Dilatation structures I. Fundamentals

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is apart of a series of papers which show that such a structure allows to do non-commutative analysis, in the sense of differential calculus, on a large class of metric spaces, some of them fractals. We also describe a formal, universal calculus with binary decorated planar trees, which underlies any dilatation structure.

The purpose of this paper is to introduce dilatation structures on metric spaces. A dilatation structure is a concept in between a group and a differential structure. Any metric space (X,d) endowed with a dilatation structure has an associated tangent bundle. The tangent space at a point is a conical group that is the tangent space has a group structure together with a one-parameter group of auto morphisms. Conical groups generalize Carnot groups, i.e nilpotent groups endowed with a graduation. Each dilatation structure leads to a non-commutative differential calculus on the metric space (X, d).

How to Prove the Riemann Hypothesis

The aim of this paper is to prove the celebrated Riemann Hypothesis. I have already discovered a simple proof of the Riemann Hypothesis. The hypothesis states that the nontrivial zeros of the Riemann zeta function have real part equal to 0.5. I assume that any such zero is s=a+bi. I use integral calculus in the first part of the proof. In the second part I employ variational calculus. Through equations (50) to (59) I consider (a) as a fixed exponent, and verify that a=0.5. From equation (60) onward I view (a) as a parameter (a <0.5) and arrive at a contradiction. At the end of the proof (from equation (73)) and through the assumption that (a) is a parameter, I verify again that a=0.5.

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The ABCs of the Mathematical Infinitology. Principles of the Modern Theory and Practice of Scientific-and-Mathematical Infinitology

In any, praiseworthy hobby, business or the craft, being appeared at the human persons for a long time process of evolution, and thanks to the mental and creative abilities growth, sometimes among the advanced people were developed such high spheres of human knowledge or personal skills or intellectual abilities, that a lot of centuries and even the millenniums came or passed away, before some difficult scientific idea or the secrets of the craft could be at last found their final decisions or they were transformed by the human individuals into such form of the representation or embodiment, available for their natural perception by people, specialists or scientists, that a team of higher skilled experts could only recognize this or that decision as a perfect standard.

It isn't necessary to go far very much for the examples! The most ancient and the unresolved task is a secret of natural prime numbers, the cornerstone of the scientific theory of their knowledge and studying was put by Eratosphen Kirensky, the Ancient Greece mathematician, being lived in the III century B.C. The knowledge by the human persons of the Great truths of the World was always, from the time of immemorial destiny, the elite of possessing advanced thinkers being had a rich life experience.