3-CIRCLES THEOREM and It's APPRICATION THEOREM

The Basic Theorem and its Consequences

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...

The Six Circles Theorem revisited

Fan-KKM Theorem in Minimal Vector Spaces and its Applications

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

On Rectification of Circles and an Extension of Beltrami’s Theorem

Introduction. The problem that the author solved has its origin in Nomography: 1 how to reduce a nomogram of aligned points to a circular nomogram? In more mathematical terms: what are local diffeomorphisms that send germs of lines to germs of circles? This question was initially posed for 2-dimensional nomograms by G.S. Khovanskii and, solved by A.G. A discipline which was discovered by the Fr...

Baire’s Theorem and Its Consequences

We prove Baire’s theorem and its standard consequences: The uniform boundedness principle, the open mapping theorem, and the closed graph theorem.