Navier-Stokes Equations—Millennium Prize Problems

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.

The Simplest Possible Fully Correct Solution of the Clay Millennium Problem about P vs. NP. A Simple Proof That P ≠ NP = EXPTIME

In the current paper, I present probably the simplest possible abstract formal proof that P ≠ NP, and NP = EXPTIME, in the context of the standard mathematical set theory of computational complexity and deterministic Turing machines. My previous publications about the solution of the P vs. NP with the same result NP = EXPTIME, to be fully correct and understandable need the Lemma 4.1 and its proof of the current paper. The arguments of the current paper in order to prove NP = EXPTME are even simpler than in my previous publications. The strategy to solve the P vs. NP problem in the current paper (and in my previous publications) is by starting with an EXPTIME-complete language (problem) and proving that it has a re-formulation as an NP-class language, thus NP = EXPTIME. The main reason that the scientific community has missed so far such a simple proof, is because of two factors 1) It has been tried extensively but in vain to simplify the solutions of NP-complete problems from exponential time algorithms to polynomial time algorithms (which would be a good strategy only if P = NP) 2) It is believed that the complexity class NP is strictly a subclass to the complexity class EXPTIME (in spite the fact that any known solution to any of the NP-complete problems is not less than exponential). The simplicity of the current solution would have been missed if 2) was to be believed true. So far the majority of the relevant scientific community has considered this famous problem not yet solved. The present results definitely solve the 3rd Clay Millennium Problem about P versus NP in a simple, abstract and transparent way that the general scientific community, but also the experts of the area, can follow, understand and therefore become able to accept.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

Erratum to “The Riemann Hypothesis-Millennium Prize Problem” [Advances in Pure Mathematics 6 (2016) 915-920]

The original online version of this article (Durmagambetov, A.A. (2016) The Riemann Hypothesis-Millennium Prize Problem. Advances in Pure Mathematics, 6, 915-920. 10.4236/apm.2016.612069) unfortunately contains a mistake. The author wishes to correct the errors in Theorem 2 of the result part.

A Short and Simple Solution of the Millennium Problem about the Navier-Stokes Equations and Similarly for the Euler Equations

This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.

The Riemann Hypothesis-Millennium Prize Problem

This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied;these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.

The Golden Ratio Theorem: A Framework for Interchangeability and Self-Similarity in Complex Systems

The Golden Ratio Theorem, deeply rooted in fractal mathematics, presents a pioneering perspective on deciphering complex systems. It draws a profound connection between the principles of interchangeability, self-similarity, and the mathematical elegance of the Golden Ratio. This research unravels a unique methodological paradigm, emphasizing the omnipresence of the Golden Ratio in shaping system dynamics. The novelty of this study stems from its detailed exposition of self-similarity and interchangeability, transforming them from mere abstract notions into actionable, concrete insights. By highlighting the fractal nature of the Golden Ratio, the implications of these revelations become far-reaching, heralding new avenues for both theoretical advancements and pragmatic applications across a spectrum of scientific disciplines.

Clarifications for the Published Article: “A Solution to the Famous Twin’s Problem” in the APM of SCIRP at 24 September of 2019

This article B is almost autonomous because it can be read independently from the first published article A [1] using only a few parts of the article A. Be-low are given instructions so to need the reader study only on few places of the article A. Also, in the part A of Introduction, here, you will find simple and useful definitions and the strategy we are going to follow as well useful new theorems (also and in Section 5, which have been produced in this solution). So the published solution of twin’s problem can now be easily understood. The inequalities (4.17), (4.18) of Article A are proved here in Section 4 by a new clear method, without the possible ambiguity of the text between the relations (4.14), (4.16) of the Article A. Also we complete the proof for the twin’s distri-bution which we use. At the end here are presented the Conclusions, the No-menclatures and the numerical control of the proof, which is probably useful as well in coding methods. For a general and convincing picture is sufficient, a study from the beginning of this article B until the end of the part A of the In-troduction here as well a general glance on the Section 5 and on the Conclu-sions below.

Riemann Hypothesis, Catholic Information and Potential of Events with New Techniques for Financial and Other Applications

In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions () which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.