Influence of the Choice of the Class of Gauge Functions on the Properties of the Henstock–Kurzweil Integral

2021 ◽  
Vol 76 (2) ◽  
pp. 65-68
Author(s):  
S. N. Kopylov
Keyword(s):  
Gauge Functions

scholarly journals The Yang-Mills heat equation with finite action in three dimensions

2022 ◽  
Vol 275 (1349) ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Heat Equation ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Three Dimensions ◽  
Dimensional Manifold ◽  
Uniqueness Of Solutions ◽  
Reconstruction Procedure ◽  
Content Type ◽  
Yang Mills ◽  
Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

scholarly journals Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Z. E. Musielak ◽  
N. Davachi ◽  
M. Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Algebraic Structure ◽  
Second Order ◽  
Lagrangian Formalism ◽  
Group Approach ◽  
Second Order Differential Equations ◽  
Gauge Functions ◽  
The Relationship

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

scholarly journals Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 379
Author(s):  
Zdzislaw E. Musielak ◽  
Niyousha Davachi ◽  
Marialis Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Calculus Of Variations ◽  
Special Functions ◽  
Lagrange Equation ◽  
Mathematical Physics ◽  
Lagrangian Formalism ◽  
Lagrange Equations ◽  
Gauge Functions ◽  
Helmholtz Conditions

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

Isometries of symmetric gauge functions

1991 ◽  
Vol 30 (1-2) ◽  
pp. 81-92 ◽  
Author(s):  
Dragomir Ž. Đoković ◽  
Chi-Kwong Li ◽  
Leiba Rodman
Keyword(s):  
Gauge Functions ◽  
Symmetric Gauge
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