A Commutative Hypercomplex Algebra with Associated Function Theory

1996 ◽  
pp. 213-227 ◽  
Author(s):  
Clyde M. Davenport
Keyword(s):  
Function Theory ◽  
Associated Function ◽  
Hypercomplex Algebra

22.—Analytic, Generalized, Hyperanalytic Function Theory and an Application to Elasticity

1975 ◽  
Vol 73 ◽  
pp. 317-331 ◽  
Author(s):  
Robert P. Gilbert ◽  
Wolfgang L. Wendland
Keyword(s):  
Integral Equations ◽  
Open Problem ◽  
Function Theory ◽  
Elliptic Systems ◽  
Representation Formula ◽  
Volterra Integral Equations ◽  
First Order ◽  
Elasticity Problems ◽  
Hypercomplex Algebra

SynopsisThough it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

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.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

scholarly journals Evaluation of Intraocular Pressure and Other Biomechanical Parameters to Distinguish between Subclinical Keratoconus and Healthy Corneas

2021 ◽  
Vol 10 (9) ◽  
pp. 1905
Author(s):  
Cristina Peris-Martínez ◽  
María Amparo Díez-Ajenjo ◽  
María Carmen García-Domene ◽  
María Dolores Pinazo-Durán ◽  
María José Luque-Cobija ◽  
...  
Keyword(s):  
Function Theory ◽  
Rank Test ◽  
Maximum Deformation ◽  
Corrected Visual Acuity ◽  
Signed Rank ◽  
Signed Rank Test ◽  
Biomechanical Parameters ◽  
Spss Software ◽  
Biomechanical Measurements ◽  
Noncontact Tonometry

(1) Purpose: To assess the main corneal response differences between normal and subclinical keratoconus (SCKC) with a Corvis® ST device. (2) Material and Methods: We selected 183 eyes of normal patients, of a mean age of 33 ± 9 years and 16 eyes of patients with SCKC of a similar mean age. We measured best corrected visual acuity (BCVA) and corneal topography with a Pentacam HD device to select the SCKC group. Biomechanical measurements were performed using the Corvis® ST device. We carried out a non-parametric analysis of the data with SPSS software (Wilcoxon signed rank-test). (3) Results: We found statistically significant differences between the control and SCKC groups in some corneal biomechanical parameters: first and second applanation time (p = 0.05 and p = 0.02), maximum deformation amplitude (p = 0.016), highest concavity radius (p = 0.007), and second applanation length and corneal velocity ((p = 0.039 and p = 0.016). (4) Conclusions: Our results show that the use of normalised biomechanical parameters provided by noncontact tonometry, combined with a discriminant function theory, is a useful tool for detecting subclinical keratoconus.

scholarly journals Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

2021 ◽  
pp. 1-26
Author(s):  
ELENA CHERKAEV ◽  
MINWOO KIM ◽  
MIKYOUNG LIM
Keyword(s):  
Composite Materials ◽  
Series Expansion ◽  
Function Theory ◽  
Effective Conductivity ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Geometric Function Theory ◽  
Boundary Integral Formulation ◽  
Geometric Function ◽  
Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Sign in / Sign up

Export Citation Format

Share Document