scholarly journals Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

2021 ◽  
Vol 8 (23) ◽  
pp. 281-296
Author(s):  
Fabrizio Colombo ◽  
David Kimsey ◽  
Stefano Pinton ◽  
Irene Sabadini
Keyword(s):  
Functional Calculus ◽  
Clifford Algebras ◽  
Function Theory ◽  
Monogenic Functions ◽  
Content Type ◽  
New Class ◽  
Zero Divisors ◽  
Slice Monogenic Functions ◽  
Class Of Functions ◽  
Clifford Numbers

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S S -functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra R n \mathbb {R}_n . The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.

scholarly journals FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

2021 ◽  
pp. 1-81
Author(s):  
Charles Batty ◽  
Alexander Gomilko ◽  
Yuri Tomilov
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Function Theory ◽  
Operator Norm ◽  
Spectral Mapping ◽  
Related Function ◽  
Norm Estimates ◽  
Sectorial Operators ◽  
Associated Function ◽  
Class Of Functions

Abstract We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

The -spectrum and the -functional calculus

2012 ◽  
Vol 142 (3) ◽  
pp. 479-500 ◽  
Author(s):  
Fabrizio Colombo ◽  
Irene Sabadini
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Resolvent Operator ◽  
Monogenic Functions ◽  
Commuting Operators ◽  
Correct Definition ◽  
Slice Monogenic Functions ◽  
Definition Of ◽  
Integral Version

In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

The effect of clam shell powder on kinetics of water absorption of jute epoxy composite

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Hemalata Jena ◽  
Abinash Panigrahi
Keyword(s):  
Water Absorption ◽  
Environmental Conditions ◽  
Epoxy Composite ◽  
Jute Fibre ◽  
Content Type ◽  
Clam Shell ◽  
New Class ◽  
Kinetics Of ◽  
Shell Powder ◽  
Marine Waste

Purpose Here, attempts have been made to explore the possible use of Marine waste as filler materials into the bio-fibre composites. Clam shell is a type of marine waste which belongs to the class of Bivalvia. It is mainly made of aragonite crystalline polymorphs. This paper aims to develop a new class of natural fibre composite in which jute fibre as reinforcement, epoxy as matrix and clam shell, as particulate microsphere filler. The study investigates the effects of different amounts of clam shell powder on the kinetics of water absorption of jute fibre-reinforced epoxy composite. Two different environmental conditions at room temperature, i.e. distilled water and seawater, are collected for this purpose. Moisture absorption reduces when clam shell is added to the jute-epoxy composite. The curve of water absorption of jute-epoxy composites with filler loading at both environmental conditions follows as Fickian behaviour. Design/methodology/approach Hand lay-up technique to fabricate the composite – Experimental observation Findings The incorporation of Clam shell filler in jute epoxy composite modified the water absorption property of the composite. Hence the present marine waste is an potential filler in jute fibre reinforced polymer composite. Originality/value The paper demonstrates a new class hybrid composite material which uses a marine waste as important phase in the bio-fibre-reinforced composite. It is a new work submitted for original research paper.

3D printing technology in musical instrument research: reviewing the potential

2018 ◽  
Vol 24 (9) ◽  
pp. 1511-1523 ◽  
Author(s):  
Antreas Kantaros ◽  
Olaf Diegel
Keyword(s):  
Additive Manufacturing ◽  
Design Methodology ◽  
Musical Instrument ◽  
Musical Instruments ◽  
Wind Instruments ◽  
Content Type ◽  
New Class ◽  
History Of ◽  
New Applications ◽  
Practical Implications

Purpose This paper aims to discuss additive manufacturing (AM) in the context of applications for musical instruments. It examines the main AM technologies used in musical instruments, goes through a history of musical applications of AM and raises the questions about the application of AM to create completely new wind instruments that would be impossible to produce with conventional manufacturing. Design/methodology/approach A literature research is presented which covers a historical application of AM to musical instruments and hypothesizes on some potential new applications. Findings AM has found extensive application to create conventional musical instruments with unique aesthetics designs. It’s true potential to create entirely new sounds, however, remains largely untapped. Research limitations/implications More research is needed to truly assess the potential of additive manufacturing to create entirely new sounds for musical instrument. Practical implications The application of AM in music could herald an entirely new class of musical instruments with unique sounds. Originality/value This study highlights musical instruments as an unusual application of AM. It highlights the potential of AM to create entirely new sounds, which could create a whole new class of musical instruments.

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

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