Boris Khots - Böcker

The author approaches an old classic problem - the existence of solutions of Navier-Stokes equations. The main objective is to model and derive of equation of continuity, Euler equation of fluid motion, energy flux equation, Navier-Stokes equations from the observer point of view and solve classic problem for this interpretation of fluid motion laws.If we have a piece of metal or a volume of liquid, the idea impresses itself upon us that it is divisible without limit, that any part of it, however small, would again have the same properties. But, wherever the methods of research in the physics of matter were refined sufficiently, limits to divisibility were reached that are not due to the inadequacy of our experiments but to the nature of the subject matter. Observability in mathematics were developed by the author based on denial of infinity idea. He introduces observers into arithmetic, and arithmetic becomes dependent on observers. And after that the basic mathematical parts also become dependent on observers. This approach permits to reconsider the fluid motion laws, analyze them and get solutions of classic problems.Table of Contents1. Introduction.2. Observability and Arithmetic.3. Observability and Vector Algebra.4. Observability and Mathematical Analysis (Calculus).5. Classic Fluid Mechanics equations and Observability.6. Observability and Thermodynamical equations.7. Observability and equation of continuity.8. Observability and Euler equation of motion of the fluid.9. Observability and energy flux and moment flux equations.10. Observability and incompressible fluids.11. Observability and Navier-Stokes equations.12. Observability and Relativistic Fluid Mechanics.13. Appendix: Review of publications of the Mathematics with Observers.14. Glossary.BibliographyIndexBiographyBoris Khots, DrSci, lives in Iowa, USA, Independent Researcher. Alma Mater - Moscow State Lomonosov University, Department of Mathematics and Mechanics (mech-math). Creator of Observer’s Mathematics. Participant of more than 30 Mathematical international congresses, conferences. In particular, participated with presentation at International Congresses of Mathematicians on 1998 (Germany), 2002 (China), 2006 (Spain), 2010 (India), 2014 (South Korea). More than 150 mathematical books and papers.

The author approaches an old classic problem - the existence of solutions of Navier-Stokes equations. The main objective is to model and derive of equation of continuity, Euler equation of fluid motion, energy flux equation, Navier-Stokes equations from the observer point of view and solve classic problem for this interpretation of fluid motion laws.If we have a piece of metal or a volume of liquid, the idea impresses itself upon us that it is divisible without limit, that any part of it, however small, would again have the same properties. But, wherever the methods of research in the physics of matter were refined sufficiently, limits to divisibility were reached that are not due to the inadequacy of our experiments but to the nature of the subject matter. Observability in mathematics were developed by the author based on denial of infinity idea. He introduces observers into arithmetic, and arithmetic becomes dependent on observers. And after that the basic mathematical parts also become dependent on observers. This approach permits to reconsider the fluid motion laws, analyze them and get solutions of classic problems.Table of Contents1. Introduction.2. Observability and Arithmetic.3. Observability and Vector Algebra.4. Observability and Mathematical Analysis (Calculus).5. Classic Fluid Mechanics equations and Observability.6. Observability and Thermodynamical equations.7. Observability and equation of continuity.8. Observability and Euler equation of motion of the fluid.9. Observability and energy flux and moment flux equations.10. Observability and incompressible fluids.11. Observability and Navier-Stokes equations.12. Observability and Relativistic Fluid Mechanics.13. Appendix: Review of publications of the Mathematics with Observers.14. Glossary.BibliographyIndexBiographyBoris Khots, DrSci, lives in Iowa, USA, Independent Researcher. Alma Mater - Moscow State Lomonosov University, Department of Mathematics and Mechanics (mech-math). Creator of Observer’s Mathematics. Participant of more than 30 Mathematical international congresses, conferences. In particular, participated with presentation at International Congresses of Mathematicians on 1998 (Germany), 2002 (China), 2006 (Spain), 2010 (India), 2014 (South Korea). More than 150 mathematical books and papers.

Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The ''mass gap'' property has been discovered by physicists from experiment, but it still has not been understood from a theoretical point of view. Proposed book describes author's approach to solution of this problem on base of Mathematics with Observers (removing from arithmetic infinity idea, taking into account Observers dependent ascending chain of embedded sets of finite decimal fractions with arithmetic operations locally coinciding with standard operations, and getting new calculus, diff geometry, etc), including interpretations of vector fields and differential forms, generalization of Yang-Mills equations, proof of mass gap existing, consideration the theory of matrix Lie groups and algebras, and this point of view gives the possibilities to make new approach and establish the existence of the Yang-Mills theory and a mass gap, Grand unified theories and Standard model of particle physics.

Observability in Mathematics were developed by authors based on denial of infinity idea. We introduce Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that the basic mathematical parts also become dependent on Observers. One of such parts is geometry. Geometry plays important role not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory, Quantum Yang-Mills theory. We call New Geometry both Observers in arithmetics and in geometry. We reconsider the basis of classic geometry (points, straight lines, planes and space) from this Mathematics point of view. The relations of connection, order, parallels (Euclid, Gauss-Bolyai-Lobachevsky, Riemann), congruence, continuity are discovered and have new properties. We show that almost all classic geometry theorems are satisfied in Mathematics with Observers geometry with probabilities less than 1. That means classic geometries are not a limiting cases of the Observer’s geometry, but are only particular cases. And new geometry opens the road to reconsider differential geometry, algebraic geometry, geometric algebra, topology, and also to reconsider geometrical applications to various parts of contemporary physics. We proved that Mathematics with Observers gives a birth a new geometry.