New arXiv submissions of complex variable theory

[Douglas Bridges]

Gidday all, from Down Under.

Having reached my 80th year, I no longer am prepared to endure long waits for verdicts of research submissions to regular journals, and so am putting only on the arXiv (with links from my website:  www.dsbridges.com) such new mathematics as I manage to obtain. My first two papers there deal with aspects of complex analysis:
 

1.  On Cauchy's Integral Theorem in Constructive Analysis   In his constructive development of complex analysis, Errett Bishop used restrictive notions of homotopy and simple connectedness. Working in Bishop-style constructive mathematics, we prove Cauchy's integral theorem using the standard notions of such properties. In consequence, Bishop's theorems in Chapters 5 of Foundations of Constructive Analysis hold under our more normal, less restrictive, definitions.  http://arxiv.org/abs/2410.11058
    
2. On the Constructive Theory of Jordan Curves     Using a definition of Jordan curve similar to that of Dieudonné, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of Jordan curves and their corresponding index functions, including the important Proposition 32 and its corollaries about lines crossing a Jordan curve at a smooth point. The final section is dedicated to proving that the index of a point with respect to a piecewise smooth Jordan curve in the complex plane is identical to the familiar winding number of the curve around that point. The paper is written within the framework of Bishop's constructive analysis.  Although the work in Sections 3--5 is almost entirely new, the paper contains a substantial amount of expository material for the benefit of the reader.  https://doi.org/10.48550/arXiv.2502.09784
    

I've also put on the website one or two corrections to books of mine, and will continue to put things thereon as I sort through old files and unfinished work.

 

 

 

Regards,
Douglas