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1.
First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform
on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions
Third, we show a set of important properties of the Clifford Fourier transform on Cl3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties
for proving an uncertainty principle for Cl3,0 multivector functions. 相似文献
2.
Heikki Orelma 《Advances in Applied Clifford Algebras》2012,22(1):143-158
In this paper we express the theory of harmonic differential forms on conformal Euclidean manifolds in terms of the so called
Clifford multivector fields. The aim is to give good definitions for d and d* operators in Clifford multivector case. Using these definitions we derive a formula for the Laplace operator. Three fundamental
examples are included in the end of the paper and connections to existing theory is discussed. 相似文献
3.
Cristina Balderrama Antonio Di Teodoro Adrian Infante 《Advances in Applied Clifford Algebras》2013,23(4):793-813
In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11]. In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order ${n, (D-\lambda)^n, \lambda \in \mathbb{R}}$ , and its conjugate ${(\overline{D} - \lambda)^n}$ in a Clifford algebra depending on parameters ${\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$ . Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications. 相似文献
4.
Min Ku Yingxiong Fu Kähler Uwe Cerejeiras Paula 《Complex Analysis and Operator Theory》2013,7(3):673-693
In this paper we consider the Riemann boundary value problem for null solutions to the iterated Dirac operator over the ball in Clifford analysis with boundary data given in $\mathbb L _{p}\left(1<p<+\infty \right)$ -space. We will use two different ways to derive its solution, one which is based on the Almansi-type decomposition theorem for null solutions to the iterated Dirac operator and a second one based on the poly-Cauchy type integral operator. 相似文献
5.
In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function ${e^{\langle \underline{x}, \underline{u} \rangle}}$ where ${\langle . , . \rangle}$ denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel. 相似文献
6.
Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions,i.e.,null solutions to a first order vector valued rotation invariant differential operator (θ) ca... 相似文献
7.
We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on L p (?+) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L 2(?+) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type. 相似文献
8.
In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{*{20}c} {\overline {DA\left( {x,u\left( x \right),Du\left( x \right)} \right)} = B\left( {x,u\left( x \right),Du\left( x \right)} \right), x \in \Omega ,} \\ {u\left( x \right) = 0, x\partial \Omega } \\ \end{array}$$ where D is a Dirac operator in Euclidean space, u(x) is defined in a bounded Lipschitz domain Ω in ? n and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space W 0 1,p(x) (Ω,C? n ) under appropriate assumptions. 相似文献
9.
This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated
extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how
the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown
to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars
and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case
when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending
on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential
operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove
that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem
of Clifford algebras context, wherein the Clifford and extended Clifford algebras
can be embedded in
which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota’s bracket, relating
it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford
algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using
the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras
from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior
products of 1-forms.
R. da Rocha is supported by CAPES 相似文献
10.
Marco Budinich 《Advances in Applied Clifford Algebras》2012,22(2):283-296
We investigate the properties of the Extended Fock Basis (EFB) of Clifford algebras [1] with which one can replace the traditional multivector expansion of ${\mathcal{C} \ell(g)}$ with an expansion in terms of simple (also: pure) spinors. We show that a Clifford algebra with 2m generators is the direct sum of 2 m spinor subspaces S characterized as being left eigenvectors of ??; furthermore we prove that the well known isomorphism between simple spinors and totally null planes holds only within one of these spinor subspaces. We also show a new symmetry between spinor and vector spaces: similarly to a vector space of dimension 2m that contains totally null planes of maximal dimension m, also a spinor space of dimension 2 m contains ??totally simple planes??, subspaces made entirely of simple spinors, of maximal dimension m. 相似文献
11.
In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere \(S^{n+1}(1)\). We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to \(S^1(r)\times S^{n-1}(\sqrt{1-r^2})\) when \(H\ne 0\) or \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to a Clifford torus \( S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right) \), for \(k=1, 2, \ldots , n-1\) when \(H=0\). 相似文献
12.
F. Brackx H. De Schepper N. De Schepper F. Sommen 《Advances in Applied Clifford Algebras》2007,17(3):311-330
Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued
Dirac operator
, where (
) forms an orthogonal basis for the quadratic space
underlying the construction of the Clifford algebra
. When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra
, which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses
on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context
and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving
both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials
are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations,
structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition
of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4]. 相似文献
13.
Cheikh Birahim Ndiaye Reiner Michael Schätzle 《Calculus of Variations and Partial Differential Equations》2014,51(1-2):291-314
In our previous work (Ndiaye and Schätzle, 2014), we proved that the flat constant mean curvature tori $$\begin{aligned} T_r := r S^1 \times \sqrt{1 - r^2} S^1 \subseteq S^3 \quad \hbox {for } 0 < r \le 1/\sqrt{2} \end{aligned}$$ minimize the Willmore energy in their conformal class in codimension one when \(r \approx 1 / \sqrt{2}\) , that is \(T_r\) is close to the Clifford torus \(T_{Cliff} = T_{1/\sqrt{2}}\) . In this article, we extend this to arbitrary codimension. Moreover we prove that the Clifford torus minimizes the Willmore energy in an open neighbourhood of its conformal class, again in arbitrary codimension, but the neighbourhood may depend on the codimension. 相似文献
14.
We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials. 相似文献
15.
We examine the operator algebra
behind the boundary integral equation method for solving transmission problems. A new type of boundary integral operator, the rotation operator, is introduced, which is more appropriate than operators of double layer type for solving transmission problems for first order elliptic partial differential equations. We give a general invertibility criteria for operators in
by defining a Clifford algebra valued Gelfand transform on
. The general theory is applied to transmission problems with strongly Lipschitz interfaces for the two classical elliptic operators
and . We here use Rellich techniques in a new way to estimate the full complex spectrum of the boundary integral operators. For
we use the associated rotation operator to solve the Hilbert boundary value problem and a Riemann type transmission problem. For the Helmholtz equation, we demonstrate how Rellich estimates give an angular spectral estimate on the rotation operator, which with the general spectral mapping properties in
translates to a hyperbolic spectral estimate for the double layer potential operator. 相似文献
16.
In this note we give an example of an ∞-hyponormal operator T whose Aluthge transform
is not (1+ɛ)-hyponormal for any ɛ > 0 and show that the sequence
of interated Aluthge transforms of T need not converge in the weak operator topology, which solve two problems in [6]. 相似文献
17.
In [17] the third author presented Moebius geometry for sub-manifolds in Sn and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in Sm+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S3, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sn. 相似文献
18.
Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions. 相似文献
19.
In this work we write the monopole equations on
which are stated in [1]. To do this we use representations of real and complex Clifford algebras and we give some explicit
solutions for these equations. 相似文献
20.
Sirkka-Liisa Eriksson Heikki Orelma Frank Sommen 《Complex Analysis and Operator Theory》2016,10(2):251-265
In this paper we consider the solutions of the equation \({\mathcal {M}}_\kappa f=0\), where \({\mathcal {M}}_\kappa \) is the so called modifier Dirac operator acting on functions \(f\) defined in the upper half-space and taking values in the Clifford algebra. We look for solutions \(f(\underline{x},x_{n})\) where the first variable is invariant under rotations. A special type of solution is generated by the so called spherical monogenic functions. These solutions may be characterize by a Vekua-type system and this system may be solved using Bessel functions. We will see that the solution of the equation \({\mathcal {M}}_\kappa f=0\) in this case will be a product of Bessel functions. 相似文献