactive point geometry machine proof system, PointGeo, is built. In this system, any arbitrary point-geometry-solvable geometric statement may be proven, along with readable information about the solution’s procedure. Additionally, users may augment the rule base by adding trustworthy rules as needed for certain issues. The implementation of the system expands the library of Coq resources on geometric algebra, which will become a significant research foundation for the fields of geometric algebra, computer science, mathematics education, and other related fields.

The noncommutative geometry of matrix polynomial algebras

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension.

A new derivation of the Minkowski metric

The four dimensional spacetime continuum, as first conceived by Minkowski, has become the dominant framework within which to describe physical laws. In this paper, we show how this four-dimensional structure is a natural property of physical three-dimensional space, if modeled with Clifford geometric algebra . We find that Minkowski spacetime can be embedded within a larger eight dimensional structure. This then allows a generalisation of the invariant interval and the Lorentz transformations. Also, with this geometric oriented approach the fixed speed of light, the laws of special relativity and a generalised form of Maxwell’s equations, arise naturally from the intrinsic properties of the algebra without recourse to physical arguments. We also find new insights into the nature of time, which can be described as two-dimensional. Some philosophical implications of this approach as it relates to the foundations of physical theories are also discussed.

Determination of Drought Tolerance Indices as Selection Criteria of Rice Genotypes under Water Deficit Conditions in Egypt

The development of rice genotypes with impressive tolerance to abiotic stress is one of the primary objectives of rice breeding programs. The current study was carried out to assess the capability of different indices to identify rice genotypes that are drought-tolerant under Egyptian conditions during the two growing seasons 2018 and 2019. Thirteen drought tolerance indices were calculated using the grain yield of twenty rice genotypes. According to the combined analysis of variance for grain yield, there were highly significant differences between genotypes (G), irrigation regimes (I), and their interactions. All drought tolerance indices, as well as grain yields under non-stress and stress conditions (Yp and Ys), exhibited significant variations among genotypes. For Yp, Ys, and all indices assessed, estimates of broad sense heritability (h2) and genetic advance as a percentage of mean (GAM%) were high. Using mean performances, drought tolerance indices, and multivariate analysis, the genotypes Giza 179, Sakha 104, IET 1444, and GZ 6296-12-1-2-1 were superior genotypes under both stress and non-stress conditions, as were the genotypes Sakha 107, IRAT 170, IR 68552-55-3-2, WAB 56-104, IRGA 318-11-6-2-6, and NERICA -4 under stress conditions. Hence, it was suggested that they can be utilized as parents in hybridization programs that aim to improve the drought tolerance of other rice genotypes in Egypt. Furthermore, mean productivity index (MP), geometric mean productivity (GMP), stress tolerance index (STI), harmonic mean (HM), and yield index (YI) were the tolerance indices that can be classified as better predictors of drought tolerance considering the yield potentials of the genotypes.