Functions of Multivector Variables

Chappell, James M.; Iqbal, Azhar; Gunn, Lachlan J.; Abbott, Derek

doi:10.1371/journal.pone.0116943

Cited by 23 publications

(41 citation statements)

References 10 publications

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“…The understanding and investigation of square roots in noncommutative CAs is still in its infancy. The published formulas are not complete or even erroneous in case of a general MV [7]. The most akin to present results are the investigations of conditions for existence of square root of −1 [12,13,22] in the Clifford-Fourier transforms and CAbased wavelet theory [14].…”

Section: Introductionmentioning

confidence: 93%

“…More important is the fact that the CA allows to extend analysis beyond complex numbers [5,13]. In paper [7], along with functions for MVs, the authors provide several formulas for square roots in Cl 3,0 . However, the general formula in [7] as shown in this article is incorrect.…”

Section: Introductionmentioning

confidence: 99%

“…In paper [7], along with functions for MVs, the authors provide several formulas for square roots in Cl 3,0 . However, the general formula in [7] as shown in this article is incorrect.…”

Section: Introductionmentioning

confidence: 99%

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Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl 3,0, Cl 2,1, Cl 1,2 and Cl 0,3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij .

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

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“…In this approach flow is computed separately in channel zones with differing flow resistance, usually the main channel and floodplains. The momentum exchange between areas of the higher and lower stream velocity, the socalled kinematic effect, is represented by rough imaginary walls at the interfaces (Sellin, 1964;Kubrak et al, 2019a, b). Despite the well-known limitations of the DCM (Myers, 1978;Fread, 1989;Soong and DePue, 1996;Pasche, 2007), the Manning formula is presently the basis for the majority of practical models for flood hazard assessments, design of hydraulic structures, and water management (Shields et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

“…Several approaches providing a physically based characterization of vegetation and the flow-vegetation interactions are available for straightforward one-dimensional discharge capacity assessments in small-to medium-sized vegetated channels. In these models, vegetation can be represented as rigid or flexible, interacting with water streams as submerged and emergent (Shields et al, 2017). There are many methods explaining each of these types of vegetation, and a comprehensive review can be found in Aberle and Järvelä (2013).…”

Section: Introductionmentioning

confidence: 99%

Predicting discharge capacity of vegetated compound channels: uncertainty and identifiability of one-dimensional process-based models

Kiczko

Västilä

Kozioł

et al. 2020

Hydrol. Earth Syst. Sci.

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Abstract. Despite the development of advanced process-based methods for estimating the discharge capacity of vegetated river channels, most of the practical one-dimensional modeling is based on a relatively simple divided channel method (DCM) with the Manning flow resistance formula. This study is motivated by the need to improve the reliability of modeling in practical applications while acknowledging the limitations on the availability of data on vegetation properties and related parameters required by the process-based methods. We investigate whether the advanced methods can be applied to modeling of vegetated compound channels by identifying the missing characteristics as parameters through the formulation of an inverse problem. Six models of channel discharge capacity are compared in respect of their uncertainty using a probabilistic approach. The model with the lowest estimated uncertainty in explaining differences between computed and observed values is considered the most favorable. Calculations were performed for flume and field settings varying in floodplain vegetation submergence, density, and flexibility, and in hydraulic conditions. The output uncertainty, estimated on the basis of a Bayes approach, was analyzed for a varying number of observation points, demonstrating the significance of the parameter equifinality. The results showed that very reliable predictions with low uncertainties can be obtained for process-based methods with a large number of parameters. The equifinality affects the parameter identification but not the uncertainty of a model. The best performance for sparse, emergent, rigid vegetation was obtained with the Mertens method and for dense, flexible vegetation with a simplified two-layer method, while a generalized two-layer model with a description of the plant flexibility was the most universally applicable to different vegetative conditions. In many cases, the Manning-based DCM performed satisfactorily but could not be reliably extrapolated to higher flows.

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On factorization of multivectors in Cl(2,1)$$ Cl\left(2,1\right) $$, by exponentials and idempotents

Hitzer

2022

Math Methods in App Sciences

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Functions of Multivector Variables

Cited by 23 publications

References 10 publications

Square root of a multivector in 3D Clifford algebras

Square root of a multivector in 3D Clifford algebras

Predicting discharge capacity of vegetated compound channels: uncertainty and identifiability of one-dimensional process-based models

On factorization of multivectors in Cl(2,1)$$ Cl\left(2,1\right) $$, by exponentials and idempotents

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