scholarly journals Variations on the Brouwer Fixed Point Theorem: A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 501
Author(s):  
Jean Mawhin
Keyword(s):  
Fixed Point ◽  
Existence Theorems ◽  
Holomorphic Functions ◽  
Cauchy Integral ◽  
Integral Theorem ◽  
Exterior Calculus ◽  
Continuous Mappings ◽  
Cauchy Integral Theorem ◽  
Basic Facts ◽  
Special Case

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in R n . The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.

The Research on the Relationship between Cauchy Integral Theorem and Complex Function Integral in Mechanics of Materials

2012 ◽  
Vol 502 ◽  
pp. 120-123
Author(s):  
Xing Rong Sun
Keyword(s):  
Integral Formula ◽  
Complex Function ◽  
Residue Theorem ◽  
Cauchy Integral ◽  
Integral Theorem ◽  
Mechanics Of Materials ◽  
Cauchy Integral Theorem ◽  
Derivative Formula ◽  
Higher Order Derivative ◽  
The Relationship

This paper is the Cauchy integral theorem and integral of complex function carried out a comparative analysis, summarized in the Cauchy integral theorem and Cauchy integral formula, higher-order derivative formula, residue theorem and the relationship between the derivations to be proved, the formula can be used in these areas, such as mechanics of materials.

scholarly journals The Solution of Embedding Problems in the Framework of GAPs with Applications on Nonlinear PDEs

2009 ◽  
Vol 2009 ◽  
pp. 1-46
Author(s):  
Reinhard Starkl
Keyword(s):  
Differential Equations ◽  
Function Theory ◽  
Nonlinear Partial Differential Equations ◽  
Complex Function ◽  
Nonlinear Pdes ◽  
Mathematical Framework ◽  
Cauchy Integral ◽  
Complex Function Theory ◽  
Integral Theorem ◽  
Cauchy Integral Theorem

The paper presents a special class of embedding problems whoes solutions are important for the explicit solution of nonlinear partial differential equations. It is shown that these embedding problems are solvable and explicit solutions are given. Not only are the solutions new but also the mathematical framework of their construction which is defined by a nonstandard function theory built over nonstandard algebraical structures, denoted as “GAPs”. These GAPs must not be neither associative nor division algebras, but the corresponding function theories built over them preserve the most important symmetries from the classical complex function theory in a generalized form: a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.

scholarly journals Curvilinear Integral Theorem for G-Monogenic Mappings in the Algebra of Complex Quaternion

2016 ◽  
Vol 6 ◽  
pp. 21-25 ◽  
Author(s):  
Tetyana Kuzmenko
Keyword(s):  
Cauchy Integral ◽  
Integral Theorem ◽  
Cauchy Integral Theorem ◽  
Complex Quaternion ◽  
Curvilinear Integral

For G-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the boundary of a domain of G-monogeneity.

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