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| viXra.org > Functions and Analysis > 2304 |
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[7] viXra:2304.0169 [pdf] replaced on 2023-05-10 11:33:09
Authors: Hans Detlef Hüttenbach
Comments: 16 Pages. Several misprints corrected.
In this paper it is shown that the Banach space of continuous, R^2- or C-valued functions on a simply connected either 2-dimensional real or 1-dimensional complex compact region can be decomposed into the topological direct sum of two subspaces, a subspace of integrable (and conformal) functions, and another one of unintegrable (and anti-conformal) functions. It is shown that complex integrability is equivalent to complex analyticity. This can be extended to real functions. The existence of a conjugation on that Banach space will be proven, which maps unintegrable functions onto integrable functions.The boundary of a 2-dimensional simply connected compact region is defined by a Jordan curve, from which it is known to topologically divide the domain into two disconnected regions. The choice of which of the two regions is to be the inside, defines the orientation.The conjugation above will be seen to be the inversion of orientation.Analyticity, integrability, and orientation on R^2 (or C) therefore are intimately related.
Category: Functions and Analysis
[6] viXra:2304.0167 [pdf] submitted on 2023-04-20 23:53:17
Authors: Shikhar Sehgal
Comments: 1 Page. All Rights Reserved, Shikhar Sehgal 2023 (Abstract added to the article by viXra Admin - Please conform)
This paper provides an overview of using Euler's identity to prove that natural logarithms of numbers approaching zero exist on the complex plane.
Category: Functions and Analysis
[5] viXra:2304.0156 [pdf] submitted on 2023-04-19 18:12:05
Authors: Biruk Alemayehu Petros
Comments: 6 Pages. Mathematically meaningful solutions for Navier Stokes equation with complete proof is provided
This study proves the existence of smooth periodic solutions for Navier- Stokes three-dimensional equations under the assumption of a given pe- riodic initial velocity vector field with positive viscosity. The solution proposed solves the equation by utilizing a Fourier series representation of periodic initial velocity vector fields and predicting the velocity vec- tor field at all times. The significance of this finding is that it con- tributes positively towards understanding the behavior of solutions of Navier-Stokes equations and suggests that smooth periodic solutions for the given problem can indeed exist under certain conditions. Addition- ally, the authors suggest that their solution can be used to settle the Clay Mathematics Millennium Prize Problem, which seeks to find a solution for Navier-Stokes equations meeting specific criteria. It is important to note, however, that this study does not provide a complete solution to the problem, but it provides a significant contribution to the understanding of the behavior of solutions of Navier-Stokes equations. Overall, this re- search demonstrates that the smooth periodic solutions for Navier-Stokes equations can exist for a given initial velocity vector field with positive viscosity, and it presents a new approach for the Navier-Stokes equation.
Category: Functions and Analysis
[4] viXra:2304.0153 [pdf] submitted on 2023-04-19 20:30:05
Authors: Saburou Saitoh
Comments: 7 Pages.
In this note, we shall refer to some serious problems for the standard complex analysis text books that may be considered as common facts for many years from the viewpoint of the division by zero calculus. We shall state clearly our opinions with the new book: V. Eiderman, An introduction to complex analysis and the Laplace transform (2022).
Category: Functions and Analysis
[3] viXra:2304.0120 [pdf] submitted on 2023-04-18 00:41:59
Authors: Saburou Saitoh
Comments: 4 Pages.
In this note, we shall refer to the equation X + (X/X) = X from our division by zero and division by zero calculus ideas against the Barukčić's idea.
Category: Functions and Analysis
[2] viXra:2304.0107 [pdf] submitted on 2023-04-16 00:49:01
Authors: Jiří Souček
Comments: 19 Pages. [Constructive criticism is welcomed]
At first we identify the main error in the formulation of the concept of the weak solution to Navier-Stokes (NS) equations which is the completely insufficient treatment of the incompressibility condition on the fluid (expressed in the standard way by div u = 0). The repair requires the complete reformulation of the NS problem. The basic concept must be the generalized motion (i.e. the generalized flow) which replaces the standard velocity field. Here we define the generalized flow on the bases of Geometric measure theory extended to the theory of Cartesian currents and weak diffeomorphisms (see [1], [2]). Then the key concept of the complete weak solution to the NS problem is defined and the two conjectures (the existence and the regularity ones) concerning the complete weak solutions are formulated. In two appendices many technical details are described (concerning e.g. Cartesian currents, homology conditions, weak diffeomorphisms, etc.). Our approach is based on the unification of the standard analysis of NS equations with the methods of Geometric measure theory and of the theory of Cartesian currents.
Category: Functions and Analysis
[1] viXra:2304.0087 [pdf] replaced on 2024-12-20 00:44:23
Authors: James C. Austin
Comments: 11 Pages.
The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by adding an extra unbounded term to the traditional definition, extending its validity to Rez>0. The Stolz-Cesàro theorem is then used to analyse zeta(z)/zeta(1-z) as a ratio of complex sequences. The results are analysed in both halves of the critical strip (0<Rez<1/2,1/2<Rez<1), yielding a contradiction when it is assumed that zeta(z)=0 in either of these halves.
Category: Functions and Analysis
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