The parabolic analytic functions and the derivative of real functions

Catoni, F.; Cannata, Roberto; Nichelatti, E.

doi:10.1007/s00006-004-0010-8

Cited by 8 publications

(12 citation statements)

References 5 publications

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“…There is also a recent interest to this topic in different areas: differential geometry [7,11,12,21,8,2], modal logic [54], quantum mechanics [28,29,57,49], space-time geometry [10,25,26,58,24,22,61], hypercomplex analysis [13,19,20]. A brief history of the topic can be found in [12] and further references are provided in the above papers.…”

Section: Background and Historymentioning

confidence: 99%

“…We already know that a similarity of a cycle with another cycle is a new cycle (4.8). The inner product of later with a third given cycle form a joint invariant of those three cycles: 11) which is build from the second-order invariant ·, · . Now we can reduce the order of this invariant by fixing C s σ3 be the real line (which is itself invariant).…”

Section: Focal Orthogonalitymentioning

confidence: 99%

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Erlangen Program at Large-1: Geometry of Invariants

Kisil¹

2010

SIGMA

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Abstract. This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL 2 (R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore-Springer-Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach.

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Section: Background and Historymentioning

confidence: 99%

Section: Focal Orthogonalitymentioning

confidence: 99%

Erlangen Program at Large-1: Geometry of Invariants

Kisil¹

2010

SIGMA

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“…In the limit ∆ → 0, the above equations give the following expression for parabolic numbers [11,13]:…”

Section: Example: Functions Of a Bidimensional Hypercomplex Variablementioning

confidence: 99%

“…[13], it has been shown that the definition of functions of a parabolic variable allows demonstrating, by means of plain algebra, the main theorems concerning the derivative of functions of a real variable. A specialization of Eqs.…”

Section: Example: Functions Of a Bidimensional Hypercomplex Variablementioning

confidence: 99%

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Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

Catoni

Cannata

Nichelatti

et al. 2005

Adv. Appl. Clifford Algebras

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Systems of hypercomplex numbers, which had been studied and developed at the end of the 19 th century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just before one of the fundamental discoveries of the 20 th century, Einstein's equivalence between space and time. Owing to this equivalence, not-defined quadratic forms have got concrete physical meaning and have been recently recognized to be in strong relationship with a system of bidimensional hypercomplex numbers. These numbers (called hyperbolic) can be considered as the most suitable mathematic language for describing the bidimensional space-time, in spite of some unfamiliar algebraic properties common to all the commutative hypercomplex systems with more than two dimensions: they are decomposable systems and there are non-zero numbers whose product is zero. With respect to the famous Hamilton quaternions, one can introduce the differential calculus for the hyperbolic numbers and for all the commutative hypercomplex systems; moreover, one can even introduce functions of hypercomplex variable.The aim of this work is to study the systems of commutative hypercomplex numbers and the functions of hypercomplex variable by describing them in terms of a familiar mathematical tool, i.e. matrix algebra.

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Erlangen Program at Large: An Overview

Kisil¹

2012

Advances in Applied Analysis

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ABSTRACT. This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the group SL 2 (R). Starting from the conformal geometry we develop analytic functions and apply these to functional calculus. Finally we link this to quantum mechanics and conclude by a list of open problems.

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The parabolic analytic functions and the derivative of real functions

Cited by 8 publications

References 5 publications

Erlangen Program at Large-1: Geometry of Invariants

Erlangen Program at Large-1: Geometry of Invariants

Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

Erlangen Program at Large: An Overview

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