561 research outputs found
The protective effect of exercise on diabetic nephropathy : the role of histidine-containing dipeptides
C
Hilbert transforms in Clifford analysis
The Hilbert transform on the real line has applications in many fields. In particular in one–dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature
Rarita-Schwinger Type Operators on Spheres and Real Projective Space
In this paper we deal with Rarita-Schwinger type operators on spheres and
real projective space. First we define the spherical Rarita-Schwinger type
operators and construct their fundamental solutions. Then we establish that the
projection operators appearing in the spherical Rarita-Schwinger type operators
and the spherical Rarita-Schwinger type equations are conformally invariant
under the Cayley transformation. Further, we obtain some basic integral
formulas related to the spherical Rarita-Schwinger type operators. Second, we
define the Rarita-Schwinger type operators on the real projective space and
construct their kernels and Cauchy integral formulas.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1106.358
Introductory clifford analysis
In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications
Segal-Bargmann-Fock modules of monogenic functions
In this paper we introduce the classical Segal-Bargmann transform starting
from the basis of Hermite polynomials and extend it to Clifford algebra-valued
functions. Then we apply the results to monogenic functions and prove that the
Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform
for monogenic functionals. This kernel is also the reproducing kernel for the
monogenic Bargmann module.Comment: 11 page
Harmonic analysis and hypercomplex function theory in co-dimension one
Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over â„ťn+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.The work of the first, second and fourth authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for
Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013
Hermite and Gegenbauer polynomials in superspace using Clifford analysis
The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard
Clifford analysis are generalized to the new framework of Clifford analysis in
superspace in a merely symbolic way. This means that one does not a priori need
an integration theory in superspace. Furthermore a lot of basic properties,
such as orthogonality relations, differential equations and recursion formulae
are proven. Finally, an interesting physical application of the super
Clifford-Hermite polynomials is discussed, thus giving an interpretation to the
super-dimension.Comment: 18 pages, accepted for publication in J. Phys.
The protective effect of exercise on diabetic nephropathy : the role of histidyl containing dipeptides
- …