**Previous months:**

2007 - 0703(1)

2009 - 0903(1)

2010 - 1003(1) - 1004(2) - 1005(2) - 1008(1)

2011 - 1106(3) - 1110(1) - 1112(1)

2012 - 1202(4) - 1203(8) - 1204(1) - 1206(3) - 1207(2) - 1210(3) - 1211(1) - 1212(2)

2013 - 1301(2) - 1302(1) - 1303(2) - 1304(4) - 1305(7) - 1306(15) - 1307(5) - 1308(3) - 1309(1) - 1310(5) - 1311(2)

2014 - 1402(1) - 1403(5) - 1404(2) - 1405(1) - 1406(1) - 1407(3) - 1408(3) - 1409(1) - 1410(1) - 1411(3) - 1412(1)

2015 - 1502(5) - 1503(3) - 1505(1) - 1506(1) - 1507(4) - 1508(5) - 1509(2) - 1510(10) - 1511(5) - 1512(1)

2016 - 1601(5) - 1602(3) - 1603(3) - 1604(7) - 1605(2) - 1606(4) - 1608(5) - 1609(4) - 1610(2) - 1611(5) - 1612(4)

2017 - 1701(8) - 1702(2) - 1703(12) - 1704(2) - 1705(7) - 1707(4) - 1708(3) - 1709(7) - 1710(5) - 1711(4) - 1712(7)

2018 - 1801(1) - 1802(4) - 1803(2) - 1804(2) - 1806(8) - 1807(6) - 1808(11) - 1809(3) - 1810(7) - 1811(5) - 1812(3)

2019 - 1901(3) - 1902(12) - 1903(8) - 1904(7) - 1905(2) - 1906(8) - 1907(3) - 1908(7) - 1909(6) - 1910(4) - 1911(5) - 1912(3)

Any replacements are listed farther down

[372] **viXra:1912.0182 [pdf]**
*submitted on 2019-12-09 09:59:19*

**Authors:** Viola Maria Grazia

**Comments:** 1 Page.

I talk about functions in particular I speak when the serie diverges and so when a function tends to infinity.

**Category:** Functions and Analysis

[371] **viXra:1912.0123 [pdf]**
*submitted on 2019-12-06 17:40:20*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

The definition of the decidable fan theorem is evaluated as not tautologous, hence refuting it and derived conjectures such as “uniform continuity theorem with continuous moduli”. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Functions and Analysis

[370] **viXra:1912.0030 [pdf]**
*submitted on 2019-12-02 11:50:41*

**Authors:** Jonathan W. Tooker

**Comments:** 22 Pages. 1 color figure

In a recent paper, the author demonstrated the existence of real numbers in the neighborhood of infinity. It was shown that the Riemann zeta function has non-trivial zeros in the neighborhood of infinity but none of those zeros lie within the critical strip. While the Riemann hypothesis only asks about non-trivial zeros off the critical line, it is also an open question of interest whether or not there are any zeros off the critical line yet still within the critical strip. In this paper, we show that the Riemann zeta function does have non-trivial zeros of this variety. The method used to prove the main theorem is only the ordinary analysis of holomorphic functions. After giving a brief review of numbers in the neighborhood of infinity, we use Robinson's non-standard analysis and Eulerian infinitesimal analysis to examine the behavior of zeta on an infinitesimal neighborhood of the north pole of the Riemann sphere. After developing the most relevant features via infinitesimal analysis, we will proceed to prove the main result via standard analysis on the Cartesian complex plane without reference to infinitesimals.

**Category:** Functions and Analysis

[369] **viXra:1911.0513 [pdf]**
*submitted on 2019-11-30 02:59:28*

**Authors:** Claude Michael Cassano

**Comments:** 4 Pages.

Any Second Order Linear Ordinary Differential Equation may be factored via two linear differential operators. The initial theorem demonstrates that for homogeneous 2nd order LODEs. Theorems that follow improve by supplying the solution of such factorization, extending the theorem to inhomogeneous LODEs, and further providing a generalization which includes the constant coefficients and Cauchy-Euler LODEs; culminating in a single-parameter LODE solution formula with suggested usage examples.

**Category:** Functions and Analysis

[368] **viXra:1911.0453 [pdf]**
*submitted on 2019-11-26 10:40:39*

**Authors:** Mohammed S. Abdo, Satish K. Panchal

**Comments:** 18 Pages.

In this paper, we investigate the existence, uniqueness and continuous dependence of solutions of fractional neutral functional differential
equations with infinite delay and the Caputo fractional derivative order, by means of the Banach's contraction principle and the Schauder's fixed point theorem.

**Category:** Functions and Analysis

[367] **viXra:1911.0343 [pdf]**
*submitted on 2019-11-20 18:10:56*

**Authors:** Daniel Thomas Hayes

**Comments:** 7 Pages.

A proposed solution to the millennium problem on the existence and smoothness of the Navier-Stokes equations.

**Category:** Functions and Analysis

[366] **viXra:1911.0115 [pdf]**
*submitted on 2019-11-06 18:56:03*

**Authors:** Saburou Saitoh

**Comments:** 9 Pages. In this paper, we will give several examples that in the general order $n$ differentials of functions we find the division by zero and by applying the division by zero calculus, we can find the good formulas for $n=0$. This viewpoint is new and curious at

In this paper, we will give several examples that in the general order $n$ differentials of functions we find the division by zero and by applying the division by zero calculus, we can find the good formulas for $n=0$. This viewpoint is new and curious at this moment for some general situation. Therefore, as prototype examples, we would like to discuss this property. Why division by zero for zero order representations for some general differential order representations of functions does happen?

**Category:** Functions and Analysis

[365] **viXra:1911.0094 [pdf]**
*submitted on 2019-11-05 14:21:38*

**Authors:** Esp. Wilson Torres Ovejero

**Comments:** 4 Pages.

La hipotesis de Riemann que posee un sin número de aplixaciones a la física, la estadística y por supuesto, a la matemática.

**Category:** Functions and Analysis

[364] **viXra:1910.0518 [pdf]**
*submitted on 2019-10-25 06:45:00*

**Authors:** Timothy W. Jones

**Comments:** 6 Pages.

We give a sequence of easy inferences that relate to the fundamental theorem of algebra. We then prove the theorem in three different ways: one way requires the least pre-requisites (the basic proof); the other two ways require results from complex analysis.

**Category:** Functions and Analysis

[363] **viXra:1910.0414 [pdf]**
*submitted on 2019-10-21 19:54:08*

**Authors:** Saburou Saitoh

**Comments:** 8 Pages. In this short note, we would like to refer to the fundamental new interpretations that for the fundamental expansion $1/(1-z) = \sum_{j=0}^{\infty} z^j$ it is valid in the sense $0=0$ for $z=1$, for the integral $\int_1^{\infty} 1/x dx $ it is zero and i

In this short note, we would like to refer to the fundamental new interpretations that for the fundamental expansion $1/(1-z) = \sum_{j=0}^{\infty} z^j$ it is valid in the sense $0=0$ for $z=1$, for the integral $\int_1^{\infty} 1/x dx $ it is zero and in the formula $\int_0^{\infty} J_0(\lambda t) dt = 1/\lambda$, it is valid with $0=0$ for $\lambda =0$ in the sense of the division by zero.

**Category:** Functions and Analysis

[362] **viXra:1910.0245 [pdf]**
*submitted on 2019-10-15 11:22:08*

**Authors:** Teo Banica

**Comments:** 200 Pages.

A complex Hadamard matrix is a square matrix $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects.

**Category:** Functions and Analysis

[361] **viXra:1910.0064 [pdf]**
*submitted on 2019-10-05 16:37:37*

**Authors:** Robert Jackson

**Comments:** Pages. The work has a 2019 copyright.

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study will discuss a new method for solving PDEs called the generating function technique (GFT), may establish a new precedence with respect to SEM. First, this study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, Cahn-Hilliard and QFT equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis

[360] **viXra:1909.0658 [pdf]**
*submitted on 2019-09-30 21:12:36*

**Authors:** Saburou Saitoh

**Comments:** 3 Pages. I think many and many mathematicians will be able to enjoy the result, indeed, over mathematicians. It will be a general interest.

In this short note, we will consider the value of the function $\exp {(ax)}/f(a)$ at $a=0$ for $f(a)=0$. This case appears for the construction of the special solution of some differential operator $f(D)$ for the polynomial case of $D$ with constant coefficients. We would like to show the power of the new method of the division by zero calculus, simply and typically.

**Category:** Functions and Analysis

[359] **viXra:1909.0572 [pdf]**
*submitted on 2019-09-26 16:06:21*

**Authors:** Bertrand Wong

**Comments:** 4 Pages.

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon. [The paper is published in an international journal.]

**Category:** Functions and Analysis

[358] **viXra:1909.0366 [pdf]**
*submitted on 2019-09-17 19:20:58*

**Authors:** Teo Banica

**Comments:** 200 Pages.

The unitary group U_N has a free analogue U_N^+, and the study of the closed subgroups G\subset U_N^+ is a problem of general interest. We review here the general theory of U_N^+ and its subgroups, with all the needed preliminaries included. We discuss as well a number of more advanced topics, selected for their beauty, and potential importance.

**Category:** Functions and Analysis

[357] **viXra:1909.0294 [pdf]**
*submitted on 2019-09-15 05:41:14*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove a version of the Beppo Levi`s lemma for the complex valued functions. This achieved by making a more stronger asumption that is assumed in Beppo Levi`s lemma. We will assume that the sum of measurable functions that is absolutely convergent almost everywhere is integrable. We will prove that it implies the asumptions of the Beppo Levi lemma, if we consider functions that are non-negative. It can be argued that our version is more suitable to applications, and we will prove a new probability law. We will show that with our asumptions in probability theory it follows that the expected value is countable additive. Moreover, it follows that in strong law of large numbers we don`t need to make any asumptions on distributions and the mean of the sample will convergence almost surely to the mean of the expected values.

**Category:** Functions and Analysis

[356] **viXra:1909.0200 [pdf]**
*submitted on 2019-09-09 16:24:17*

**Authors:** Saburou Saitoh

**Comments:** 6 Pages. A very surprising and mysterious property at the point at infinity.

Okumura's Disc Series Can Beyond the Crucial Point of D\"aumler-Puha's Horn Torus Models for the Riemann Sphere

**Category:** Functions and Analysis

[355] **viXra:1909.0189 [pdf]**
*submitted on 2019-09-10 05:20:53*

**Authors:** Kouider Mohammed Ridha

**Comments:** 3 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in represent numerical method which is the Modified Bisection Algorithm(MBA) referred to by Tanakan, (2013, [9]) for finding the multi-roots of a function. Hence in this study we programming the MBA for multi-roots with the R software.

**Category:** Functions and Analysis

[354] **viXra:1908.0542 [pdf]**
*submitted on 2019-08-26 13:41:43*

**Authors:** Bulat Khabibullin, Enzhe Menshikova

**Comments:** 4 Pages.

We discuss the existence of an envelope of a function from a certain subclass of function space. Here we restrict ourselves to considering the model space of functions locally integrable with respect to the Lebesgue measure in a domain from the finite dimensional Euclidean space

**Category:** Functions and Analysis

[353] **viXra:1908.0511 [pdf]**
*submitted on 2019-08-25 14:34:25*

**Authors:** Bulat N. Khabibullin, Enzhe Menshikova

**Comments:** 5 Pages.

Let u and M are two non-trivial subharmonic functions in a domain D in the complex plane. We investigate two related but different problems. The first is to find the conditions on the Riesz measures of functions u and M respectively under which there exists a non-trivial subharmonic function h on D such that u+h< M. The second is the same question, but for a harmonic function h on D.
The answers to these questions are given in terms of the special affine balayage of measures introduced in our recent previous works. Applications of this technique concern the description of distribution of zeros for holomorphic functions f on the domain D satisfying the restriction |f|< exp M.

**Category:** Functions and Analysis

[352] **viXra:1908.0489 [pdf]**
*submitted on 2019-08-25 02:41:13*

**Authors:** Hiroki Tahara

**Comments:** 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic Exegeses and define the Quranic correctness as the unique existence of Tahara iota map.In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.

**Category:** Functions and Analysis

[351] **viXra:1908.0436 [pdf]**
*submitted on 2019-08-22 02:32:21*

**Authors:** Bulat N. Khabibullin, Enzhe Menshikova

**Comments:** 19 Pages.

We investigate some properties of balayage of measures and their potentials on domains or open sets in finite-dimensional Euclidean space. Main results are Duality Theorems for potentials of balayage of measures, for Arens-Singer and Jensen measures and potentials, and also a new extended and generalized variant of Poisson-Jensen formula for balayage of measure and their potentials.

**Category:** Functions and Analysis

[350] **viXra:1908.0434 [pdf]**
*submitted on 2019-08-20 06:47:41*

**Authors:** Kouider Mohammed Ridha

**Comments:** 5 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan, (2013, [13]) superficially and didn't know her as a numerical method for finding the roots of a function. Hence in this study we define her as a new numerical method with error bound and the number iterations necessary. Finally we present a new MBM for multi-roots with the R software.

**Category:** Functions and Analysis

[349] **viXra:1908.0413 [pdf]**
*submitted on 2019-08-19 11:11:26*

**Authors:** Atabey Mahmudov

**Comments:** 3 Pages.

In this article, we introduce Cauchy's integral formula and proving by using analiticity of function inside of disk

**Category:** Functions and Analysis

[348] **viXra:1908.0296 [pdf]**
*submitted on 2019-08-15 06:47:48*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove an explicit bound for the isotropic constant in the symmetric case.

**Category:** Functions and Analysis

[347] **viXra:1908.0061 [pdf]**
*submitted on 2019-08-03 16:21:46*

**Authors:** Federico Espil

**Comments:** 2 Pages.

Shortly we can derive the Cauchy's residue theorem (its general form) just by integration of a Taylor Series "without" making any radius go to zero,even without the limit circumference idea take place. The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem

**Category:** Functions and Analysis

[346] **viXra:1907.0491 [pdf]**
*submitted on 2019-07-25 19:05:50*

**Authors:** Josh O'Connor

**Comments:** 18 Pages.

This paper generalises the limit definitions of calculus to define differintegrals of complex order, calculates some differintegrals of elementary functions, and introduces the notion of a fractional differential equation. An application to quantum theory is explored, and we conclude with some operator algebra. Functions in this paper will only have one variable.

**Category:** Functions and Analysis

[345] **viXra:1907.0454 [pdf]**
*submitted on 2019-07-25 04:57:09*

**Authors:** Naum E. Salis

**Comments:** 15 Pages.

The purpose of this paper is to try to replicate what happens in C on spaces where there are more then one of immaginary units. All these spaces, in our definition, will have the same Hilbert structure. At first we will introduce the sum and product operations on C(H):=RxH (where H is an Hilbert space), then we'll investigate on its algebraic properties. In our construction we lose only the associative of multiplication regardless of H, exept when dim H=1 (in this case RxH = C), and this is why we say "weak extension". One of the most important result of this study is the Weak Integrity Theorem according to which in particular conditions there exist zero divisors. The next result is the Foundamental Theorem according to which for all z in C(H) there exists w in C(H) such that z=w^2. Afterwards we will study tranformations between these spaces which keep operation (that's why we will call them C-morphisms). At the end we will look at the "commutative" functions, i.e. maps C(H) to C(H') which can be rapresented by complex transformations C to C

**Category:** Functions and Analysis

[344] **viXra:1907.0111 [pdf]**
*submitted on 2019-07-08 06:03:23*

**Authors:** Stephen C. Pearson.

**Comments:** 48 Pages.

This particular submission contains (inter alia) copies of three (3) monographs, whose purpose is to further elaborate upon various topics having been enunciated in the author's previous set of submissions, namely - (a) "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PARTS 1/10 to 10/10."; (b) "Supplementary Notes pertaining to a Specific Quaternion Analogue of the Cauchy-Goursat Theorem.", which have been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[343] **viXra:1906.0569 [pdf]**
*submitted on 2019-06-30 18:51:51*

**Authors:** Saburou Saitoh

**Comments:** 14 Pages. The division by zero calculus requests the essential arrangements for equations and inequalities in analytic functions.

In this paper, we will examine the division by zero calculus from the viewpoints of equations and inequalities as a starting new idea.

**Category:** Functions and Analysis

[342] **viXra:1906.0509 [pdf]**
*submitted on 2019-06-27 03:36:56*

**Authors:** Galyna Kriukova

**Comments:** 6 Pages. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152601

Hidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.

**Category:** Functions and Analysis

[341] **viXra:1906.0415 [pdf]**
*submitted on 2019-06-23 05:10:07*

**Authors:** 3.Alexander Domoshnitsky, Oleg Kupervasser, Hennadii Kutomanov

**Comments:** 6 Pages.

In this paper, new explicit tests for exponential stability of systems of second order equations are proposed. Our approach is based on nonoscillation of solutions of the corresponding diagonal scalar second order delay differential equations.

**Category:** Functions and Analysis

[340] **viXra:1906.0329 [pdf]**
*submitted on 2019-06-19 05:59:05*

**Authors:** Pith Peishu Xie

**Comments:** 2 Pages.

The Operator axioms have deduced number systems. In this paper, we conjecture some inequalities in Operator axioms. The general inequalities show the value of Operator axioms.

**Category:** Functions and Analysis

[339] **viXra:1906.0237 [pdf]**
*submitted on 2019-06-13 14:10:23*

**Authors:** Jonathan W. Tooker

**Comments:** 66 Pages. This paper is undergoing some syntactical changes/improvements. The paper is fine as is, but there are some issues which are currently being improved. In the meantime, readers are directed to the finalized verisons of viXra:1811.0222 and viXra:1809.0234

Recent analysis has uncovered a broad swath of previously unconsidered real numbers called real numbers in the neighborhood of infinity. Here we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior. The main results of this paper include (1) to prove that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity. The methods used in this analysis include nothing other than basic arithmetic, a little trigonometry, and Euclidean geometry. In addition to the zeros used to disprove the Riemann hypothesis in earlier work, here we present yet more zeros which independently constitute the negation of the Riemann hypothesis.

**Category:** Functions and Analysis

[338] **viXra:1906.0236 [pdf]**
*submitted on 2019-06-13 14:11:34*

**Authors:** Jonathan W. Tooker

**Comments:** 5 Pages.

In this brief note, we propose a set of operations for the affinely extended real number called infinity. Under the terms of the proposition, we show that the Riemann zeta function has infinitely many non-trivial zeros on the complex plane.

**Category:** Functions and Analysis

[337] **viXra:1906.0185 [pdf]**
*submitted on 2019-06-11 20:12:46*

**Authors:** Saburou Saitoh

**Comments:** 11 Pages. We propose new problems in several complex analysis from the viewpoint of division by zero calculus.

In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from one dimensional case.

**Category:** Functions and Analysis

[336] **viXra:1906.0163 [pdf]**
*submitted on 2019-06-11 02:34:24*

**Authors:** Andrej Liptaj

**Comments:** 8 Pages.

Derivative of a function can be expressed in terms of integration over a small neighborhood of the point of differentiation, so-called differentiation by integration method. In this text a maximal generalization of existing results which use one-dimensional integrals is presented together with some interesting non-analytic weight functions.

**Category:** Functions and Analysis

[335] **viXra:1906.0148 [pdf]**
*submitted on 2019-06-09 19:44:28*

**Authors:** Saburou Saitoh

**Comments:** 4 Pages. I gave a new type isoperimetric inequality and propose several fundamental open problems.

In this paper, as a direct application of Q. Guan's result on the conjugate analytic Hardy $H_2$ norm we will derive a new type isoperimetric inequality for Dirichlet integrals of analytic functions.

**Category:** Functions and Analysis

[334] **viXra:1905.0273 [pdf]**
*submitted on 2019-05-17 12:01:57*

**Authors:** Stephen C. Pearson.

**Comments:** 20 Pages.

This particular submission contains (inter alia) a copy of the author's original paper, which was completed on 6th March 2019 and thus comprises a total of 16 handwritten A4 pages. Subsequently, in view of its contents, it is being presented as an addendum to the author's previous set of submissions, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PARTS 1/10 to 10/10", which have been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[333] **viXra:1905.0181 [pdf]**
*submitted on 2019-05-12 08:58:16*

**Authors:** Henry Wong

**Comments:** 2 Pages.

In this paper we evaluate certain Gaussian-type integrals via contour integration in the complex plane along with the application of Cauchy-Goursat theorem.

**Category:** Functions and Analysis

[332] **viXra:1904.0414 [pdf]**
*submitted on 2019-04-21 10:18:41*

**Authors:** Teo Banica

**Comments:** 26 Pages.

We study the intermediate liberation problem for the real and complex unitary and reflection groups, namely $O_N,U_N,H_N,K_N$. For any of these groups $G_N$, the problem is that of understanding the structure of the intermediate quantum groups $G_N\subset G_N^\times\subset G_N^+$, in terms of the recently introduced notions of ``soft'' and ``hard'' liberation. We solve here some of these questions, our key ingredient being the generation formula $H_N^{[\infty]}=

**Category:** Functions and Analysis

[331] **viXra:1904.0408 [pdf]**
*submitted on 2019-04-22 00:32:30*

**Authors:** Saburou Saitoh

**Comments:** 73 Pages. Please kindly give me suggestions and comments to the paper.

In this survey paper, we will introduce the importance of the division by zero and its great impact to elementary mathematics and mathematical sciences for some general people. For this purpose, we will give its global viewpoint in a self-contained manner by using the related references.

**Category:** Functions and Analysis

[330] **viXra:1904.0380 [pdf]**
*submitted on 2019-04-19 20:00:44*

**Authors:** Pedro Hugo García Peláez

**Comments:** 2 Pages.

Integral seno por coseno que tiene como solución un determinado número de Fibonacci.
La fórmula sirve tanto para hallar integrales de línea de funciones tipo x*y sobre trayectorias curvas si queremos que tenga como solución un número de Fibonacci. Como para integrales de campos vectoriales como un campo de fuerzas en trayectorias curvas.

**Category:** Functions and Analysis

[329] **viXra:1904.0360 [pdf]**
*submitted on 2019-04-18 13:19:38*

**Authors:** Jesús Álvarez Lobo

**Comments:** 2 Pages. MSC2010: 58C05

A new definition of the number e is presented by the integral of a function that involves
an infinite product of nested radicals whose indexes form the sequence 1, 2, 3, ...
____________________________________________________________________

**Category:** Functions and Analysis

[328] **viXra:1904.0259 [pdf]**
*submitted on 2019-04-13 08:45:34*

**Authors:** H. C. Rhaly Jr.

**Comments:** 3 Pages.

A countable subcollection of the Endl-Jakimovski generalized Ces\`{a}ro matrices of positive order is seen to inherit posinormality, coposinormality, and hyponormality from the Ces\`{a}ro matrix of the same order.

**Category:** Functions and Analysis

[327] **viXra:1904.0138 [pdf]**
*submitted on 2019-04-06 08:36:03*

**Authors:** Jesús Sánchez

**Comments:** 3 Pages.

As we know, the natural logarithm at zero diverges, towards minus infinity:
lim┬(x→0)〖Ln(x)〗=-∞
But, as happens with other functions or series that diverge at some points, it has a Ramanujan or Cauchy principal value (a finite value) associated to that point. In this case, it will be calculated to be:
lim┬(x→0)〖Ln(x)〗=-γ
Being γ the Euler-Mascheroni constant 0.577215... It will be shown that Ln(0) tends to the negative of the sum of the harmonic series (that of course, diverges). But the harmonic series has a Cauchy principal value that is γ, the Euler-Mascheroni constant. So the finite associated value to Ln(0) will be calculated as - γ .

**Category:** Functions and Analysis

[326] **viXra:1904.0052 [pdf]**
*submitted on 2019-04-03 20:31:13*

**Authors:** Saburou Saitoh

**Comments:** 12 Pages. In Section 1, we will introduce the horn torus model by V.V. Puha and in Section 1.1, by modifying the Puha mapping, we introduce D\"aumler's horn torus model. In Section 1.2 we introduce division by zero and division by zero calculus with up-to-date

In this paper, we will introduce a beautiful horn torus model by Puha and D\"aumler for the Riemann sphere in complex analysis attaching the zero point and the point at infinity. Surprisingly enough, we can introduce analytical structure of conformal to the model. Here, some basic opinions on the D\"aumler's horn torus model will be stated as the basic ones in mathematics.

**Category:** Functions and Analysis

[325] **viXra:1903.0488 [pdf]**
*submitted on 2019-03-27 21:04:39*

**Authors:** Saburou Saitoh

**Comments:** 18 Pages. In this paper, we will introduce the division by zero calculus in complex analysis for one variable at the first stage in order to see the elementary properties.

In this paper, we will introduce the division by zero calculus in complex analysis for one variable at the first stage in order to see the elementary properties.

**Category:** Functions and Analysis

[324] **viXra:1903.0432 [pdf]**
*submitted on 2019-03-24 23:28:16*

**Authors:** Saburou Saitoh

**Comments:** 10 Pages. In this paper, we will show a very interesting interpretation of singular integrals by the division by zero calculus. This may be considered as the basic relation of ZERO and INFINITY through integrals. Furthermore, we will see a similar nature of singula

What are the singular integrals? Singular integral equations are presently encountered
in a wide range of mathematical models,
for instance in acoustics, fluid dynamics, elasticity
and fracture mechanics.
Together with these models, a variety of methods
and applications for these integral equations has been developed.
In this paper, we will give the interpretation for the Hadamard finite part of singular integrals by means of the division by zero calculus.

**Category:** Functions and Analysis

[323] **viXra:1903.0421 [pdf]**
*submitted on 2019-03-23 11:17:49*

**Authors:** Federico Espil

**Comments:** 9 Pages.

The problem of integration technique over integrands of the form f(t)/t^n, can be solved by differentiation(n times) by using Leibniz's rule to get rid of t^n, that leads to integrate back (n times) to end the game which it's harder than the original problem.This work focuses on the derivation of the formula (Espil's theorem) which is a perfect tool to avoid that hard task. It allows to change the difficult n iterated integrals into a more outstanding easier problem which consists of n -1 derivatives.The Espil's theorem is a generalization of the Dirichlet integral.

**Category:** Functions and Analysis

[322] **viXra:1903.0409 [pdf]**
*submitted on 2019-03-22 11:29:29*

**Authors:** Teo Banica

**Comments:** 12 Pages.

We investigate the liberation question for the compact Lie groups, by using various ``soft'' and ``hard'' methods, based respectively on joint generation with a free quantum group, and joint generation with a free torus. The soft methods extend the ``easy'' methods, notably by covering groups like $SO_N,SU_N$, and the hard methods partly extend the soft methods, notably by covering the real and complex tori themselves.

**Category:** Functions and Analysis

[321] **viXra:1903.0371 [pdf]**
*submitted on 2019-03-20 23:56:47*

**Authors:** Saburou Saitoh

**Comments:** 9 Pages. In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from the one dimensional case.

In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from the one dimensional case.

**Category:** Functions and Analysis

[320] **viXra:1903.0326 [pdf]**
*submitted on 2019-03-17 07:33:40*

**Authors:** Federico Espil

**Comments:** 6 Pages.

The problem of fraction decomposition it's easy to solve by using the cover up method, when there are no repeated linear factors in the denominator .
Nevertheless it could turn into a hard work if these factors are raised to a high power, where the cover up method doesn't work. This technique shows how to calculate these coefficients without solving large systems of equations with a clever rearrangement of the numerator.

**Category:** Functions and Analysis

[319] **viXra:1903.0315 [pdf]**
*submitted on 2019-03-18 05:25:09*

**Authors:** Fayowole David Ayadi

**Comments:** 3 Pages.

The laws of Physics and some other related courses are generally written as differential equations. Therefore, all of science
and engineering use differential equations to some extent. A good knowledge of differential equations will be an integral
part of your study in science and/or engineering classes. You can think of mathematics as the language of science, and
differential equations are one of the most important parts of this language as far as science and engineering are concerned.

**Category:** Functions and Analysis

[318] **viXra:1903.0231 [pdf]**
*submitted on 2019-03-12 22:17:08*

**Authors:** Federico Espil

**Comments:** 5 Pages.

Shortly from the Espil's theorem, we can derive the generalized Dirichlet
integral for any natural value when the hole integrand is raised to the n-th power.

**Category:** Functions and Analysis

[317] **viXra:1902.0508 [pdf]**
*submitted on 2019-02-28 06:04:36*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 5/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 4/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[316] **viXra:1902.0499 [pdf]**
*submitted on 2019-02-28 09:47:12*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 6/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 5/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[315] **viXra:1902.0496 [pdf]**
*submitted on 2019-02-28 11:53:12*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 7/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 6/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[314] **viXra:1902.0493 [pdf]**
*submitted on 2019-02-28 12:30:18*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 8/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 7/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[313] **viXra:1902.0492 [pdf]**
*submitted on 2019-02-28 15:18:28*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 9/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 8/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[312] **viXra:1902.0491 [pdf]**
*submitted on 2019-02-28 15:55:02*

**Authors:** Stephen C. Pearson.

**Comments:** 31 Pages.

This particular submission contains a copy [PART 10/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 9/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[311] **viXra:1902.0488 [pdf]**
*submitted on 2019-02-27 06:41:13*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains (inter alia) a copy [PART 1/10] of the author's original paper, which was completed on 5th March 2001 and thus comprises a total of 316 handwritten foolscap pages. Bearing in mind that it is a sequel to the author's previous set of submissions, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PARTS 1/6 to 6/6", its purpose is to enunciate various definitions and theorems, which pertain to the following topics, i.e. (a) the classification of quaternion hypercomplex functions; (b) further calculus of quaternion hypercomplex functions; (c) series expansions of quaternion hypercomplex functions. Many of the concepts presented therein are analogous to well established notions from real and complex variable analysis with any divergent results being due to the non-commutativity of quaternion products.

**Category:** Functions and Analysis

[310] **viXra:1902.0483 [pdf]**
*submitted on 2019-02-27 10:33:16*

**Authors:** Stephen C. Pearson.

**Comments:** 8 Pages.

This particular submission is an addendum to the author's previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 1/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[309] **viXra:1902.0478 [pdf]**
*submitted on 2019-02-27 12:41:38*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 2/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 1/10", which has been published under the 'VIXRA' Mathematics subheading:-'Functions and Analysis'.

**Category:** Functions and Analysis

[308] **viXra:1902.0472 [pdf]**
*submitted on 2019-02-27 14:44:08*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 3/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 2/10", which has been published under the 'VIXRA' Mathematics subheading:-'Functions and Analysis'.

**Category:** Functions and Analysis

[307] **viXra:1902.0466 [pdf]**
*submitted on 2019-02-28 05:25:15*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 4/10] of the author's original paper and is therefore a continuation of his previous submission, namely - "A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PART 3/10", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[306] **viXra:1902.0223 [pdf]**
*submitted on 2019-02-12 18:39:18*

**Authors:** Wolfgang W. D\"aumler, Hiroshi Okumura, Vyacheslav V. Puha, Saburou Saitoh

**Comments:** 16 Pages. We will introduce a beautiful horn torus model by Puha and D\"aumler for the Riemann sphere in complex analysis attaching the zero point and the point at infinity. Surprisingly enough, we can introduce analytical structure of conformality to the model.

In this paper, we will introduce a beautiful horn torus model by Puha and D\"aumler for the Riemann sphere in complex analysis attaching the zero point and the point at infinity. Surprisingly enough, we can introduce analytical structure of conformality to the model.

**Category:** Functions and Analysis

[305] **viXra:1901.0341 [pdf]**
*submitted on 2019-01-23 13:23:22*

**Authors:** H. C. Rhaly Jr.

**Comments:** 3 Pages.

Necessary and sufficient conditions are given for a special subclass of the factorable matrices to be hyponormal operators on $\ell^2$. Two examples are then given of polynomials that generate hyponormal weighted mean operators, and one that does not. Paired with the main result presented here, various computer software programs may then be used as an aid for classifying operators in that subclass as hyponormal or not.

**Category:** Functions and Analysis

[304] **viXra:1901.0294 [pdf]**
*submitted on 2019-01-20 04:27:04*

**Authors:** Jesús Sánchez

**Comments:** 9 Pages.

In this paper it will calculated that the Ramanujan summation of the Ln(n) series is:
lim┬█(n→∞)(Ln(1)+Ln(2)+Ln(3)+⋯Ln(n))=Ln(-γ)=Ln(γ)+(2k+1)πi
Being γ the Euler-Mascheroni constant 0.577215... The solution is valid for every integer number k (it has infinite solutions). The series are divergent because Ln(n) tends to infinity as n tends to infinity. But, as in other divergent series, a summation value can be associated to it, using different methods (Cesàro, Abel or Ramanujan).
If we take the logarithm of the absolute value (this is, we take only the real part of the solution), the value corresponds to the smooth continuation to the y axis of the curve that calculates the partial sums at every point, as we will see in the paper.
lim┬█(n→∞)(Ln|1|+Ln|2|+Ln|3|+⋯Ln|n|)=Ln|-γ|=Ln|γ|

**Category:** Functions and Analysis

[303] **viXra:1901.0134 [pdf]**
*submitted on 2019-01-10 21:01:16*

**Authors:** Mark C Marson

**Comments:** 18 Pages.

To gain true understanding of a subject it can help to study its origins and how its theory and practice changed over the years – and the mathematical field of calculus is no exception. But calculus students who do read accounts of its history encounter something strange – the claim that the theory which underpinned the subject for long after its creation was wrong and that it was corrected several hundred years later, in spite of the fact that the original theory never produced erroneous results. I argue here that both this characterization of the original theory and this interpretation of the paradigm shift to its successor are false. Infinitesimals, used properly, were never unrigorous and the supposed rigor of limit theory does not imply greater correctness, but rather the (usually unnecessary) exposition of hidden deductive steps. Furthermore those steps can, if set out, constitute a proof that original infinitesimals work in accordance with limit theory – contrary to the common opinion that the two approaches represent irreconcilable philosophical positions. This proof, demonstrating that we can adopt a unified paradigm for calculus, is to my knowledge novel although its logic may have been employed in another context. I also claim that non-standard analysis (the most famous previous attempt at unification) only partially clarified the situation because the type of infinitesimals it uses are critically different from original infinitesimals.

**Category:** Functions and Analysis

[302] **viXra:1812.0345 [pdf]**
*submitted on 2018-12-19 12:39:47*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 14 Pages. Submitted to the journal Annals of PDE. Comments welcome.

This paper represents an attempt to give a solution of Navier-Stokes equations under the assumptions $(A)$ of the problem as described by the Clay Mathematics Institute. After elimination of the pressure, we obtain the fundamental equations function of the velocity vector $u$ and vorticity vector $\Omega=curl(u)$, then we deduce the new equations for the description of the motion of viscous incompressible fluids, derived from the Navier-Stokes equations, given by:
\begin{eqnarray}
\nu \Delta \Omega -\frac{\partial \Omega}{\partial t}=0 \nonumber \\
\Delta p=-\sum^{i=3}_{i=1}\sum^{j=3}_{j=1}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i} \nonumber
\end{eqnarray}
Then, we give a proof of the solutions of the Navier-Stokes equations $u$ and $p$ that are smooth functions and $u$ verifies the condition of bounded energy.

**Category:** Functions and Analysis

[301] **viXra:1812.0321 [pdf]**
*submitted on 2018-12-18 12:16:21*

**Authors:** Markus Sprecher

**Comments:** 6 Pages.

Positivity of the Fourier transform of a convolution mask can be used to define an inverse convolution and show that the spatial dependency decays exponentially. In this document, we consider, for an arbitrary order, the shortest possible convolution mask which transforms samples of a function to Cardinal B-spline coefficients and show that it is unique and has indeed a positive Fourier transform. We also describe how the convolution mask can be computed including some code.

**Category:** Functions and Analysis

[300] **viXra:1812.0178 [pdf]**
*submitted on 2018-12-10 14:37:27*

**Authors:** Terrence P. Murphy

**Comments:** 4 Pages.

This paper presents an uncommon variation of proof by induction. We call it deferred induction by recursion. To set up our proof, we state (but do not prove) the Zeta Induction Theorem. Using that theorem, we provide an elementary proof of the Riemann Hypothesis. To be clear, we make no claim as to the usefulness of the Zeta Induction Theorem to the theory of the Riemann Zeta Function. In fact, we poke a bit of fun at the theorem in our Introduction (and, indirectly, in our Title).

**Category:** Functions and Analysis

[299] **viXra:1811.0510 [pdf]**
*submitted on 2018-11-29 10:08:17*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the logic of the definition of the k-triangular function in set theory and find it tautologous, hence confirming it as a theorem.

**Category:** Functions and Analysis

[298] **viXra:1811.0496 [pdf]**
*submitted on 2018-11-28 06:20:10*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 19 Pages.

Some versions of Dieudonne-type
convergence and uniform boundedness theorems are proved, for k-triangular and regular lattice group-valued set functions. We use
sliding hump techniques and direct methods. We extend earlier results, proved in the real case.

**Category:** Functions and Analysis

[297] **viXra:1811.0281 [pdf]**
*submitted on 2018-11-19 04:01:17*

**Authors:** Fayowole David Ayadi

**Comments:** 2 Pages.

This work is an alternate method of evaluating absolute (modulus) value.

**Category:** Functions and Analysis

[296] **viXra:1811.0244 [pdf]**
*submitted on 2018-11-15 06:38:41*

**Authors:** Yogesh J. Bagul

**Comments:** 4 Pages. In this paper , a mathematical mistake is discovered and another simple proof of the theorem is proposed.

In this short review note we show that the new proof of theorem
1.1 given by Zheng Jie Sun and Ling Zhu in the paper Simple proofs of the
Cusa-Huygens-type and Becker-Stark-type inequalities is logically incorrect
and present another simple proof of the same.

**Category:** Functions and Analysis

[295] **viXra:1811.0222 [pdf]**
*submitted on 2018-11-14 17:09:09*

**Authors:** Jonathan W. Tooker

**Comments:** 11 Pages.

We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals we prove that numbers in the neighborhood of infinity are ordinary real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity.

**Category:** Functions and Analysis

[294] **viXra:1810.0313 [pdf]**
*submitted on 2018-10-19 06:28:06*

**Authors:** Fayowole David Ayadi

**Comments:** 3 Pages.

Abstract:I can still remember my expression and feeling when we were asked to show that sup(A + B) = sup(A) + sup(B). It was an herculean task because the concept was too difficult to grasp with the use of approximation property until I discovered an easy route. In a bid to restrict my papers to just few pages, I will focus more on examples than theorems.

**Category:** Functions and Analysis

[293] **viXra:1810.0312 [pdf]**
*submitted on 2018-10-19 06:32:53*

**Authors:** Fayowole David Ayadi, Olabiyi Tobi David, Oluwajoba Godsfavour Favour, Oluwusi Faith Tolu, Isaleye Dorcas, Olorunisola Femi Stephen

**Comments:** 13 Pages.

Throughout these discussions the numbers epsilon > 0 and delta > 0 should be thought of as very small numbers. The aim of this part is to provide a working definition for the integral of a bounded function f(x) on the interval [a, b]. We will see that the real number "f(x)dx" is really the limit of sums of areas of rectangles.

**Category:** Functions and Analysis

[292] **viXra:1810.0308 [pdf]**
*submitted on 2018-10-19 12:22:45*

**Authors:** Zaid Laadjal

**Comments:** 4 Pages.

In this work, we apply the fixed point theorems, we study the existence and uniqueness of solutions for Langevin differential equations involving two ractional orders with multi-point boundary conditions on the half-line.

**Category:** Functions and Analysis

[291] **viXra:1810.0303 [pdf]**
*submitted on 2018-10-20 03:37:18*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[290] **viXra:1810.0170 [pdf]**
*submitted on 2018-10-10 15:15:29*

**Authors:** Zaid Laadjal

**Comments:** 4 Pages.

In this paper, we study the existence and uniqueness of solutions for Langevin differential equations of Riemman-Liouville fractional derivative with boundary value conditions on the half-line. By a classical fixed point theorems, several new existence results of solutions are obtained.

**Category:** Functions and Analysis

[289] **viXra:1810.0169 [pdf]**
*submitted on 2018-10-10 15:21:57*

**Authors:** Zaid Laadjal

**Comments:** 5 Pages.

In this paper, we investigate the existence and uniqueness of solutions for the following fractional Langevin equations with boundary conditions $$\left\{\begin{array}{l}D^{\alpha}( D^{\beta}+\lambda)u(t)=f(t,u(t)),\text{ \ \ \ }t\in(0,+\infty),\\ \\u(0)=D^{\beta}u(0)=0,\\ \\ \underset{t\rightarrow+\infty}{\lim}D^{\alpha-1}u(t)=\underset{t\rightarrow+\infty}{\lim}D^{\alpha +\beta-1}u(t)=au(\xi),\end{array}\right.$$ where $1<\alpha \leq2$ and$\ 0<\beta \leq1,$ such that $1<\alpha +\beta \leq2,$ with $\ a,b\in\mathbb{R},$ $\xi \in\mathbb{R}^{+},$\ and $D^{\alpha}$, $D^{\beta }$ are the Riemman-Liouville fractional derivative. Some new results are obtained by applying standard fixed point theorems.

**Category:** Functions and Analysis

[288] **viXra:1810.0168 [pdf]**
*submitted on 2018-10-10 15:28:53*

**Authors:** Zaid Laadjal

**Comments:** 6 Pages.

In this work, we use the fixed point theorems, we investigate the existence and uniqueness of solutions for a class of fractional Langevin equations with boundary value conditions on an infinite interval.

**Category:** Functions and Analysis

[287] **viXra:1809.0481 [pdf]**
*submitted on 2018-09-24 03:45:57*

**Authors:** Michael Atiyah

**Comments:** 5 Pages.

The Riemann Hypothesis is a famous unsolved problem dating from 1859. This paper will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).

**Category:** Functions and Analysis

[286] **viXra:1809.0234 [pdf]**
*submitted on 2018-09-11 22:00:35*

**Authors:** Jonathan Tooker

**Comments:** 66 Pages.

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition fo the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Category:** Functions and Analysis

[285] **viXra:1809.0171 [pdf]**
*submitted on 2018-09-08 15:03:38*

**Authors:** Edigles Guedes

**Comments:** 3 Pages.

I derive an infinite product for the ratio of k-th power and factorial.

**Category:** Functions and Analysis

[284] **viXra:1808.0641 [pdf]**
*submitted on 2018-08-29 12:01:02*

**Authors:** Edigles Guedes

**Comments:** 5 Pages.

I derive some Ser's infinite product for exponential function and exponential of the digamma function; as well as an integral representation for the digamma function.

**Category:** Functions and Analysis

[283] **viXra:1808.0576 [pdf]**
*submitted on 2018-08-26 10:55:02*

**Authors:** Adel Farhat, Anouar Ben Mabrouk

**Comments:** 20 Pages.

In the present paper, new multifractal analysis of vector valued Ahlfors type measures is developed. Mutual multifractal generalizations f fractal measures such as Hausdorff and packing have been introduced with associated dimensions. Essential properties of these measures have been shown using convexity arguments.

**Category:** Functions and Analysis

[282] **viXra:1808.0515 [pdf]**
*submitted on 2018-08-22 14:21:38*

**Authors:** Edigles Guedes

**Comments:** 5 Pages.

I derive an integral representation for the Barnes G-function among other things.

**Category:** Functions and Analysis

[281] **viXra:1808.0514 [pdf]**
*submitted on 2018-08-22 14:23:28*

**Authors:** Edigles Guedes

**Comments:** 2 Pages.

I derive an infinite product for gamma function and infinite series for log gamma function.

**Category:** Functions and Analysis

[280] **viXra:1808.0233 [pdf]**
*submitted on 2018-08-16 09:49:42*

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

I derive some infinite product representations for the exponential function.

**Category:** Functions and Analysis

[279] **viXra:1808.0207 [pdf]**
*submitted on 2018-08-15 11:22:40*

**Authors:** Edigles Guedes

**Comments:** 9 Pages.

I derived an identity involving gamma functions and sine function at rational argument; hence, the representation of infinite product arose.

**Category:** Functions and Analysis

[278] **viXra:1808.0202 [pdf]**
*submitted on 2018-08-15 21:33:00*

**Authors:** Ruslan Sharipov

**Comments:** Designed for double sided printing, US Letter size, 35 pages, 5 color figures

Tetrahedral discretizations of the multielectron Schrödinger operator are suggested. They is based on tetrahedral triangulations of domains in R^{3}. Theoretical results proving that these discretizations are able to approximate energy levels of electrons in atoms and molecules are obtained.

**Category:** Functions and Analysis

[277] **viXra:1808.0154 [pdf]**
*submitted on 2018-08-12 21:21:29*

**Authors:** Seong Won Cha

**Comments:** 22 Pages.

We will show interesting properties of two-sided Laplace transform, mainly of positive even functions. Further, we will also prove that the Laguerre inequalities and generalized Laguerre inequalities are true and finally, the Riemann hypothesis is true.

**Category:** Functions and Analysis

[276] **viXra:1808.0136 [pdf]**
*submitted on 2018-08-10 10:32:29*

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

I used an identity for cosine function involving finite sum of the gamma functions; hence, the representation of infinite product arose.

**Category:** Functions and Analysis

[275] **viXra:1808.0116 [pdf]**
*submitted on 2018-08-10 07:35:26*

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

I corrected the Theorem 21 of previous paper, obtaining an identity for sine function at rational argument involving finite sum of the gamma functions; hence, the representation of infinite product arose.

**Category:** Functions and Analysis

[274] **viXra:1808.0053 [pdf]**
*submitted on 2018-08-04 12:22:26*

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

I derive an identity for the decomposition of the Pochhammer's symbol.

**Category:** Functions and Analysis

[273] **viXra:1807.0532 [pdf]**
*submitted on 2018-07-31 08:39:29*

**Authors:** Edigles Guedes

**Comments:** 5 Pages.

I derive some finite product representations of gamma functions for the Pochhammer's symbol at rational argument.

**Category:** Functions and Analysis

[272] **viXra:1807.0475 [pdf]**
*submitted on 2018-07-28 20:40:47*

**Authors:** Edigles Guedes

**Comments:** 15 pages.

I derived identities for some surd numbers, involving gamma functions; thence, I have represented them as infinite products.

**Category:** Functions and Analysis

[271] **viXra:1807.0324 [pdf]**
*submitted on 2018-07-20 12:04:12*

**Authors:** Zaid Laadjal

**Comments:** Pages.

In this paper, we study an open problem; where we obtained several results for a Lyapunov-type and Hartman-Wintner-type inequalities for a Hadamard fractional differential equation on a general interval [a;b],(1≤a<b) with the boundary value conditions.

**Category:** Functions and Analysis

[270] **viXra:1807.0228 [pdf]**
*submitted on 2018-07-11 05:35:49*

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

We derive some identities for limit of the exponential for digamma function, k-power and exponential function, involving gamma functions and Pochhammer symbols.

**Category:** Functions and Analysis

[269] **viXra:1807.0227 [pdf]**
*submitted on 2018-07-11 05:38:23*

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

I derive some news identities for limit of the exponential of Pi/8, involving Pochhammer symbols and secant function.

**Category:** Functions and Analysis

[268] **viXra:1807.0135 [pdf]**
*submitted on 2018-07-07 01:53:55*

**Authors:** Viktor Strohm

**Comments:** 4 Pages.

The motion of a point along an ellipse under the action of a generalized force is investigated.
Result: differential equation of second-order curves with respect to the focus, differential equation of curves of the second order with respect to the center, general differential equation of second order curves. Several examples of the application of these equations are proposed.

**Category:** Functions and Analysis

[267] **viXra:1806.0464 [pdf]**
*submitted on 2018-06-30 13:06:43*

**Authors:** Thinh D. Nguyen

**Comments:** 1 Page.

We only point out that the work of algorithmic algebra community is not enough, at least so far.

**Category:** Functions and Analysis

[266] **viXra:1806.0444 [pdf]**
*submitted on 2018-06-28 10:42:13*

**Authors:** Hassine Saidane

**Comments:** 8 Pages.

Based on the observation that several physical, biological and social proceesses seem to be optimizing an objective function such as an action or a utility, the Central Principle of Science was deemed to be Optimization. Indeed, optimization proved to be an efficient tool for uncovering several scientific laws and proving some scientific theories. In this paper, we use this paradigm to identify the location of the nontrivial zeros of the Riemann Zeta function (RZF).This approach enabled the formulation of this problem as a constrained optimization problem where a simple objective function referred to here as the “Push-Pull Action” is maximized. The solution of the resulting constrained nonlinear optimization problem proved that nontrivial zeros of RZF are located on the critical line. In addition to proving the Riemann Hypothesis, this approach unveiled a plausible law of “Maximum Action of Push-Pull” that seems to be driving RZF to its equilibrium states at the different heights where it reaches its nontrivial zeros. We also show that this law applies to functions exhibiting the same properties as RZF.

**Category:** Functions and Analysis

[265] **viXra:1806.0360 [pdf]**
*submitted on 2018-06-24 12:50:58*

**Authors:** Thinh Nguyen

**Comments:** 17 Pages.

The multi-homogeneous B´ezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the B´ezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous B´ezout number is actually NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multihomogeneous B´ezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP⊇NP.

**Category:** Functions and Analysis

[264] **viXra:1806.0326 [pdf]**
*submitted on 2018-06-22 12:30:06*

**Authors:** Tejas Chandrakant Thakare

**Comments:** 3 Pages. Please feel free to comment on this study

Using method of integration as the limit of sum we can easily evaluate sum of an infinite series in which 1/n is common from every term such that n→∞ (n∈N). However in this method we do some rigorous calculations before integration. In this paper, in order to minimize the labor involved in this process I propose an alternative new method for finding the sum of an infinite series in which 1/n is common from every term such that n→∞.

**Category:** Functions and Analysis

[263] **viXra:1806.0239 [pdf]**
*submitted on 2018-06-17 23:43:19*

**Authors:** Michael Parfenov

**Comments:** 18 Pages.

This paper is the third paper of the cycle devoted to the theory of essentially adequate quaternionic differentiability. It is established that the quaternionic holomorphic (ℍ -holomorphic) functions, satisfying the essentially adequate generalization of Cauchy-Riemann’s equations, make up a very remarkable class: generally non-commutative quaternionic multiplication behaves as commutative in the case of multiplication of ℍ -holomorphic functions. Everyone can construct such ℍ-holomorphic functions by replacing a complex variable as a single whole by a quaternionic one in expressions for complex holomorphic functions, and thereafter verify their commutativity. This property, which is confirmed by a lot of ℍ-holomorphic functions, gives conclusive evidence that the developed theory is true. The rules for quaternionic differentiation of combinations of ℍ-holomorphic functions find themselves similar to those from complex analysis: the formulae for differentiation of sums, products, ratios, and compositions of H-holomorphic functions as well as quaternionic power series, are fully identical to their complex analogs. The example of using the deduced rules is considered and it is shown that they reduce essentially the volume of calculations. The base notions of complex Maclaurin series expansions are adapted to the quaternion case.

**Category:** Functions and Analysis

[262] **viXra:1806.0067 [pdf]**
*submitted on 2018-06-07 04:22:20*

**Authors:** Claude Michael Cassano

**Comments:** 9 Pages.

Theorems establishing exact solution for any linear ordinary differential equation of arbitrary order (homogeneous and inhomogeneous) are presented and proven.

**Category:** Functions and Analysis

[261] **viXra:1806.0047 [pdf]**
*submitted on 2018-06-06 04:42:10*

**Authors:** Claude Michael Cassano

**Comments:** 18 Pages.

Further development of exactly solving second order linear ordinary differential equations, and related non-linear ordinary differential equations.

**Category:** Functions and Analysis

[260] **viXra:1804.0405 [pdf]**
*submitted on 2018-04-26 11:14:24*

**Authors:** Adel Farhat, Anouar Ben Mabrouk

**Comments:** 19 Pages.

In the present work we are concerned with some density estimations of vector valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures satisfying a weak quasi-Alfors regularity. Mixed multifractal generalizations of densities are then introduced and studied in a framework of relative mixed multifractal analysis.

**Category:** Functions and Analysis

[259] **viXra:1804.0264 [pdf]**
*submitted on 2018-04-20 06:18:07*

**Authors:** Andrej Liptaj

**Comments:** 10 Pages.

Abstract I focus in this text on the construction of functions f_{j} with the delta property C_{i}\left(f_{j}\right)=\delta_{i,j}, where C_{i} are operators which associate to a function its i-th raw moment. A formal method for their construction is found, however results are divergent, from what a non-existence of such functions is conjectured. This also prevents an elegant series expansion with order-by-order moment matching. For a finite interval some partial results are presented: a method of expansion into raw moment series for finite number of moments and a “non-delta” method based on computing Legendre-expansion coefficients from moments (an already known method [1]). As by-product some coefficients formulas are found: coefficients for expanding a Hermite function into the Taylor series and coefficients for expanding into the Taylor series an element of a Fourier series (i.e. common formula for sine and cosine) thus formally merging the two (sine and cosine) Fourier sub-series into one.

**Category:** Functions and Analysis

[258] **viXra:1803.0498 [pdf]**
*submitted on 2018-03-22 20:24:32*

**Authors:** John Herapath, Quincy Howard Xavier, Carl Wigert

**Comments:** 1 Page.

In this document, we present several important insights concerning the Riemann Zeta
function and the locations of its zeros. More importantly, we prove that we
should be awarded the $1 000 000 prize for proving or disproving the Riemann
hypothesis

**Category:** Functions and Analysis

[257] **viXra:1803.0001 [pdf]**
*submitted on 2018-03-01 03:59:30*

**Authors:** Andrej Liptaj

**Comments:** 9 Pages.

A novel method of function expansion is presented. It is based on matching the definite integrals of the derivatives of the function to be approximated by appropriate polynomials. The method is fully integral-based, it is easy to construct and it presumably slightly outperforms Taylor series in the convergence rate.

**Category:** Functions and Analysis

[296] **viXra:1911.0343 [pdf]**
*replaced on 2019-12-07 19:06:40*

**Authors:** Daniel Thomas Hayes

**Comments:** 8 Pages.

A proposed solution to the millennium problem on the existence and smoothness of the Navier-Stokes equations.

**Category:** Functions and Analysis

[295] **viXra:1911.0343 [pdf]**
*replaced on 2019-11-30 04:29:26*

**Authors:** Daniel Thomas Hayes

**Comments:** 8 Pages.

A proposed solution to the millennium problem on the existence and smoothness of the Navier-Stokes equations.

**Category:** Functions and Analysis

[294] **viXra:1911.0343 [pdf]**
*replaced on 2019-11-24 18:45:27*

**Authors:** Daniel Thomas Hayes

**Comments:** 7 Pages.

**Category:** Functions and Analysis

[293] **viXra:1910.0518 [pdf]**
*replaced on 2019-10-31 05:38:42*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages. Many improvements.

We motivate and give a proof of the fundamental theorem of algebra using high school algebra.

**Category:** Functions and Analysis

[292] **viXra:1910.0518 [pdf]**
*replaced on 2019-10-29 10:01:27*

**Authors:** Timothy W. Jones

**Comments:** 9 Pages. Following comments received improvements have been made.

We give a sequence of inferences from high school algebra that can be related to the fundamental theorem of algebra (FTA). The sequence builds to an easy proof of this theorem. In passing we mention two proofs given in complex analysis courses; these theorems use Rouche and Liouville. These proofs, although short, require differential and integral calculus for complex variables. They do, however, have simple concepts that can help with a simpler, albeit longer, proof. The proof given here is easy enough for good high school students. It may also pique their curiosity about more advanced complex and real analysis courses.

**Category:** Functions and Analysis

[291] **viXra:1910.0518 [pdf]**
*replaced on 2019-10-27 08:35:29*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages. Some further edits per suggested comments.

We give a sequence of easy inferences from typical topics in high school algebra that relate to the fundamental theorem of algebra (FTA). The sequence builds to an easy proof of FTA. In passing we mention two proofs typically given in complex analysis courses. These proofs, although short, require developing differential and integral calculus for complex variables. The proof given here is leisurely and easy -- enough for good high school and typical calculus students.

**Category:** Functions and Analysis

[290] **viXra:1910.0518 [pdf]**
*replaced on 2019-10-26 11:10:26*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages. A few edits and elaborations added.

We give a sequence of easy inferences from typical topics in high school algebra that relate to the fundamental theorem of algebra (FTA). The sequence builds to an easy proof of FTA. In passing we mention two proofs typically given in complex analysis courses. These proofs, although short, require developing differential and integral calculus for complex variables. The proof given here is leisurely and easy -- enough for good high school and typical calculus students.

**Category:** Functions and Analysis

[289] **viXra:1910.0064 [pdf]**
*replaced on 2019-11-06 19:11:42*

**Authors:** Robert Jackson

**Comments:** 15 Pages. contact rljacksonmd@gmail.com

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study discusses a new method for solving PDEs called the generating function technique (GFT) which may establish a new precedence with respect to SEM. First, the study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, QFT and Good Boussinesq equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis

[288] **viXra:1910.0064 [pdf]**
*replaced on 2019-10-19 11:43:00*

**Authors:** Robert Jackson

**Comments:** 15 Pages.

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study discusses a new method for solving PDEs called the generating function technique (GFT) which may establish a new precedence with respect to SEM. First, the study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, QFT and Good Boussinesq equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis

[287] **viXra:1910.0064 [pdf]**
*replaced on 2019-10-15 10:26:40*

**Authors:** Robert Jackson

**Comments:** 15 Pages.

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study will discuss a new method for solving PDEs called the generating function technique (GFT) which may establish a new precedence with respect to SEM. First, this study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, QFT and Good Boussinesq equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis

[286] **viXra:1910.0064 [pdf]**
*replaced on 2019-10-10 16:25:43*

**Authors:** Robert Jackson

**Comments:** 15 Pages. contact rljacksonmd@gmail.com

The current gold standard for solving nonlinear partial differential equations, or PDEs, is the simplest equation method, or SEM. As a matter of fact, another prior technique for solving such equations, the G'/G-expansion method, appears to branch from the simplest equation method (SEM). This study will discuss a new method for solving PDEs called the generating function technique (GFT) which may establish a new precedence with respect to SEM. First, this study shows how GFT relates to SEM and the G'/G-expansion method. Next, the paper describes a new theorem that incorporates GFT, Ring and Knot theory in the finding of solutions to PDEs. Then the novel technique is applied in the derivation of new solutions to the Benjamin-Ono, QFT and Good Boussinesq equations. Finally, the study concludes via a discourse on the reasons why the technique is likely better than SEM and G'/G-expansion method, the scope and range of what GFT could ultimately accomplish, and the elucidation of a putative new branch of calculus, called "diversification".

**Category:** Functions and Analysis

[285] **viXra:1909.0294 [pdf]**
*replaced on 2019-09-16 03:56:42*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove a version of the Beppo Levi`s lemma for the complex valued functions. This achieved by making a more stronger asumption that is assumed in Beppo Levi`s lemma. We will assume that the sum of measurable functions that is absolutely convergent almost everywhere is integrable. We will prove that it implies the asumptions of the Beppo Levi lemma, if we consider functions that are non-negative. It can be argued that our version is more suitable to applications, and we will prove a new probability law. We will show that with our asumptions in probability theory it follows that the expected value is countable additive. Moreover, it follows that in strong law of large numbers we don`t need to make any asumptions on distributions and the mean of the sample will convergence almost surely to the mean of the expected values.

**Category:** Functions and Analysis

[284] **viXra:1908.0434 [pdf]**
*replaced on 2019-09-01 13:45:41*

**Authors:** Kouider Mohammed Ridha

**Comments:** 6 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan superficially and he didn't know her as a numerical method for finding the roots of a function. Hence in this study we define a new numerical method base on MBM with error bound and the number iterations necessary. Finally we present our new MBM for multi-roots with the R software.

**Category:** Functions and Analysis

[283] **viXra:1908.0434 [pdf]**
*replaced on 2019-08-21 08:37:00*

**Authors:** Kouider Mohammed Ridha

**Comments:** 5 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan, (2013, [3]) superficially and didn't know her as a numerical method for finding the roots of a function. Hence in this study we define her as a new numerical method with error bound and the number iterations necessary. Finally we present a new MBM for multi-roots with the R software.

**Category:** Functions and Analysis

[282] **viXra:1906.0415 [pdf]**
*replaced on 2019-06-23 06:52:35*

**Authors:** Alexander Domoshnitsky, Oleg Kupervasser, Hennadii Kutomanov

**Comments:** 6 Pages.

In this paper, new explicit tests for exponential stability of systems of second order equations are proposed. Our approach is based on nonoscillation of solutions of the corresponding diagonal scalar second order delay differential equations.

**Category:** Functions and Analysis

[281] **viXra:1906.0329 [pdf]**
*replaced on 2019-06-21 07:53:29*

**Authors:** Pith Peishu Xie

**Comments:** 2 Pages.

The Operator axioms have deduced number systems. In this paper, we conjecture some inequalities in Operator axioms. The general inequalities show the value of Operator axioms.

**Category:** Functions and Analysis

[280] **viXra:1906.0236 [pdf]**
*replaced on 2019-07-06 10:52:26*

**Authors:** Jonathan W. Tooker

**Comments:** 5 Pages.

In this brief note, we propose a set of operations for the affinely extended real number called infinity. Under the terms of the proposition, we show that the Riemann zeta function has infinitely many non-trivial zeros on the complex plane.

**Category:** Functions and Analysis

[279] **viXra:1906.0163 [pdf]**
*replaced on 2019-06-20 06:50:03*

**Authors:** Andrej Liptaj

**Comments:** 8 Pages.

Derivative of a function can be expressed in terms of integration over a small neighborhood of the point of differentiation, so-called differentiation by integration method. In this text a maximal generalization of existing results which use one-dimensional integrals is presented together with some interesting non-analytic weight functions.

**Category:** Functions and Analysis

[278] **viXra:1901.0341 [pdf]**
*replaced on 2019-01-24 13:38:55*

**Authors:** H. C. Rhaly Jr.

**Comments:** 4 Pages.

Necessary and sufficient conditions are given for a special subclass of the factorable matrices to be hyponormal operators on $\ell^2$. Three examples are then given of polynomials that generate hyponormal weighted mean operators, and one example that does not. Paired with the main result presented here, various computer software programs may then be used as an aid for classifying operators in that subclass as hyponormal or not.

**Category:** Functions and Analysis

[277] **viXra:1812.0345 [pdf]**
*replaced on 2018-12-20 08:27:21*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 14 Pages. Submitted to the journal Annals of PDE. Comments welcome.

This paper represents an attempt to give a solution of the Navier-Stokes equations under the assumptions (A) of the problem as described by the Clay Mathematics Institute. After elimination of the pressure, we obtain the fundamental equations function of the velocity vector u and vorticity vector \Omega=curl(u), then we deduce the new equations for the description of the motion of viscous incompressible fluids, derived from the Navier-Stokes equations, given by:
\nu \Delta \Omega -\frac{\partial \Omega}{\partial t}=0
\Delta p=-\sum^{i=3}_{i=1}\sum^{j=3}_{j=1}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}
Then, we give a proof of the solution of the Navier-Stokes equations u and p that are smooth functions and u verifies the condition of bounded energy.

**Category:** Functions and Analysis

[276] **viXra:1812.0178 [pdf]**
*replaced on 2018-12-12 11:29:13*

**Authors:** Terrence P. Murphy

**Comments:** 4 Pages.

This paper presents an uncommon variation of proof by induction. We call it deferred induction by recursion. To set up our proof, we state (but do not prove) the Zeta Induction Theorem. We then assume that theorem is true and provide an elementary proof of the Riemann Hypothesis (showing their equivalence).

**Category:** Functions and Analysis

[275] **viXra:1811.0222 [pdf]**
*replaced on 2019-10-27 16:08:48*

**Authors:** Jonathan W. Tooker

**Comments:** 32 Pages.

We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals, we prove that numbers in the neighborhood of infinity are ordinary real numbers of the type detailed in Euclid's Elements. We show that real numbers in the neighborhood of infinity obey the Archimedes property of real numbers. The main result is an application in complex analysis. We show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity.

**Category:** Functions and Analysis

[274] **viXra:1811.0222 [pdf]**
*replaced on 2019-08-17 15:44:20*

**Authors:** Jonathan W. Tooker

**Comments:** 25 Pages.

We give a geometric definition of real numbers. We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals, we prove that numbers in the neighborhood of infinity are ordinary real numbers. We show that real numbers in the neighborhood of infinity obey the Archimedes property of real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line.

**Category:** Functions and Analysis

[273] **viXra:1811.0222 [pdf]**
*replaced on 2019-06-27 21:04:14*

**Authors:** Jonathan W. Tooker

**Comments:** 14 Pages. significant changes from v1

We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals, we prove that numbers in the neighborhood of infinity are ordinary real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line for every application in quantum field theory.

**Category:** Functions and Analysis

[272] **viXra:1811.0222 [pdf]**
*replaced on 2018-12-10 09:38:19*

**Authors:** Jonathan W. Tooker

**Comments:** 12 Pages.

We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals we prove that numbers in the neighborhood of infinity are ordinary real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity.

**Category:** Functions and Analysis

[271] **viXra:1811.0222 [pdf]**
*replaced on 2018-11-23 21:14:57*

**Authors:** Jonathan W. Tooker

**Comments:** 12 Pages.

We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals we prove that numbers in the neighborhood of infinity are ordinary real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity.

**Category:** Functions and Analysis

[270] **viXra:1811.0222 [pdf]**
*replaced on 2018-11-18 01:28:07*

**Authors:** Jonathan W. Tooker

**Comments:** 11 Pages.

**Category:** Functions and Analysis

[269] **viXra:1811.0222 [pdf]**
*replaced on 2018-11-16 21:03:52*

**Authors:** Jonathan W. Tooker

**Comments:** 11 Pages. fixed a catastophic error associated with Def 1.3 in v1

**Category:** Functions and Analysis

[268] **viXra:1810.0308 [pdf]**
*replaced on 2019-04-04 22:58:32*

**Authors:** Zaid Laadjal

**Comments:** 4 Pages.

In this work, we apply the fixed point theorems, we study the existence and uniqueness of solutions for Langevin differential equations involving two ractional orders with multi-point boundary conditions on the half-line.

**Category:** Functions and Analysis

[267] **viXra:1810.0303 [pdf]**
*replaced on 2019-05-18 11:00:44*

**Authors:** Johan Aspegren

**Comments:** 3 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[266] **viXra:1810.0303 [pdf]**
*replaced on 2019-05-03 10:42:47*

**Authors:** Johan Aspegren

**Comments:** 3 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[265] **viXra:1810.0303 [pdf]**
*replaced on 2018-10-24 04:39:25*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[264] **viXra:1810.0303 [pdf]**
*replaced on 2018-10-22 08:53:49*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[263] **viXra:1810.0303 [pdf]**
*replaced on 2018-10-20 18:05:33*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[262] **viXra:1809.0234 [pdf]**
*replaced on 2019-08-03 19:27:14*

**Authors:** Jonathan W. Tooker

**Comments:** 48 Pages. Greatly improved v7

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we will show that the new representation has special properties which allow for a modification to the transformation law for the variation which preserves, in certain cases, the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. We use the modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Category:** Functions and Analysis

[261] **viXra:1809.0234 [pdf]**
*replaced on 2019-06-27 11:15:56*

**Authors:** Jonathan W. Tooker

**Comments:** 47 Pages. Greatly improved in v6

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Category:** Functions and Analysis

[260] **viXra:1809.0234 [pdf]**
*replaced on 2018-11-06 23:43:46*

**Authors:** Jonathan W. Tooker

**Comments:** 71 Pages. Greatly improved in v5

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Category:** Functions and Analysis

[259] **viXra:1809.0234 [pdf]**
*replaced on 2018-09-20 05:27:16*

**Authors:** Jonathan W. Tooker

**Comments:** 67 Pages.

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Category:** Functions and Analysis

[258] **viXra:1809.0234 [pdf]**
*replaced on 2018-09-14 12:29:27*

**Authors:** Jonathan W. Tooker

**Comments:** 66 Pages.

**Category:** Functions and Analysis

[257] **viXra:1809.0234 [pdf]**
*replaced on 2018-09-13 09:31:38*

**Authors:** Jonathan W. Tooker

**Comments:** 66 Pages.

**Category:** Functions and Analysis

[256] **viXra:1808.0136 [pdf]**
*replaced on 2018-08-27 11:22:36*

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

I used an identity for cosine function involving finite product of the gamma functions; hence, the representation of infinite product arose.

**Category:** Functions and Analysis

[255] **viXra:1807.0324 [pdf]**
*replaced on 2018-07-28 16:23:07*

**Authors:** Zaid Laadjal

**Comments:** Pages. After having some modifications on this preprint with other authors, it was published in J. Math. Inequal., Vol. 13, no. 3, (2019), 789–799., Doi: dx.doi.org/10.7153/jmi-2019-13-54

In this paper, we studied an open problem, where using two different methods, we obtained several results for a Lyapunov-type and Hartman-Wintner-type inequalities for a Hadamard fractional differential equation on a general interval [a;b],(1≤a<b) with the boundary value conditions.

**Category:** Functions and Analysis

[254] **viXra:1806.0444 [pdf]**
*replaced on 2018-07-01 07:28:05*

**Authors:** Hassine Saidane

**Comments:** 8 Pages.

Abstract. Based on the observation that several physical, biological and social processes seem to be optimizing an objective function such as an action or a utility, the Central Principle of Science was deemed to be Optimization. Indeed, optimization proved to be an efficient tool for uncovering several scientific laws and proving some scientific theories. In this paper, we use this paradigm to identify the location of the nontrivial zeros of the Riemann Zeta function (RZF). This approach enabled the formulation of this problem as a constrained optimization problem where a simple objective function referred to here as the “Push-Pull Action” is maximized. The solution of the resulting constrained nonlinear optimization problem proved that nontrivial zeros of RZF are located on the critical line. In addition to proving the Riemann Hypothesis, this approach unveiled a plausible law of “Maximum Action of Push-Pull” that seems to be driving RZF to its equilibrium states at the different heights where it reaches its nontrivial zeros. We also show that this law applies to functions exhibiting the same properties as RZF.
Keywords: Zeta function, Riemann Hypothesis, Constrained Optimization

**Category:** Functions and Analysis

[253] **viXra:1806.0082 [pdf]**
*replaced on 2018-07-26 19:07:43*

**Authors:** Jonathan W. Tooker

**Comments:** 5 Pages. two figures

This paper examines some familiar results from complex analysis in the framework of hypercomplex analysis. It is usually taught that the oscillatory behavior of sine waves means that they have no limit at infinity but here we derive definite limits. Where a central element in the foundations of complex analysis is that the complex conjugate of a C-number is not analytic at the origin, we introduce the tools of hypercomplex analysis to show that the complex conjugate of a *C-number is analytic at the origin.

**Category:** Functions and Analysis

[252] **viXra:1806.0082 [pdf]**
*replaced on 2018-06-09 05:42:18*

**Authors:** Jonathan W. Tooker

**Comments:** 4 Pages. two figures

This paper examines some familiar results from complex analysis in the framework of hypercomplex analysis. It is usually taught that the oscillatory behavior of sine waves means that they have no limit at infinity but here we derive definite limits. Where a central element in the foundations of complex analysis is that the complex conjugate of a $\mathbb{C}$-number is not analytic at the origin, we introduce the tools of hypercomplex analysis to show that the complex conjugate of a $^\star\mathbb{C}$-number is analytic at the origin.

**Category:** Functions and Analysis

[251] **viXra:1804.0264 [pdf]**
*replaced on 2018-04-23 04:22:49*

**Authors:** Andrej Liptaj

**Comments:** 10 Pages.

I focus in this text on the construction of functions f_{j} with the delta property C_{i}\left(f_{j}\right)=\delta_{i,j}, where C_{i} are operators which associate to a function its i-th raw moment. A formal method for their construction is found, however results are divergent, from what a non-existence of such functions is conjectured. This also prevents an elegant series expansion with order-by-order moment matching. For a finite interval some partial results are presented: a method of expansion into raw moment series for finite number of moments and a “non-delta” method based on computing Legendre-expansion coefficients from moments (an already known method [1]). As by-product some coefficients formulas are found: coefficients for expanding a Hermite function into the Taylor series and coefficients for expanding into the Taylor series an element of a Fourier series (i.e. common formula for sine and cosine) thus formally merging the two (sine and cosine) Fourier sub-series into one.

**Category:** Functions and Analysis

[250] **viXra:1803.0001 [pdf]**
*replaced on 2018-03-05 04:58:02*

**Authors:** Andrej Liptaj

**Comments:** 8 Pages.

A method of function expansion is presented. It is based on matching the definite integrals of the
derivatives of the function to be approximated by a series of (scaled) Bernoulli polynomials. The method is fully integral-based, easy to construct and presumably slightly outperforms Taylor series in the convergence rate. Text presents already known results.

**Category:** Functions and Analysis

[249] **viXra:1803.0001 [pdf]**
*replaced on 2018-03-01 15:48:19*

**Authors:** Andrej Liptaj

**Comments:** 7 Pages.

A novel method of function expansion is presented. It is based on matching the definite integrals of
the derivatives of the function to be approximated by a series of (scaled) Bernoulli polynomials. The
method is fully integral-based, easy to construct and presumably slightly outperforms Taylor series in the convergence rate.

**Category:** Functions and Analysis