Erratum: Evaluation of a class of integrals occurring in mathematical physics via a higher order generalization of the principal value

1990 ◽  
Vol 68 (6) ◽  
pp. 549-549
Author(s):  
K. T. R. Davies ◽  
R. W. Davies
Keyword(s):  
Mathematical Physics ◽  
Higher Order ◽  
Principal Value

Evaluation of a class of integrals occurring in mathematical physics via a higher order generalization of the principal value

1989 ◽  
Vol 67 (8) ◽  
pp. 759-765 ◽  
Author(s):  
K. T. R. Davies ◽  
R. W. Davies
Keyword(s):  
Single Particle ◽  
Numerical Evaluation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Simple Algorithm ◽  
Mathematical Physics ◽  
Higher Order ◽  
Divergent Integral ◽  
Principal Value

The notion of the principal value of an integral is generalized to treat higher order singularities. The principal value of an integral can be considered the "convergent part" of a divergent integral, an interpretation that is almost trivial for simple poles, but more meaningful for higher order poles. Application of this concept leads to a simple algorithm that may be applied to the evaluation of a class of integrals arising in mathematical physics. Many of these integrals frequently occur in the analytic and numerical evaluation of folding functions arising from the product of single-particle Green's functions.

scholarly journals A Note on the Fractional Generalized Higher Order KdV Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Yongyi Gu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Nonlinear Differential Equations ◽  
Mathematical Physics ◽  
Higher Order ◽  
Complex Method ◽  
Applied Method ◽  
De Vries

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.

THE MATHEMATICS OF PRINCIPAL VALUE INTEGRALS AND APPLICATIONS TO NUCLEAR PHYSICS, TRANSPORT THEORY, AND CONDENSED MATTER PHYSICS

1996 ◽  
Vol 06 (06) ◽  
pp. 833-885 ◽  
Author(s):  
K.T.R. DAVIES ◽  
M.L. GLASSER ◽  
V. PROTOPOPESCU ◽  
FRANK TABAKIN
Keyword(s):  
Nuclear Physics ◽  
Transport Theory ◽  
Condensed Matter ◽  
Higher Order ◽  
Condensed Matter Physics ◽  
Matrix Formalism ◽  
Principal Value ◽  
Multiple Integrals ◽  
T Matrix ◽  
Double Integrals

A review of developments in the mathematics and methods for principal value (PV) integrals is presented. These topics include single-pole formulas for simple and higher-order PVs, simple and higher-order poles in double integrals, and products of simple poles in general multiple integrals. Two generalizations of the famous Poincaré-Bertrand (PB) theorem are studied. We then review the following topics: dispersion relations for the advanced, retarded, and causal Green’s functions; Titchmarsh’s theorem; applications of the PB theorem to two- and three-particle loop integrals; and the R and T matrix formalism. Also, various applications of the PV methods to nuclear physics, transport theory, and condensed matter physics are studied. In the appendices several methods for evaluating PV integrals, including the Haftel-Tabakin procedure for calculating the R and T matrices, are reviewed.

scholarly journals Stable Optical Solitons for the Higher-Order Non-Kerr NLSE via the Modified Simple Equation Method

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1986
Author(s):  
Noha M. Rasheed ◽  
Mohammed O. Al-Amr ◽  
Emad A. Az-Zo’bi ◽  
Mohammad A. Tashtoush ◽  
Lanre Akinyemi
Keyword(s):  
Short Pulse ◽  
Soliton Solutions ◽  
Kerr Nonlinearity ◽  
Mathematical Physics ◽  
Higher Order ◽  
Simple Equation ◽  
Equation Method ◽  
Computational Work ◽  
The Difference ◽  
Modified Simple Equation Method

This paper studies the propagation of the short pulse optics model governed by the higher-order nonlinear Schrödinger equation (NLSE) with non-Kerr nonlinearity. Exact one-soliton solutions are derived for a generalized case of the NLSE with the aid of software symbolic computations. The modified Kudryashov simple equation method (MSEM) is employed for this purpose under some parametric constraints. The computational work shows the difference, effectiveness, reliability, and power of the considered scheme. This method can treat several complex higher-order NLSEs that arise in mathematical physics. Graphical illustrations of some obtained solitons are presented.

scholarly journals Higher Order Riesz Transforms in the Ultraspherical Setting as Principal Value Integral Operators

2011 ◽  
Vol 70 (4) ◽  
pp. 511-539 ◽  
Author(s):  
Jorge J. Betancor ◽  
Juan C. Fariña ◽  
Lourdes Rodríguez-Mesa ◽  
Ricardo Testoni
Keyword(s):  
Integral Operators ◽  
Higher Order ◽  
Riesz Transforms ◽  
Principal Value

scholarly journals Transformation formula of higher order integrals

2000 ◽  
Vol 68 (2) ◽  
pp. 155-164 ◽  
Author(s):  
Tao Qian ◽  
Tongde Zhong
Keyword(s):  
Singular Integrals ◽  
Higher Order ◽  
Transformation Formula ◽  
Integration By Parts ◽  
Hadamard Principal Value ◽  
Principal Value ◽  
Stokes Formula ◽  
Definition Of

AbstractBy using integration by parts and Stokes' formula the authors give a new definition of the Hadamard principal value of higher order singular integrals on the complex hypersphere in Cn. Then the transformation formula for the higher order singular integrals is deduced.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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