scholarly journals Canonical quantization on the half-line and in an interval based upon an alternative concept for the momentum in a space with boundaries

2021 ◽  
Vol 3 (3) ◽  
Author(s):  
M. H. Al-Hashimi ◽  
U.-J. Wiese
Keyword(s):  
Canonical Quantization ◽  
Half Line ◽  
Alternative Concept

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

scholarly journals Existence of solutions for functional boundary value problems at resonance on the half-line

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Bingzhi Sun ◽  
Weihua Jiang
Keyword(s):  
Boundary Value Problems ◽  
Banach Spaces ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Half Line ◽  
Projection Scheme ◽  
At Resonance ◽  
Functional Boundary

Abstract By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with $\operatorname{dim}\operatorname{Ker}L = 1$ dim Ker L = 1 . And an example is given to show that our result here is valid.

Solomon and Saturn I, 89 a, “prologa prim”: An Exercise in Monastic Reading Practice

2020 ◽  
Vol 138 (4) ◽  
pp. 618-648
Author(s):  
Rachel A. Burns
Keyword(s):  
Old English ◽  
Half Line ◽  
Lectio Divina ◽  
Reading Practice ◽  
John 1 ◽  
Grammatical Analysis ◽  
Subsequent Episode ◽  
In The Beginning ◽  
The Devil ◽  
Different Levels

AbstractThis article reassesses the grammatically problematic half-line prologa prima (l. 89a) in the Old English wisdom poem Solomon and Saturn I, and suggests that it ought to be emended to the grammatically viable reading of “prologa prim”. Line 89 a introduces a passage in which the words of the Pater Noster become anthropomorphised as warriors and attack the devil. I will argue that “prologa prim” is an exegetical exercise, informed by grammatical theory and liturgical practice, designed for an audience of monastic readers. This multivalent half-line offers different levels of meaning when read according to different permutations of language and metaphor, in a process analogous to the interpretation of scripture according to the influential model of fourfold exegesis. When read literally, as ‘the first of the initial letters’, “prologa prim” indicates the unfolding and time-bound process of reading. Previous scholars (Anlezark 2009; Anderson 1998) have noted the allusive references in line 89 a to Greek logos (‘word’) and Old English prim (‘first hour’, ‘Prime office’), but not their full significance. Through these allusions, the reader shifts from a literal reading to a spiritual and metaphorical reading of the half-line, achieving a diachronic perspective of the Pater Noster’s recitation across time, and finally an atemporal perspective, reading in line 89 a a paraphrase of John 1:1, “In the beginning was the Word”. In conjunction with the subsequent episode of the battle, line 89 a forms an exemplum of the monastic practice of lectio divina. This example of ‘monastic poetics’ (O’Camb 2014; Niles 2019) moves from grammatical analysis to a vision of the Word.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

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