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My newest findings in Mathematics, May 2021: - a new formula for the Euler-Mascheroni and Lugo constants based on the calculation of the Jensen coefficients and also the Taylors' and Turan's for the Taylor series of the Riemann Xi-function - The Even Derivatives of the Xi function at (s=1/2) based on the non-trivial zeros and Jensen coeffic. - Findings: assertion of the Riemann Hypothesis' consequences and the hyperbolicity of the Jensen polynomials for the Taylor series of Xi at s=1/2 +ti.
January 2005 - April 2007
Universidad Pontificia Bolivariana
- Researcher in Optical Metrology
How to avoid and resolve an integral that involves the use of the Jacobi functions for computing the coefficient a_0 or also the Jensen C_0 of the Taylor series of the Xi function at s=1/2 +ti?The Riemann Hypothesis helps to do that! And more consequences. The biggest surprise or greatest finding is that the Taylor series of the Riemann Xi functio...
The Bernoulli numbers can be useful in special summations formulas that define the coefficients a_2n and b_n known in the references as set of coefficients that represent the Taylor series of the Riemann Xi function.
An algorithm for computing an approximate model for the Riemann zeta function at several values, especially when testing the integers known n=2, n=3, n=4, maximum n=10. The values are good approximations of known values of Riemann Zeta. The algorithm does not expose the Riemann Xi at all as the Gamma function needs to be examined in Matlab properly...
Finding: the first twenty-one coefficients C_n of Jensen polynomials with shift N=0 and the Taylor even coefficients a_2n needed for a quite enough representation for the Riemann Xi-function in Taylor series form, being the coefficients very useful for a new representation series for the Euler-Mascheroni constant in my current research work
A complete table for the first twenty-one coefficients that describe two fascinating consequences within the Riemann Hypothesis' scope: 1) The formulation of a never seen before formula (summation series) for the Euler-Mascheroni constant based strictly in 'pi' and the Taylor coefficients a_2n, Turán moments b_n or b_m and the coefficients of the...
In the systems of reconstruction 3D of structured light or interferometric, the spatial phase unwrapping by traditional methods constitutes a difficult problem to solve. The inherent noise to the surface, low visibility of the fringes patterns or the computational method for fringe pattern analysis can affect the discontinuous phase map, producing...
Some interesting 3‐D parameters how: amplitude parameters, spatial parameters, hybrid parameters and functional parameters; are presented in this paper for characterizing surface topography of a steel plate in the carbon (A36), which was subjected to process of corrosion. The surface will be reconstructed with the technique of laser triangulation...
In the perfilometry by means of structured light there are two techniques that are the most used: fringe projection and Moiré. The reconstruction of the object surface with inherently discontinuous or prominent slopings, is usually a difficult problem by using standard phase unwrapping technique i.e. the spatial unwrapping. The alternative of spati...
This work presents a comparative study between the methods of spatial unwrapping by means of reliability parameter and modified temporal unwrapping as a solution to the phase discontinuities presents in surface perfilometry by fringe projection.
The reconstruction of the object surface with inherently discontinuous regions is usually a difficult problem by using standard phase unwrapping technique i.e. the spatial unwrapping. To recover the range data of such a surface, Huntley and Saldner recently proposed a temporal phase unwrapping procedure (TPUP) and Peng X. et al. based on their work...
For a model on Matlab I have been implementing, a formulation for the Riemann Zeta function, I require more precision per Stieltjes (in number of Stieltjes and decimal places used), I have values with great precision from the Gamma(k=1) to Gamma(k=10) however, I require to know the next sequences of Stieltjes with more than 9 decimal places from Gamma(k=11) to Gamma(k=100) or greater values. Are there any complete data with more than 9 digits of precision in some website or other kind of resource? It is part of understanding a phenomenon related to the number of decimal places and Stieltjes used in some summations that I can evidence.
I have a matrix with a pattern, it starts from an index given by a value for the integers, i.e, i=0, 1, 2,3.... As I know the matrix which multiplies a vector of data (also with defined values based on special constants), which alternative coming from the known techniques of linear algebra lets operate on the rows in order to transform the original matrix to its inverse? (Is still valid the Gauss reduction techniques or others modern?) Not sure if an inverse matrix with infinite elements could be gotten easily, as theoretically the definition of square matrix does not apply or we need to imagine an infinite square matrix (infinite rows and columns as in M, below).
I know the old methods, but the modern algebra or formal lets define properly the operations for a matrix like this, from which I need get its inverse:
M= ( ( 1 0 0 0 .... 0....) for i=0
(1 2 4 8.... (2 at j).... ) for i=1
(1 2x2 (2 at 4) (2 at 6) ...... ) for i=2
(1 2x3 (( 2 at 2) X (3 at 2)) (2 at 3( x (3 at 3)...... ) for i=3
In general, I pursuit for methods to determine properly the inverse of this matrix or for all kind of matrices, if applies. the matrix M lets calculate some terms involving the Bernoulli numbers from a special coefficients G0, G1, G2, .... which I need to define thanks to the inverse. Inverse might be possible.
Dear members, I have got a summation formula which I would like to be able to reduce or find some useful equivalent expression for that with certain sense. The summation is:
S = Sum( ((k+1). S_k) / (k - m + 1)! ) when the sum starts from k=m-1 to Infinite. Where ! is the factorial operator. Being S_k the Stieltjes constants.
Thanks in advance!
Dear members, I am focused on a particular problem based on a series at the complex variable s which contains summations of triangle numbers, tetrahedral, pentatope and others sequences in the development of the coefficients. I mean, the summation involves such coefficients and the Stieltjes constants as well.
The formulation of such numbers (triangle, tetrahedral, etc) is well known, but I am interested more in to know if the Galois theory or groups of Galois and other branches related to it could specify theorems for transforming expressions with polynomials to rational functions or other expressions.
I am pursuing a philosophical interpretation of the geometry of such numbers for my particular problem related to the Riemann Zeta function formulation as they are present there. It would be a key for defining coefficients and resolving some terms within this context.
Also, a theory about Zeta transform and Mellin transform would be great related to the topics I am describing, not the introductory definition I can find on Internet for these topics. I would like to be able to connect triangle, tetrahedral and other numbers, their sequences with the variable s in an advanced algebra, then, I am able to learn more about that.
Dear researchers, I am very interested in making more progress in the direction of the functional analysis and number theory. After the second stage of my private research regarding the Riemann Zeta function and Xi-function, I have been able to validate one fascinating consequence of the Riemann Hypothesis, probably never seen before, and it is related to the set of the Taylor even coefficients a_2n and Jensen C_n (and the Turán moments b_m or b_n calculated by professors Csordas, Norfolk and Varga, almost 36 years ago). I have validated these b_m, and their impact in the definition of the famous coefficients of the Jensen polynomials with shift N=0 and Taylor coefficients a_2n which evidently (or undoubtedly) lets formulate a representation by a series for the well-known and important Euler-Mascheroni constant, and also the Lugo's. I do not have any contract with any institution (university) nor studying any master/PhD, but I considere that I could show an significant set of results that would open the door for future researchs not only in mathematics, but also in physics. As I live in Europe, electronics engineer graduate from UPB, several years ago (2007) and my passion for searching and validating models ( I am good at Matlab and other programms), I can work in a research project if I had such priceless opportunity. The reason for investigating in functional analysis and number theory when being an electronics engineer is because I like mathematics being applied to several fields, I believe in the possibility to determine new formulas that impact in exact science and engineering. Nowadays, I feel alone in my own research and I have tried to contact several magazines for showing these golden results and to be able to coming back to the college world, unfortunately, time and some circumstances during my years as engineer trying to pursuit some chance in research have affected my way to do what I would love to do the best: researching and completing my PhD. That is why I post this question about who could help me to come back to an institution and be able to build an interesting path of results and findings I know I can give.
For now, I have reached a tremendous data and fascinating equations that only could be exposed once finished my 10 or 13 pages of current article, as I am aware of the formal stages of a publication. I would like to contact not only magazines, but also prestigious universities whose groups or members could get inmersed in my ideas and results.
Let me know if there were good information and contacts via inbox or replying to this post.
Thanks in advance.
Graduate from UPB, Medellín, Colombia
Technical representative, current city: Szczecin, Poland
I have read about some information on the Internet regarding the Lorentz contraction factor.
It can be said that photo-sphere is the most significant region of black holes in the Universe. For Lorentz contraction factor Beta on the photo-sphere it must hold
Beta = (1 - v ^2 / (c ^2) ) ^ (1/2) = 0.57735
Some sources suggest that it is the Euler-Mascheroni constant.
I am interested in the background or deduction of that factor, i.e., the previous expressions or analysis around that factor.
I would like to know a complete list (with detailed use) of the application of the Euler Mascheroni constant, hopefully in current research in physics and others fields of exact science.
I would like to know special formulas in cosmology, quantum mechanics, statistics, etc., where the constant appeared with a very good explanation of why it is or at least valid arguments of the equations that contain that constant.
I am exploring its use.
I am calculating some values related to the Taylor series of the Riemann Xi function and I am very interested in to have good approximations of the eight (8) 10th, 12 or other derivatives order of the Riemann Xi function evaluated at the s = 1/2 + 0.i , i.e., the even derivatives of Xi at (1/2) if it was possible. I have the values for the second, fourth and sixth derivatives, which are well-known, however, it is difficult to find some greater derivatives. Let me know by which book or website I could read such data, it is very useful for a proof of computational evaluation of summations involving the coefficients of the Taylor series and the Jensen polynomials which I am developing.
How to work properly in the development of an integral like the Abel Plana defined on this image:
I am interested in to have a set of steps for attacking the problem of developing the integral and to determine a criterion of convergence for any complex value s, I mean, when the integral could have some specifical behavior at, for example, s=1/2 + i t where I am interested in to study it.
I am interested in the proper evaluation of that integral only with formal steps into complex analysis.
The Abel PLana formula appears also on https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula
I have been proving, by valid data and evidence, that the non trivial zeros of the Riemann Zeta function let compute the even derivatives of Xi at s=1/2 and the coefficients of the Jensen polynomials of the Taylor series of Xi about s=1/2 +it as part of the consequences of the Riemann Hypothesis' assertion.