On solutions of a discretized heat equation in discrete Clifford analysis

Baaske, Franka; Bernstein, Swanhild; Ridder, Hilde De; Sommen, Franciscus

doi:10.1080/10236198.2013.831407

Cited by 13 publications

(42 citation statements)

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“…'.w/. The restriction of the Cayley map (12) to the subspaces Re j e nCj (j D 1, 2, : : : , n) of ƒ 2 .R n,n / will be of special interest in the proof of Proposition 2.1, mainly because for the (paravector) subspaces H n 1,n j of H n 1,n the map ' : Re j e nCj ! H n 1,n j is evidently onto.…”

Section: Compactification Through the Cayley Mapmentioning

confidence: 99%

A conformal group approach to the Dirac–Kähler system on the lattice

Faustino

2017

Math Methods in App Sciences

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Abstract. Starting from the representation of the (n − 1) + n−dimensional Lorentz pseudo-sphere on the projective space PR n,n , we propose a method to derive a class of solutions underlying to a Dirac-Kähler type equation on the lattice. We make use of the Cayley transform ϕ(w) = 1 + w 1 − w to show that the resulting group representation arise from the same mathematical framework as the conformal group representation in terms of the general linear group GL (2, Γ(n − 1, n − 1) ∪ {0}). That allows us to describe such class of solutions as a commutative n−ary product, involving the quasi-monomials ϕ (z j ) − x j h (x j ∈ hZ) with membership in the paravector space R ⊕ Re j e n+j .

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Section: Compactification Through the Cayley Mapmentioning

confidence: 99%

A conformal group approach to the Dirac–Kähler system on the lattice

Faustino

2017

Math Methods in App Sciences

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“…At the same time there has been interest in studying evolution problems in the context of hypercomplex variables, namely discretized variants for the heat equation (cf. [1]) and for the Cauchy-Kovaleskaya extension (cf. [8]).…”

Section: Introductionmentioning

confidence: 99%

Relativistic Wave Equations on the Lattice: An Operational Perspective

Faustino

2019

Trends in Mathematics

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Dedicated to Professor Wolfgang Sprößig on occasion of his 70th birthday.Abstract. This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolutiontype operador from the knowledge of the Exponential Generating Function (EGF), carrying a degree lowering operator Lt = L(∂t). We also use certain operational properties of the discrete Fourier transform over the n−dimensional Brioullin zone Q h = − π h , π h n -a toroidal Fourier transform in disguise -to describe the discrete counterparts of the continuum wave propagators,∆ − m 2 respectively, as discrete convolution operators. In this way, a huge class of discretized time-evolution problems of differential-difference and difference-difference type may be studied in the spirit of hypercomplex variables.

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“…A first discretization is a discrete Fueter's theorem, allowing A topic for future research and a nice application would involve a factorization of the discrete Heat equation (see [12]) by means of a parabolic Dirac operator, which may be considered as an extension of the Dirac operator with an additional time dimension, and applying (a slightly modified version of ) Fueters theorem to obtain solutions to the first-order Clifford-Heat equation. Moreover, the use of axially monogenic special functions such as the Clifford-Hermite polynomials may be a starting point for the study of wavelet transforms in the discrete setting, as has been performed in the continuous setting in [15].…”

Section: Conclusion and Future Researchmentioning

confidence: 99%

“…The former, in turn, lead to discrete versions of the spherical Bessel function, which appears in the theory of the Helmholtz operator, while the latter play a part in solutions to the discrete Heat equation (e.g. [12]), which can be interpreted as an extension of the Dirac equation with an additional time dimension. Clifford-Hermite polynomials are also being used in the definition of continuous wavelet transforms.…”

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confidence: 99%

Fueter's theorem in discrete Clifford analysis

Ridder

Sommen

2015

Math Methods in App Sciences

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In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector e j is split into a forward and backward basis vector: e j D e C j C e j . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f. 0 , 1 / left-monogenic in two variables 0 and 1 and for a left-monogenic P k . /, the m-dimensional function kC m 1 2 f. 0 , /P k . / is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua-type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial-exponential functions and the discrete Clifford-Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.Keywords: discrete Clifford analysis; axial monogenicity; Fueter theorem IntroductionIn 1935, the Swiss mathematician R. Fueter described in his paper This classical result was later generalized to R 0,m for m odd by Sce [2], for m even by Qian [3,4], and in [5], Sce's result was generalized by Sommen as follows: if m is an odd positive integer and P k .x/ is a homogeneous monogenic polynomial of degree k in R m , then is also monogenic in Q . The goal of this paper is to discretize Sommen's result to the setting of discrete Clifford analysis. Clifford analysis is a natural generalization of complex analysis to higher dimensions. For a detailed description, we refer to [6][7][8]. For the sake of completion, we recall some of the basic notions. It is constructed by associating to the standard Euclidean space R m the real Clifford algebra R 0,m , generated by the canonical basis e 1 , : : : , e m . These generators satisfy the multiplication rule e i e j C e j e i D 2 ı ij . For a basis of R 0,m , we consider for each set A D fj 1 , : : : , j k g Â f1, : : : , mg the element e A D e j1 : : : e jk with 1 6 j 1 < j 2 < : : : < j k 6 m, together with e ; D 1, the identity element.The Euclidean space R m is embedded in R 0,m by identifying the point . Discrete Clifford analysis [9][10][11] has emerged in recent years as both a direct discretization of Euclidean Clifford analysis and a higherdimensional analogue of discrete complex analysis. There are, although, various choices related to the construction of a discrete Dirac operator. We work in the so-called discrete Hermitian setting, where the use of both forward and backward differences in the definition of the discrete Dirac operator @ translates into a direct factorization of the discrete (star) Laplacian: D @ 2 . As in Euclidean Clifford analysis, the construction of discrete monogenic functions, that is, functions in the kernel of the discrete Dirac operator, is a central topic in discrete Clifford analysis, and various techniques have already been developed or directly discretized: a discrete CauchyKovalevskaya extension theorem, a discrete Fischer decompo...

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On solutions of a discretized heat equation in discrete Clifford analysis

Cited by 13 publications

References 13 publications

A conformal group approach to the Dirac–Kähler system on the lattice

A conformal group approach to the Dirac–Kähler system on the lattice

Relativistic Wave Equations on the Lattice: An Operational Perspective

Fueter's theorem in discrete Clifford analysis

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