79 research outputs found
Regular Moebius transformations of the space of quaternions
Let H be the real algebra of quaternions. The notion of regular function of a
quaternionic variable recently presented by G. Gentili and D. C. Struppa
developed into a quite rich theory. Several properties of regular quaternionic
functions are analogous to those of holomorphic functions of one complex
variable, although the diversity of the quaternionic setting introduces new
phenomena. This paper studies regular quaternionic transformations. We first
find a quaternionic analog to the Casorati-Weierstrass theorem and prove that
all regular injective functions from H to itself are affine. In particular, the
group Aut(H) of biregular functions on H coincides with the group of regular
affine transformations. Inspired by the classical quaternionic linear
fractional transformations, we define the regular fractional transformations.
We then show that each regular injective function from the Alexandroff
compactification of H to itself is a regular fractional transformation.
Finally, we study regular Moebius transformations, which map the unit ball B
onto itself. All regular bijections from B to itself prove to be regular
Moebius transformations.Comment: 12 page
Matrix representations of a special polynomial sequence in arbitrary dimension
This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.Fundação para a Ciência e a Tecnologia (FCT
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Four Dimensional Integrable Theories
There exist many four dimensional integrable theories. They include self-dual
gauge and gravity theories, all their extended supersymmetric generalisations,
as well the full (non-self-dual) N=3 super Yang-Mills equations. We review the
harmonic space formulation of the twistor transform for these theories which
yields a method of producing explicit connections and metrics. This formulation
uses the concept of harmonic space analyticity which is closely related to that
of quaternionic analyticity. (Talk by V. Ogievetsky at the G\"ursey Memorial
Conference I, Istanbul, June 1994)Comment: 11 pages, late
Nonlinear Dirac operator and quaternionic analysis
Properties of the Cauchy-Riemann-Fueter equation for maps between
quaternionic manifolds are studied. Spaces of solutions in case of maps from a
K3-surface to the cotangent bundle of a complex projective space are computed.
A relationship between harmonic spinors of a generalized nonlinear Dirac
operator and solutions of the Cauchy-Riemann-Fueter equation are established.Comment: Cosmetic changes onl
From 2D conformal to 4D self-dual theories: quaternionic analyticity
It is shown that self-dual theories generalize to four dimensions both the
conformal and analytic aspects of two-dimensional conformal field theories. In
the harmonic space language there appear several ways to extend complex
analyticity (natural in two dimensions) to quaternionic analyticity (natural in
four dimensions). To be analytic, conformal transformations should be realized
on , which appears as the coset of the complexified conformal group
modulo its maximal parabolic subgroup. In this language one visualizes the
twistor correspondence of Penrose and Ward and consistently formulates the
analyticity of Fueter.Comment: 24 pages, LaTe
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
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 ℝn+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
Hypercomplex polynomials, vietoris’ rational numbers and a related integer numbers sequence
This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For S a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.The work of the first and third 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 PEstOE/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.info:eu-repo/semantics/publishedVersio
Riemann-Hilbert problems for monogenic functions in axially symmetric domains
We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of (D−α)ϕ=0, α∈R, where R denotes the field of real numbers
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