**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(16) - 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) - 1706(1) - 1707(4) - 1708(3) - 1709(9) - 1710(5) - 1711(4) - 1712(7)

2018 - 1801(1) - 1802(6) - 1803(2)

Any replacements are listed farther down

[264] **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

[263] **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

[262] **viXra:1802.0267 [pdf]**
*submitted on 2018-02-19 17:56:17*

**Authors:** Ayal Sharon

**Comments:** 18 Pages.

Euler's formula is used to derive a trigonometric version of the Dirichlet series $\zeta(s)=\sum n^{-s}$, which is divergent in the half-plane $\sigma \le 1$, wherein $s \in \mathbb{C}$ and $s=\sigma +it$. Abel's lemma and Dirichlet's test incorrectly hold that trigonometric $\zeta(s)$ is convergent in the critical strip $0<\sigma \le 1$ at $t\ne0$, because they fail to consider a divergent monotonically decreasing series (e.g. the harmonic series) in combination with a bounded oscillating function having an increasing period duration (e.g. $f(t, n) = \sin(t \cdot \ln(n))$).

**Category:** Functions and Analysis

[261] **viXra:1802.0191 [pdf]**
*submitted on 2018-02-14 18:58:19*

**Authors:** Zeraoulia Rafik

**Comments:** 06 Pages. Thank's to the special function

In this note we present a new special function such that behaves
more like error function and since we arn’t able to express it as elementary
function using previous standard functions ,we only give it’s simple expression
in some range values using numerical approximation . and we will show how
it helps to get values of complicated integral which they arn’t available at
wolfram alpha and in the same time we will show it’s relationship with error
function and cumulative distribution function .

**Category:** Functions and Analysis

[260] **viXra:1802.0126 [pdf]**
*submitted on 2018-02-10 07:24:37*

**Authors:** Han Geurdes, Koji Nagata, Tadao Nakamura, Ahmed Farouk

**Comments:** 11 Pages. None

In the paper it is demonstrated that Bells theorem is an unprovable theorem. This inconsistency is similar to concrete mathematical incompleteness. The inconsistency is purely mathematical. Nevertheless the basic physics requirements of a local model are fulfilled.

**Category:** Functions and Analysis

[259] **viXra:1802.0120 [pdf]**
*submitted on 2018-02-10 14:44:28*

**Authors:** Zeraoulia Rafik

**Comments:** 23 Pages. I wish my results w'd be considerable for any futur refeered journal

In this note we present some new results about the analyticity of the functional-differential equation $ f'=e^{{f}^{-1}}$ at $ 0$ with $f^{-1}$ is a compositional inverse of $f$ , and the growth rate of $f_-(x)$ and $f_+(x)$ as $x\to \infty$ , and we will check the analyticity of some functional equations which they were studied before and had a relashionship with the titled functional-differential and we will conclude our work with a conjecture related to Borel- summability and some interesting applications of some divergents generating function with radius of convergent equal $0$ in number theory

**Category:** Functions and Analysis

[258] **viXra:1802.0094 [pdf]**
*submitted on 2018-02-08 07:08:19*

**Authors:** Jesús Álvarez Lobo

**Comments:** 2 Pages. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 22. Spanish.

Upper bound for the product of the sum of the reciprocals of n real numbers greater than or equal to 1 by the product of those increased by 1, and some variants.
Se establece una cota superior para el producto del sumatorio de los recíprocos de n números reales mayores o iguales que 1 por el producto de éstos incrementados en 1, y para algunas variantes.

**Category:** Functions and Analysis

[257] **viXra:1802.0021 [pdf]**
*submitted on 2018-02-02 16:57:10*

**Authors:** Jesús Álvarez Lobo

**Comments:** 10 Pages.

Usually, the complexity of a fractional function increases significantly in its second derivative, so the calculation of the second derivative can be tedious and difficult to simplify and evaluate its value at a point, especially if the abscise isn't an integer.
However, to determine whether a point at which cancels the first derivative of a function is a relative extremum (maximum or minimum) of it, is not necessary to know the value of the second derivative at the point but only its sign.
Motivated by these facts, we define a signum function for the second derivative of fractional functions in the domain of the roots of the first derivative of the function.
The method can dramatically simplify the search for maximum and minimum points in fractional functions and can be implemented by means of a simple algorithm.

**Category:** Functions and Analysis

[256] **viXra:1801.0096 [pdf]**
*submitted on 2018-01-08 07:56:30*

**Authors:** Martin Nicholson

**Comments:** 6 Pages.

In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters.

**Category:** Functions and Analysis

[255] **viXra:1712.0539 [pdf]**
*submitted on 2017-12-20 06:47:39*

**Authors:** Martin Nicholson

**Comments:** 8 Pages.

We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation $\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$ and to some other formulas.

**Category:** Functions and Analysis

[254] **viXra:1712.0519 [pdf]**
*submitted on 2017-12-19 19:49:52*

**Authors:** Seong Won Cha

**Comments:** 11 Pages.

We show that some interesting properties of the bilateral Laplace transform of even and positive functions both on the line z=x+iy0 and on a circle. We also show the Riemann hypothesis is true using these properties.

**Category:** Functions and Analysis

[253] **viXra:1712.0478 [pdf]**
*submitted on 2017-12-15 08:30:49*

**Authors:** Martin Nicholson

**Comments:** 10 Pages.

Several Fourier transformations of functions of one and two variables are evaluated and
then used to derive some integral and series identities. It is shown that certain two-
dimensional Mordell integrals factorize into product of two integrals and that the square
of the absolute value of the Mordell integral can be reduced to a single one-dimensional
integral. Some connections to elliptic functions and lattice sums are discussed.

**Category:** Functions and Analysis

[252] **viXra:1712.0463 [pdf]**
*submitted on 2017-12-16 01:01:52*

**Authors:** Carl Wigert, Quincy-Howard Xavier

**Comments:** 1 Page.

In this paper, we define very small numbers and very very small numbers and use them to construct derivatives as ratios of real numbers. We then use that result to rigorously prove that the chain rule treats derivatives as fractions being multiplied.

**Category:** Functions and Analysis

[251] **viXra:1712.0355 [pdf]**
*submitted on 2017-12-08 19:58:12*

**Authors:** Seong Won Cha

**Comments:** 9 Pages.

This is a brief report before writing a full paper.
We proved the Riemann hypothesis using the properties of the bilateral Laplace transform.

**Category:** Functions and Analysis

[250] **viXra:1712.0113 [pdf]**
*submitted on 2017-12-04 21:50:14*

**Authors:** D Williams

**Comments:** 8 Pages.

An overview of some types of multiplicative infinitesimal calculi is given. Analogs of standard results ("Simpson's" Product, "Maclurin's" Product, fundamental theorems, etc) are shown. An area that deserves more attention.

**Category:** Functions and Analysis

[249] **viXra:1712.0019 [pdf]**
*submitted on 2017-12-02 12:52:22*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

It is showed that a large class of functions defined by integrals verify the Riemann Hypothesis.

**Category:** Functions and Analysis

[248] **viXra:1711.0356 [pdf]**
*submitted on 2017-11-18 15:54:02*

**Authors:** Matanari Shimoinuda

**Comments:** 12 Pages.

The group X, which is proposed by A.Connes, is an interesting thing for number theory. Let's think of the trace of a regular representation U on X of the idele class. However it is hard to compute it since X is non-compact. In this article, we try to show that the trace is computable.

**Category:** Functions and Analysis

[247] **viXra:1711.0298 [pdf]**
*submitted on 2017-11-13 20:01:20*

**Authors:** D Williams

**Comments:** 3 Pages.

Some "continuous" (that is, over real numbers in the interval (0,1)) infinite products are given with their finite product approximations. THESE PRODUCTS DESERVE MORE STUDY.

**Category:** Functions and Analysis

[246] **viXra:1711.0297 [pdf]**
*submitted on 2017-11-13 20:05:52*

**Authors:** D Williams

**Comments:** 3 Pages.

Some examples of dx-less integrals are given with their finite sum approximations. They appear to have use in estimating long-term values of certain stochastic recursive functions. A request is made for determining convergence of such integrals.

**Category:** Functions and Analysis

[245] **viXra:1711.0257 [pdf]**
*submitted on 2017-11-09 15:57:32*

**Authors:** D Williams

**Comments:** 10 Pages.

An improved version of Stirling's Formula (which I call Neylon's Approximation) for n! is constructed using Product Integrals.

**Category:** Functions and Analysis

[244] **viXra:1710.0246 [pdf]**
*submitted on 2017-10-22 16:35:58*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 2 Pages. Riemann-Zeta Note.

None.

**Category:** Functions and Analysis

[243] **viXra:1710.0140 [pdf]**
*submitted on 2017-10-12 11:04:15*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 4 Pages.

We prove some Schur and limit theorems
for lattice group-valued k-triangular set functions with respect to filter convergence, by means of sliding hump-type techniques.
As consequences, we deduce some Vitali-Hahn-Saks and Nikodym-type theorems.

**Category:** Functions and Analysis

[242] **viXra:1710.0126 [pdf]**
*submitted on 2017-10-11 21:03:23*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 2 Pages. Mathematical certainty often does not translate; but here the stringent analytical means of it's establishment are presented.

Mathematical certainty is defined in terms of sets and deterministic variables; in terms of the error root mean squared deviation.

**Category:** Functions and Analysis

[241] **viXra:1710.0083 [pdf]**
*submitted on 2017-10-08 03:04:30*

**Authors:** Carlos Oscar Rodríguez Leal

**Comments:** 16 Pages. Paper writting in spanish. Paper presented at the VII International Congress of Numerical Methods, CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico.

In this work I develop numerical algorithms that can be applied directly to differential equations of the general form f (t, x, x ) = 0, without the need to cleared x . My methods are hybrid algorithms between standard methods of solving differential equations and methods of solving algebraic equations, with which the variable x is numerically cleared.
The application of these methods ranges from the ordinary differential equations of order one, to the more general case of systems of m equations of order n. These algorithms are applied to the solution of different physical-mathematical
equations.
Finally, the corresponding numerical analysis of existence, uniqueness, stability, consistency and convergence is made, mainly for the simplest case of a single ordinary differential equation of the first order.

**Category:** Functions and Analysis

[240] **viXra:1710.0036 [pdf]**
*submitted on 2017-10-03 21:20:37*

**Authors:** Hong Lai Zhu

**Comments:** 18 Pages.

In this paper, four kinds of Z Transformations are proposed to get many laws of general solutions of mth-order linear and nonlinear partial differential equations with n variables. Some general solutions of first-order linear partial differential equations, which cannot be obtained by using the characteristic equation method, can be solved by the Z Transformations. By comparing, we find that the general solutions of some first-order partial differential equations got by the characteristic equation method are not complete.

**Category:** Functions and Analysis

[239] **viXra:1709.0442 [pdf]**
*submitted on 2017-09-30 11:38:21*

**Authors:** Antoine Warnery

**Comments:** 11 Pages. French

The purpose of this study is to explore the mathematical principle of causality.

**Category:** Functions and Analysis

[238] **viXra:1709.0393 [pdf]**
*submitted on 2017-09-26 07:41:30*

**Authors:** Andrej Liptaj

**Comments:** 8 Pages. Text presents known results.

Multiplicative coefficients of a series of Bessel functions of the first kind can be adjusted so as to match desired values corresponding to a derivatives of a function to be expanded. In this way Neumann series of Bessel functions is constructed. Text presents known results.

**Category:** Functions and Analysis

[237] **viXra:1709.0357 [pdf]**
*submitted on 2017-09-23 12:37:20*

**Authors:** Johan Aspegren

**Comments:** 3 Pages.

In this article we will give a proof that the Kakeya tube conjecture implies the Kakeya conjecture.

**Category:** Functions and Analysis

[236] **viXra:1709.0310 [pdf]**
*submitted on 2017-09-20 13:49:06*

**Authors:** Misha Mikhaylov

**Comments:** 2 Pages.

This is the Russian version of my previous publication.

**Category:** Functions and Analysis

[235] **viXra:1709.0305 [pdf]**
*submitted on 2017-09-20 13:07:26*

**Authors:** Misha Mikhaylov

**Comments:** 2 Pages.

This sum for natural values is, of course, already calculated by Bernoulli himself – at least modern or relatively recent authors that deal with it usually refer to take into account Bernoulli numbers. But, apparently, this method is rather cumbersome. Therefore, there can be suggested another, easier way to do this, but without claiming of its superfluous rigidity.

**Category:** Functions and Analysis

[234] **viXra:1709.0304 [pdf]**
*submitted on 2017-09-20 07:24:33*

**Authors:** Richard J. Mathar

**Comments:** 48 Pages. Most of the content is the source code listing

Boys' Function F_m(z) that appears in the quantum mechanics of Gaussian Type Orbitals is
a special case of Kummer's confluent hypergeometric function. We evaluate its integral representation
of a product of a power and an exponential function over the unit interval
with the numerical Gauss-Jacobi quadrature. We provide an implementation in C for real values
of the argument z which basically employs a table of the weights and abscissae of the
quadrature rule for integer quantum numbers m <= 129.

**Category:** Functions and Analysis

[233] **viXra:1709.0281 [pdf]**
*submitted on 2017-09-18 20:00:56*

**Authors:** Johan Aspegren

**Comments:** 3 Pages.

In this paper we will prove a theorem that implies the Kakeya conjecture.

**Category:** Functions and Analysis

[232] **viXra:1709.0113 [pdf]**
*submitted on 2017-09-10 07:44:48*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained.

**Category:** Functions and Analysis

[231] **viXra:1709.0047 [pdf]**
*submitted on 2017-09-05 05:45:21*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure.

**Category:** Functions and Analysis

[230] **viXra:1708.0123 [pdf]**
*submitted on 2017-08-11 10:14:24*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations.

**Category:** Functions and Analysis

[229] **viXra:1708.0006 [pdf]**
*submitted on 2017-08-02 03:18:34*

**Authors:** E. U. Agom, M. S. Atureta

**Comments:** 4 Pages. ijsr.net publication

In this paper, we present a unified minimal compartmental model to estimate mathematically the concentration of a Therapeutic Agent injected intravenously in a steady state into Human tissues divided into two compartments; the blood and tissues. The model takes into consideration most, if not all physiological factors of the Human system in conformity with the physical realities vis-a-vis the Therapeutic Agent concentration before uptake by the compartments. The models were a system of first order non-homogeneous ordinary differential equations. And, the result from the models gives a zero concentration in both the blood and the tissues before the advent of the Therapeutic agent.

**Category:** Functions and Analysis

[228] **viXra:1708.0005 [pdf]**
*submitted on 2017-08-02 03:37:25*

**Authors:** E. U. Agom, A. M. Badmus

**Comments:** 6 Pages. ijesi.org paper

In this paper, we use Adomian Decomposition Method to numerically analyse second order nonlinear ordinary differential equations and implement the continuous algorithm in a discrete domain. This is facilitated by Maple package. And, the results from the two test problems used shows that the Adomian Decomposition Method is almost as the classical solutions.

**Category:** Functions and Analysis

[227] **viXra:1707.0246 [pdf]**
*submitted on 2017-07-18 08:02:22*

**Authors:** Eman.M.El-Nakeeb, Hewayda ElGhawalby, A.A.Salama, S.A.El-Hafeez

**Comments:** 13 Pages.

In this paper, we aim to apply the concepts of the neutrosophic crisp sets and its operations to the classical mathematical morphological operations, introducing what we call "Neutrosophic Crisp Mathematical Morphology". Several operators are to be developed, including the neutrosophic crisp dilation, the neutrosophic crisp erosion, the neutrosophic
crisp opening and the neutrosophic crisp closing.Moreover, we extend the definition of some morphological filters using the neutrosophic crisp sets concept. For instance, we introduce the neutrosophic crisp boundary extraction, the neutrosophic crisp Top-hat and the neutrosophic crisp Bottom- hat filters.The idea behind the new introduced operators and filters is to act on the image in the neutrosophic crisp domain instead of the spatial domain.

**Category:** Functions and Analysis

[226] **viXra:1707.0155 [pdf]**
*submitted on 2017-07-11 08:27:57*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 27 Pages.

A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement.

**Category:** Functions and Analysis

[225] **viXra:1707.0131 [pdf]**
*submitted on 2017-07-09 17:37:53*

**Authors:** Edigles Guedes

**Comments:** 3 Pages.

In this paper, I demonstrate one new infinite product representation for cosine
function, one new power series representation for tangent function and amazing identities
involving radical.

**Category:** Functions and Analysis

[224] **viXra:1707.0130 [pdf]**
*submitted on 2017-07-09 17:41:03*

**Authors:** Edigles Guedes, Cícera Guedes

**Comments:** 6 Pages.

In this paper, we demonstrate some limit's formulae for gamma function and
binomial coefficient among other things.

**Category:** Functions and Analysis

[223] **viXra:1706.0008 [pdf]**
*submitted on 2017-06-01 15:48:08*

**Authors:** Wilson Torres Ovejero

**Comments:** 12 Pages.

In mathematics, today there are six problems that take up to 200 years without solution, one of them is the demonstration of the Riemann hypothesis; Today I present a possible demonstration of this hypothesis that has spent 10 years of my life in its possible demonstration and I hope to have achieved it.

**Category:** Functions and Analysis

[222] **viXra:1705.0410 [pdf]**
*submitted on 2017-05-29 06:39:46*

**Authors:** Hong Lai Zhu

**Comments:** 71 Pages.

This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct.

**Category:** Functions and Analysis

[221] **viXra:1705.0399 [pdf]**
*submitted on 2017-05-28 00:50:44*

**Authors:** Andrzej Peczkowski

**Comments:** 15 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OY axis that crosses the OX axis at any angle.

**Category:** Functions and Analysis

[220] **viXra:1705.0398 [pdf]**
*submitted on 2017-05-28 00:55:17*

**Authors:** Andrzej Peczkowski

**Comments:** 14 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OX axis that crosses the OY axis at any angle.

**Category:** Functions and Analysis

[219] **viXra:1705.0397 [pdf]**
*submitted on 2017-05-28 01:04:24*

**Authors:** Andrzej Peczkowski

**Comments:** 17 Pages.

This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Part 3. Axes OX and OY intersect at any angle

**Category:** Functions and Analysis

[218] **viXra:1705.0249 [pdf]**
*submitted on 2017-05-16 08:26:20*

**Authors:** Andrej Liptaj

**Comments:** 6 Pages.

A set of functions which allows easy derivative-matching is proposed. Several examples of approximations are shown.

**Category:** Functions and Analysis

[217] **viXra:1705.0165 [pdf]**
*submitted on 2017-05-09 17:00:33*

**Authors:** Nicholas R. Wright

**Comments:** 6 Pages.

We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards.

**Category:** Functions and Analysis

[216] **viXra:1705.0028 [pdf]**
*submitted on 2017-05-02 15:33:07*

**Authors:** Morad Ahmad, Shaher Momani, Omar Abu Arqub, Mohammed Al-Smadi, Ahmed Alsaedi

**Comments:** 13 Pages.

In this paper, a powerful computational algorithm is developed for the solution of classes of singular second-order, three-point Volterra integrodifferential equations in favorable reproducing kernel Hilbert spaces. The solutions is represented in the form of series in the Hilbert space W₂³[0,1] with easily computable components. In finding the computational solutions, we use generating the orthogonal basis from the obtained kernel functions such that the orthonormal basis is constructing in order to formulate and utilize the solutions. Numerical experiments are carried where two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. Error estimates are proven that it converge to zero in the sense of the space norm. Several computational simulation experiments are given to show the good performance of the proposed procedure. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to multipoint singular boundary value problems restricted by Volterra operator.

**Category:** Functions and Analysis

[215] **viXra:1704.0282 [pdf]**
*submitted on 2017-04-21 20:56:48*

**Authors:** En-Lin Liu

**Comments:** 6 Pages. Quite trivial research XD

This article is concerned with the scattering problem for the defocusing nonlinear Schrödinger
equations (NLS) with a power nonlinear |u|^p u where 2/n < p < 4/n. We show that for any
initial data in H^{0,1}
x the solution will eventually scatter, i.e. U(-t)u(t) tends to some function
u+ as t tends to innity.

**Category:** Functions and Analysis

[214] **viXra:1704.0030 [pdf]**
*submitted on 2017-04-04 03:53:37*

**Authors:** Andrej Liptaj

**Comments:** 11 Pages.

Inspired by Taylor polynomials, several other approximations based on derivative-matching are proposed.

**Category:** Functions and Analysis

[213] **viXra:1703.0295 [pdf]**
*submitted on 2017-03-31 06:45:36*

**Authors:** Andrej Liptaj

**Comments:** 7 Pages.

A general recursive and limit formula for higher order derivatives of the inverse function is presented. The formula is next used in couple of mathematical applications: expansion of the inverse function into Taylor series, solving equations, constructing random numbers with a given distribution from uniformly distributed randomnumbers and expanding a function in the neighborhood of a given point in an alternative way to the Taylor expansion.

**Category:** Functions and Analysis

[212] **viXra:1703.0261 [pdf]**
*submitted on 2017-03-28 03:24:35*

**Authors:** J.A.J. van Leunen

**Comments:** 4 Pages.

Field equations occur in many physical theories. Most dynamic fields share a set of first and second order partial differential equations and differ in the kinds of artifacts that cause discontinuities. The paper restricts to first and second order partial differential equations. These equations can describe the interaction between the field and pointlike artifacts. The paper treats periodic and one-shot triggers in maximally three spatial dimensions. The paper applies quaternionic differential calculus. It uses the quaternionic nabla operator. This configuration implements the storage of dynamic geometric data as a combination of a proper timestamp and a three-dimensional spatial location in a quaternionic storage container. The storage format is Euclidean. The paper introduces warps and clamps as new types of super-tiny objects that constitute higher order objects.

**Category:** Functions and Analysis

[211] **viXra:1703.0253 [pdf]**
*submitted on 2017-03-27 00:05:14*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has prevented a novel scheme of Universal Evolution Model.

**Category:** Functions and Analysis

[210] **viXra:1703.0184 [pdf]**
*submitted on 2017-03-19 09:23:11*

**Authors:** F.L.B. Périat

**Comments:** 2 Pages.

This could help science going faster.

**Category:** Functions and Analysis

[209] **viXra:1703.0135 [pdf]**
*submitted on 2017-03-13 17:20:52*

**Authors:** Naoya Isobe

**Comments:** 13 Pages.

Global in time solvability of incompressive Navier-Stokes initial value problem in the whole space is proved using time transformation analysys.

**Category:** Functions and Analysis

[208] **viXra:1703.0134 [pdf]**
*submitted on 2017-03-13 17:22:16*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Global in Time Solvability of Incompressive NSIVP in the Whole Space).

**Category:** Functions and Analysis

[207] **viXra:1703.0133 [pdf]**
*submitted on 2017-03-13 17:59:21*

**Authors:** Naoya Isobe

**Comments:** 15 Pages.

Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys.

**Category:** Functions and Analysis

[206] **viXra:1703.0132 [pdf]**
*submitted on 2017-03-13 18:00:32*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Global in Time Solvability of Incompressive NSIVP in periodic Space).

**Category:** Functions and Analysis

[205] **viXra:1703.0121 [pdf]**
*submitted on 2017-03-13 06:11:06*

**Authors:** Naoya Isobe

**Comments:** 12 Pages.

Global in time solvability of incompressive Navier-Stokes initial value problem in the whole space is proved using time transformation analysys.

**Category:** Functions and Analysis

[204] **viXra:1703.0120 [pdf]**
*submitted on 2017-03-13 06:17:44*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Global in Time Solvability of Incompressive NSIVP in the Whole Space).

**Category:** Functions and Analysis

[203] **viXra:1703.0119 [pdf]**
*submitted on 2017-03-13 06:19:44*

**Authors:** Naoya Isobe

**Comments:** 14 Pages.

Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys.

**Category:** Functions and Analysis

[202] **viXra:1703.0118 [pdf]**
*submitted on 2017-03-13 06:21:47*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Global in Time Solvability of Incompressive NSIVP in periodic Space).

**Category:** Functions and Analysis

[201] **viXra:1703.0073 [pdf]**
*submitted on 2017-03-07 21:27:59*

**Authors:** Jonathan Tooker

**Comments:** 5 Pages. Five figures. Uploading rough draft for posterity.

We discuss the Riemann zeta function and make an argument against the Riemann hypothesis. While making the argument in the classical formalism, we discuss the argument as it relates to the theory of infinite complexity. We extend Riemann's own (planar) analytic continuation $\mathbb{R}\to\mathbb{C}^2$ into (bulk) hypercomplexity with $\mathbb{C}^2\to\,^\star\mathbb{C}$.

**Category:** Functions and Analysis

[200] **viXra:1702.0119 [pdf]**
*submitted on 2017-02-09 07:58:42*

**Authors:** Matthew Marko

**Comments:** 114 pages including supplementary code

The author demonstrates a stable Lagrangian solid modeling method, tracking the interactions of solid mass particles, rather than using a meshed grid. This numerical method avoids the problem of tensile instability often seen with Smooth Particle Applied Mechanics by having the solid particles apply stresses expected with Hooke's law, as opposed to using a smoothing function for neighboring solid particles. This method has been tested successfully with a bar in tension, compression, and shear, as well as a disk compressed into a flat plate, and the numerical model consistently matched the analytical Hooke's law as well as Hertz contact theory for all examples. The solid modeling numerical method was then built into a 2-D model of a pressure vessel, which was tested with liquid water particles under pressure and simulated with Smoothed Particle Hydrodynamics. This simulation was stable, and demonstrated the feasibility of Lagrangian specification modeling for Fluid Solid Interactions.

**Category:** Functions and Analysis

[199] **viXra:1702.0039 [pdf]**
*submitted on 2017-02-03 03:27:16*

**Authors:** Carl-Gustav Hedenby

**Comments:** 1 Page. -

The author proves Euler's formula for the imaginary exponential without reverting to series expansions.

**Category:** Functions and Analysis

[198] **viXra:1701.0617 [pdf]**
*submitted on 2017-01-26 01:27:43*

**Authors:** Ilya Chernykh

**Comments:** 8 Pages. In Russian language

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size.

**Category:** Functions and Analysis

[197] **viXra:1701.0511 [pdf]**
*submitted on 2017-01-15 16:19:44*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 5/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 4/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[196] **viXra:1701.0510 [pdf]**
*submitted on 2017-01-15 17:01:34*

**Authors:** Stephen C. Pearson.

**Comments:** 24 Pages.

This particular submission contains a copy [PART 6/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 5/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[195] **viXra:1701.0505 [pdf]**
*submitted on 2017-01-15 14:34:25*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 3/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 2/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[194] **viXra:1701.0504 [pdf]**
*submitted on 2017-01-15 15:44:13*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 4/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 3/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[193] **viXra:1701.0502 [pdf]**
*submitted on 2017-01-15 11:15:18*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains (inter alia) a copy [PART 1/6] of the author's original paper, which was completed on 31st March 1984 and thus comprises a total of 161 handwritten foolscap pages. Subsequently, its purpose is to enunciate various definitions and theorems, which pertain to the following topics, i.e. (a) the algebra of quaternion hypercomplex numbers; (b) functions of a single quaternion hypercomplex variable; (c) the concepts of limit and continuity applied to such functions; (d) the elementary principles of differentiation and integration applied to 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

[192] **viXra:1701.0501 [pdf]**
*submitted on 2017-01-15 12:58:04*

**Authors:** Stephen C. Pearson.

**Comments:** 42 Pages.

This particular submission contains a copy [PART 2/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 1/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'.

**Category:** Functions and Analysis

[191] **viXra:1701.0324 [pdf]**
*submitted on 2017-01-08 04:19:38*

**Authors:** Adam Chmaj

**Comments:** 6 Pages. Original 2014 version of the result is posted here. Some minor corrections are left to the reader.

The existence of traveling waves for the fractional Burgers equation is established, using an operator splitting trick. This solves a 1998 open problem.

**Category:** Functions and Analysis

[190] **viXra:1612.0413 [pdf]**
*submitted on 2016-12-30 16:56:35*

**Authors:** Sinisa Bubonja

**Comments:** 9 Pages.

In our previous work [1], we defined the method for computing general limits of functions at their singular points and showed that it is useful for calculating divergent integrals, the sum of divergent series and values of functions in their singular points. In this paper, we have described that method and we will use it to calculate the area of Torricelli's trumpet or Gabriel's horn, the sum of the reciprocals of the primes and factorials of negative integers.

**Category:** Functions and Analysis

[189] **viXra:1612.0394 [pdf]**
*submitted on 2016-12-29 16:05:22*

**Authors:** Arthur Shevenyonov

**Comments:** 10 Pages. novel foundations

An early formal glimpse at a survey of results yet to be revealed bridging topics as diverse as, the extensions of Cauchy functional equation, Taylor expansion, ABC conjecture, and Fermat LP to name but a few.

**Category:** Functions and Analysis

[188] **viXra:1612.0245 [pdf]**
*submitted on 2016-12-14 08:14:24*

**Authors:** Edigles Guedes

**Comments:** 3 Pages.

In this paper, the author proved new expansions in series for tangent and secant functions.

**Category:** Functions and Analysis

[187] **viXra:1612.0238 [pdf]**
*submitted on 2016-12-14 01:05:36*

**Authors:** Daniel Thomas Hayes

**Comments:** 7 Pages.

The problem on the existence and smoothness of the Navier--Stokes equations is considered.

**Category:** Functions and Analysis

[186] **viXra:1611.0368 [pdf]**
*submitted on 2016-11-26 20:05:27*

**Authors:** Edigles Guedes

**Comments:** 7 Pages.

In this paper, I demonstrate one new infinite product for binomial coefficient
and news Euler's and Weierstrass's infinite product for Gamma function among other things.

**Category:** Functions and Analysis

[185] **viXra:1611.0073 [pdf]**
*submitted on 2016-11-05 14:49:21*

**Authors:** Matthew Marko

**Comments:** 13 Pages, English

This algorithm is designed to perform Discrete Fourier Transforms (DFT) to convert temporal data into spectral data. What is unique about this DFT algorithm is that it can produce spectral data at any user-defined resolution; existing DFT methods such as FFT are limited in resolution proportional to the temporal resolution. This algorithm obtains the Fourier Transforms by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate.

**Category:** Functions and Analysis

[184] **viXra:1611.0056 [pdf]**
*submitted on 2016-11-04 07:11:54*

**Authors:** O. P. Ferreira, S. Z. Németh

**Comments:** 10 Pages.

The extended second order cones were introduced by G. Zhang and S. Z. N\'emeth for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. G. Zhang and
S. Z. Németh found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods.

**Category:** Functions and Analysis

[183] **viXra:1611.0049 [pdf]**
*submitted on 2016-11-03 23:41:33*

**Authors:** Edigles Guedes

**Comments:** 15 Pages.

In this paper, I demonstrate one infinite product for binomial coefficient, Euler's
and Weierstrass's infinite product for Pochhammer's symbol, limit formula for Pochhammer's
symbol, limit formula for exponential function, Euler's and Weierstrass's infinite product for
Newton's binomial and exponential function, among other things.

**Category:** Functions and Analysis

[182] **viXra:1611.0002 [pdf]**
*submitted on 2016-11-01 01:50:33*

**Authors:** Kenneth C. Johnson

**Comments:** 13 Pages.

This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for approximation orders 2, 4, 6, and 8, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed.

**Category:** Functions and Analysis

[181] **viXra:1610.0278 [pdf]**
*submitted on 2016-10-23 16:35:25*

**Authors:** Yigal Gurevich

**Comments:** 8 Pages.

The method of multiple scales is applied and the second order two-scale approximation is calculated for a linear dispersive wave equation with a small perturbation proportional to the amplitude cubed.

**Category:** Functions and Analysis

[180] **viXra:1610.0244 [pdf]**
*submitted on 2016-10-21 04:28:14*

**Authors:** David Eelbode, Eckhard Hitzer

**Comments:** 16 Pages. published in Advances in Applied Clifford Algebras, 26(3), pp. 953-968 (2016), Online First: 22 Oct. 2015, DOI: 10.1007/s00006-015-0600-7.

In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit i∈C is replaced by a multivector square root of −1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator.

**Category:** Functions and Analysis

[179] **viXra:1609.0349 [pdf]**
*submitted on 2016-09-25 01:25:44*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force.

**Category:** Functions and Analysis

[178] **viXra:1609.0141 [pdf]**
*submitted on 2016-09-11 08:38:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1.

**Category:** Functions and Analysis

[177] **viXra:1609.0074 [pdf]**
*submitted on 2016-09-06 15:08:57*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

We present a solution for the Navier-Stokes equations for incompressible case with any smooth (C^∞) initial velocity given a pressure and external force in n = 3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, n≥1.

**Category:** Functions and Analysis

[176] **viXra:1609.0006 [pdf]**
*submitted on 2016-09-01 04:47:12*

**Authors:** Michael Parfenov

**Comments:** 46 Pages.

Abstract: The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory.

**Category:** Functions and Analysis

[175] **viXra:1608.0432 [pdf]**
*submitted on 2016-08-31 08:23:50*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

We present two proofs of theorems on solutions of the Navier-Stokes equations for incompressible case with a conservative external force in n = 3 spatial dimensions. Without major difficulties, it can be adapted to any spatial dimension, n>=1.

**Category:** Functions and Analysis

[174] **viXra:1608.0394 [pdf]**
*submitted on 2016-08-29 06:04:21*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 2 Pages.

Using sliding hump-type techniques, we prove some Schur, Vitali-Hahn-Saks and Nikodým-type theorems for lattice group-valued k-triangular set functions.

**Category:** Functions and Analysis

[173] **viXra:1608.0229 [pdf]**
*submitted on 2016-08-21 11:27:59*

**Authors:** Fu Yuhua

**Comments:** 12 Pages.

Unlike Dirac operator method, this paper discusses one dimensional method for Clifford analysis, namely the n-dimensional problem is simplified into n problems of one dimension, even reduced to only one problem of one dimension. For example, the typhoon track is a two-dimensional problem related to latitude and longitude, but as forecasting typhoon track, it can be simplified into two problems of one dimension: forecasting longitude and forecasting latitude respectively. Again, the stock index is effected by various factors, however as forecasting stock index we may assume that it is only a function of time. In order to improve the effect of one dimensional method, we can change finding the solution suitable for whole space or a domain, into finding the solution suitable for one point only with single point method. As applying one dimensional method, the fractal model is very effective.

**Category:** Functions and Analysis

[172] **viXra:1608.0155 [pdf]**
*submitted on 2016-08-16 03:44:17*

**Authors:** L. A. N. de Paula

**Comments:** 5 Pages.

A large number of methods have been proposed for solving nonlinear differential equations. The
Jacobi elliptic function method and the f-expansion methods are generalizations from a few of
them. These methods produce not only single-solitons but also multi-soliton solutions. In this work
we applied the f -expansion method and found novel solutions besides those known for three main
equations of the kind sine-Gordon: Triple Sine-Gordon (TSG), Double Sine-Gordon (DSG) and
Simple Sine-Gordon (SSG).

**Category:** Functions and Analysis

[171] **viXra:1608.0065 [pdf]**
*submitted on 2016-08-05 21:34:41*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 4 Pages.

This article explores a case of signal based chaos generation, using the Carotid-Kundalini function, shown in literature to possess fractal artifacts. Specifically, we set the input to a two tone signal, with the frequency ratio between the sinusoids acting as the control parameter. We explore the iterative map using the time derivatives, and upon plotting the bifurcation plot, observe the chaotic nature of the generated signal. Phase portraits are plotted for different orders, and presence of rich patterns are observed. True to the nonlinear nature, the frequency spectrum shows a horde of new frequency components generated at the output. Lyapunov Exponents also quantitatively confirm the presence of generated chaos in the Carotid-Kundalini signal.

**Category:** Functions and Analysis

[170] **viXra:1606.0344 [pdf]**
*submitted on 2016-06-30 08:34:29*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 2 Pages.

Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations.

**Category:** Functions and Analysis

[169] **viXra:1606.0324 [pdf]**
*submitted on 2016-06-28 16:56:02*

**Authors:** Furkan Semih Dundar

**Comments:** 5 Pages.

We give a theory of $n$-scale previously called as $n$ dimensional time scale. In previous approaches to the theory of time scales, multi dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. Here we define an $n$-scale as an arbitrary closed subset of $\mathbb R^n$. Modified forward and backward jump operators, $\Delta$-derivatives and multiple integrals on $n$-scales are defined.

**Category:** Functions and Analysis

[168] **viXra:1606.0180 [pdf]**
*submitted on 2016-06-17 22:42:03*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** Functions and Analysis

[167] **viXra:1606.0141 [pdf]**
*submitted on 2016-06-14 08:42:21*

**Authors:** Richard J. Mathar

**Comments:** 16 Pages.

The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x) K_1^v(x) dx
are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are
Modified Bessel Functions. There are essentially 15 types of the 4-fold product of which
10 are discussed here.
Partial integration of each of these types leads to correlations between these integrals. The main result are
(forward) recurrences of the integrals with respect to the exponent j.

**Category:** Functions and Analysis

[166] **viXra:1605.0290 [pdf]**
*submitted on 2016-05-28 22:52:24*

**Authors:** Eric Stucky

**Comments:** 6 Pages.

In this document, we define a fully formalized notion of the ``Implied Integral'' which works for a broad variety of symbolic expressions. We show that it is equal to the ordinary integral up to a generalized constant term, and we show that this generalized constant is in some sense optimal.

**Category:** Functions and Analysis

[165] **viXra:1605.0236 [pdf]**
*submitted on 2016-05-22 15:47:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions.

**Category:** Functions and Analysis

[164] **viXra:1604.0376 [pdf]**
*submitted on 2016-04-29 06:12:02*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 13 Pages.

We give necessary and sufficient
conditions for exchange of limits of
double-indexed families, taking values in sets
endowed with an abstract structure of convergence, and for preservation
of continuity or semicontinuity of the limit family, with respect to filter convergence.
As consequences, we give some filter
limit theorems and some characterization of
continuity and semicontinuity of
the limit of a pointwise convergent family of set functions.

**Category:** Functions and Analysis

[163] **viXra:1604.0369 [pdf]**
*submitted on 2016-04-28 08:05:07*

**Authors:** Claude Michael Cassano

**Comments:** 4 Pages.

The Homogeneous , Reduction of Order Homogeneous , and Inhomogeneous Second Order Linear Ordinary Differential Equations Solution may be established in one analytical proof-process.

**Category:** Functions and Analysis

[162] **viXra:1604.0244 [pdf]**
*submitted on 2016-04-15 22:16:18*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. In english.

We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[161] **viXra:1604.0235 [pdf]**
*submitted on 2016-04-15 04:23:52*

**Authors:** Danil Krotkov

**Comments:** 12 Pages.

We derive some new formulas, connecting some series with Möbius function with Sine Integral and Cosine Integral functions, give the formal proof for full version of Stirling's formula; investigate the values of new Dirichlet series function at natural numbers >1 and it's behavior at the pole s=1, connecting it with elementary constants.

**Category:** Functions and Analysis

[160] **viXra:1604.0204 [pdf]**
*submitted on 2016-04-13 02:17:14*

**Authors:** Bing He

**Comments:** 16 Pages.

Let \begin{equation*}
A_{q}^{(\alpha)}(a;z)=\sum_{k=0}^{\infty}\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}},
\end{equation*}
where $\alpha >0,~0<q<1.$ In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire
function $A_{q}^{(\alpha)}(a;z)$ are all real and established some results on the zeros of $A_{q}^{(\alpha)}(a;z)$ which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that $A_{q}^{(\alpha)}(q^l;z),~l\geq 2$ has only infinitely many negative zeros that gives a partial answer to Zhang's question. In addition, we establish some results on zeros of certain entire functions involving the Rogers-Szeg\H{o} polynomials and the Stieltjes-Wigert polynomials.

**Category:** Functions and Analysis

[159] **viXra:1604.0107 [pdf]**
*submitted on 2016-04-05 12:31:45*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[158] **viXra:1604.0001 [pdf]**
*submitted on 2016-04-01 02:42:01*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. 1 figure

In this paper we use the general steerable one-sided Clifford Fourier transform (CFT), and relate the classical convolution of Clifford algebra-valued signals over $\R^{p,q}$ with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the CFTs of the factor functions. In full generality do we express the classical convolution of Clifford algebra signals in terms of a linear combination of Mustard convolutions, and vice versa the Mustard convolution of Clifford algebra signals in terms of a linear combination of classical convolutions.

**Category:** Functions and Analysis

[157] **viXra:1603.0413 [pdf]**
*submitted on 2016-03-30 21:39:45*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 12 Pages.

Motivated by the call for a more flexible, easy-to-tune means of generating chaos, the present work elaborates upon the development of the Ramanujan Theta Function as a source of chaos. By setting the variables of this function to sinusoids of competing frequencies the chaotic output signal is generated, with the ratio of the input frequencies serving as control parameter. The characteristics of such chaos generated are studied using the iterative map and bifurcation plots, and the presence of chaos is ascertained using Lyapunov Exponents and Kolmogorov Entropy. Following this, the route from order to chaos of the proposed system are studied using three techniques – phase portrait, Fourier Spectra and wavelet analysis. In the phase portraits, it is seen that for non-chaotic regimes, phase portraits are orderly with definite number of loops, whereas for chaotic regimes, trajectories are spread all over the phase space, suggesting ergodicity and giving the phase portrait a rich, ornamental look. The Fourier spectra highlighted the discrete frequency components in non-chaotic regimes, with well formed sidebands, whereas in chaotic regimes, a lot of new frequency components are seen, giving the spectral profile a ‘grassy’ appearance. Finally, a hyperbolic wavelet, termed the Solitary Wavelet seen to possess vanishing higher order moments with a negative logarithmic slope, is used as the basis to perform wavelet analysis. The results reveal that the rhythmic periodicity observed in large scale values for non-chaotic regimes is significantly absent for chaotic regimes, with variations in the trends of new pulses emerging alongside the main pulse train. It is seen that the wavelet analyses combine the best features of phase portrait analyses (ergodicity detected by peak sporadicity and pulse variance), and Fourier Spectral analyses (new frequency component generation seen by observing dominance at various scales, and new peaks at lower scales corresponding to new pulses), while revealing additional features such as fractal nature, not seen in the other two analysis tools. In summary, the present article ushers in a novel perspective pertaining to signal oriented chaos, calling for a change in the way bifurcation plots and iterative maps are perceived, as also the means to generate and control such chaos. It is hoped that the wavelet analysis, highlighting both spectral and temporal aspects of the signal, emerges as a reliable and assertive qualitative means to identify, detect and to an extent, characterize the nature of chaos, either stand-alone, or in conjunction with tools such as phase portraits. Progress in such an area will eventually drive chaos analyses away from Lyapunov Exponents, which are most useful in system based chaos where initial conditions are well known, and are at best computed with approximations from output chaotic signals, using methods such as the Rosenstein Algorithm.

**Category:** Functions and Analysis

[156] **viXra:1603.0250 [pdf]**
*submitted on 2016-03-17 10:41:39*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 19 Pages.

A most deep solution to the fifteent problem of Smale, the Navier-Stokes equations in three spatial dimentions. The answer is negative.

**Category:** Functions and Analysis

[155] **viXra:1603.0050 [pdf]**
*submitted on 2016-03-04 05:27:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. Three examples are given.

**Category:** Functions and Analysis

[154] **viXra:1602.0246 [pdf]**
*submitted on 2016-02-20 06:15:20*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Article in construction. Number of pages with total prepared for 2 idioms.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0(x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[153] **viXra:1602.0235 [pdf]**
*submitted on 2016-02-19 06:00:15*

**Authors:** Oleh Kyrhan

**Comments:** 6 Pages.

The article discusses the representation of discrete functions defined in an analytic form without the use of approximations, namely the Heaviside function, identity function, the Dirac delta function and the prime-counting function.

**Category:** Functions and Analysis

[152] **viXra:1602.0044 [pdf]**
*submitted on 2016-02-04 07:36:02*

**Authors:** Eckhard Hitzer

**Comments:** 19 Pages. Submitted to Adv. in Appl. Cliff. Algs., 1 figure.

In this paper we use the general steerable two-sided Clifford Fourier transform (CFT), and relate the classical convolution of Clifford algebra-valued signals over $\R^{p,q}$ with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the CFTs of the factor functions. In full generality do we express the classical convolution of Clifford algebra signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of Clifford algebra signals in terms of finite linear combinations of classical convolutions.

**Category:** Functions and Analysis

[151] **viXra:1601.0289 [pdf]**
*submitted on 2016-01-26 11:04:20*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 7 Pages.

We give some Korovkin-type theorems on
convergence and estimates of rates of approximations for nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some comparison result between different types of convergence and applications to integral type
and discrete operators.

**Category:** Functions and Analysis

[150] **viXra:1601.0283 [pdf]**
*submitted on 2016-01-26 05:07:27*

**Authors:** Eckhard Hitzer

**Comments:** 15 Pages. Submitted to S. Bernstein and P. Cerejeiras (eds), Complex Variables and Elliptic Equations (Taylor & Francis), Special Issue entitled “Clifford analysis – theory and applications”, 26 Jan. 2016.

In this paper we use the steerable space-time Fourier transform (SFT), and relate the classical convolution of the algebra for space-time $Cl(3,1)$-valued signals over the space-time vector space $\R^{3,1}$, with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space-time signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of space-time signals in terms of finite linear combinations of classical convolutions.

**Category:** Functions and Analysis

[149] **viXra:1601.0189 [pdf]**
*submitted on 2016-01-18 00:35:19*

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

The Newton's second law and third law has proven to be wrong, his mistake is the derivative operation the limitation resulting. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative algorithm. The Inverse Derivative is a kind of extension of the derivative computation, it is based on the derivative calculated generated, using most of the derivative formulas to converted adjusted. In the field of physics and mechanical
engineering, Inverse Derivative may be a large number of employed, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[148] **viXra:1601.0165 [pdf]**
*submitted on 2016-01-15 05:15:47*

**Authors:** Eckhard Hitzer

**Comments:** 14 Pages. Submitted to special issue of Adv. in Appl. Cliff. Algs., 1 color figure.

In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over $\R^2$ with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.

**Category:** Functions and Analysis

[147] **viXra:1601.0129 [pdf]**
*submitted on 2016-01-12 15:46:52*

**Authors:** Wu Sheng-Ping

**Comments:** 1 Page.

I find the mistake in Cauchy's theorem of Complex variables.

**Category:** Functions and Analysis

[146] **viXra:1601.0039 [pdf]**
*submitted on 2016-01-05 22:52:20*

**Authors:** GuagSan Yu

**Comments:** 6 Pages. This paper is Chinese

The Newton second law and the law of third have beened wrong by the proof, his mistake be caused by the localization of the derivative. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative computation method. The inverse derivative is a kind of extension of the derivative computation, it creation been according to derivative the computation, the adoption is through regulatory and conversion the computation formula of the mostly derivative. In the physics and mechanics engineering realm, the Inverse Derivative be may adopted in largeness, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[236] **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

[235] **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

[234] **viXra:1802.0126 [pdf]**
*replaced on 2018-02-12 23:41:44*

**Authors:** Han Geurdes, Koji Nagata, Tadao Nakamura, Ahmed Farouk

**Comments:** 12 Pages.

In the paper it is demonstrated that Bells theorem is an unprovable theorem. This inconsistency is similar to concrete mathematical incompleteness. The inconsistency is purely mathematical. Nevertheless the basic physics requirements of a local model are fulfilled.

**Category:** Functions and Analysis

[233] **viXra:1801.0096 [pdf]**
*replaced on 2018-01-16 06:00:50*

**Authors:** Martin Nicholson

**Comments:** 8 Pages. Presentation is improved, a theorem, a corollary and some references are added

In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters.

**Category:** Functions and Analysis

[232] **viXra:1709.0357 [pdf]**
*replaced on 2018-01-22 03:40:51*

**Authors:** Johan Aspegren

**Comments:** 3 Pages.

In this article we will give a proof that the Kakeya tube conjecture implies the Kakeya conjecture.

**Category:** Functions and Analysis

[231] **viXra:1709.0113 [pdf]**
*replaced on 2017-09-15 04:26:30*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained.

**Category:** Functions and Analysis

[230] **viXra:1709.0113 [pdf]**
*replaced on 2017-09-13 00:01:34*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained.

**Category:** Functions and Analysis

[229] **viXra:1709.0047 [pdf]**
*replaced on 2017-09-19 07:48:27*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure.

**Category:** Functions and Analysis

[228] **viXra:1709.0047 [pdf]**
*replaced on 2017-09-11 14:27:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure.

**Category:** Functions and Analysis

[227] **viXra:1708.0123 [pdf]**
*replaced on 2017-09-07 19:37:37*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

Describe a fluid in three-dimensional circular motion with at most one spatial variable by rectangular coordinate, beyond time, and concludes on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure.

**Category:** Functions and Analysis

[226] **viXra:1708.0123 [pdf]**
*replaced on 2017-08-15 06:33:11*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations.

**Category:** Functions and Analysis

[225] **viXra:1708.0123 [pdf]**
*replaced on 2017-08-12 11:42:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations.

**Category:** Functions and Analysis

[224] **viXra:1707.0155 [pdf]**
*replaced on 2017-09-01 11:06:05*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 31 Pages.

A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid in movement.

**Category:** Functions and Analysis

[223] **viXra:1707.0155 [pdf]**
*replaced on 2017-08-27 13:57:42*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 31 Pages.

A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid in movement.

**Category:** Functions and Analysis

[222] **viXra:1707.0155 [pdf]**
*replaced on 2017-08-22 12:28:29*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 30 Pages. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi

A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement.

**Category:** Functions and Analysis

[221] **viXra:1707.0155 [pdf]**
*replaced on 2017-08-21 07:31:25*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 30 Pages. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi

A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement.

**Category:** Functions and Analysis

[220] **viXra:1707.0155 [pdf]**
*replaced on 2017-07-24 06:41:52*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 28 Pages. This paper need some change. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi

**Category:** Functions and Analysis

[219] **viXra:1707.0155 [pdf]**
*replaced on 2017-07-20 09:28:40*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 27 Pages.

**Category:** Functions and Analysis

[218] **viXra:1705.0410 [pdf]**
*replaced on 2017-05-30 20:59:45*

**Authors:** Hong Lai Zhu

**Comments:** 71 Pages.

This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct.

**Category:** Functions and Analysis

[217] **viXra:1705.0165 [pdf]**
*replaced on 2018-02-14 22:45:03*

**Authors:** Nicholas R. Wright

**Comments:** 6 Pages. Replaced "Pareto" with "Pareto Improvement"

We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto Improvement exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards.

**Category:** Functions and Analysis

[216] **viXra:1705.0165 [pdf]**
*replaced on 2017-10-11 04:08:40*

**Authors:** Nicholas R. Wright

**Comments:** 6 Pages. Replaced "irrationality of square" with "geometric rate" and Cambria Math font.

We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards.

**Category:** Functions and Analysis

[215] **viXra:1703.0135 [pdf]**
*replaced on 2017-03-29 06:22:52*

**Authors:** Naoya Isobe

**Comments:** 13 Pages.

Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis.

**Category:** Functions and Analysis

[214] **viXra:1703.0134 [pdf]**
*replaced on 2017-03-29 06:21:16*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis).

**Category:** Functions and Analysis

[213] **viXra:1703.0133 [pdf]**
*replaced on 2017-03-29 06:19:34*

**Authors:** Naoya Isobe

**Comments:** 15 Pages.

Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis.

**Category:** Functions and Analysis

[212] **viXra:1703.0132 [pdf]**
*replaced on 2017-03-29 06:17:44*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis).

**Category:** Functions and Analysis

[211] **viXra:1703.0121 [pdf]**
*replaced on 2017-03-29 06:11:05*

**Authors:** Naoya Isobe

**Comments:** 12 Pages.

Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis.

**Category:** Functions and Analysis

[210] **viXra:1703.0120 [pdf]**
*replaced on 2017-03-29 06:09:07*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis).

**Category:** Functions and Analysis

[209] **viXra:1703.0119 [pdf]**
*replaced on 2017-03-29 06:06:14*

**Authors:** Naoya Isobe

**Comments:** 14 Pages.

Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys.

**Category:** Functions and Analysis

[208] **viXra:1703.0118 [pdf]**
*replaced on 2017-03-29 06:02:51*

**Authors:** Naoya Isobe

**Comments:** 1 Page.

Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis)

**Category:** Functions and Analysis

[207] **viXra:1701.0617 [pdf]**
*replaced on 2017-05-18 05:57:40*

**Authors:** Ilya Chernykh

**Comments:** 11 Pages. Now in English

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used as a cardinality measure for fine comparison between infinite countable sets in metric space which respects the intuitive notion of the set's size.

**Category:** Functions and Analysis

[206] **viXra:1701.0617 [pdf]**
*replaced on 2017-05-12 07:46:53*

**Authors:** Ilya Chernykh

**Comments:** 10 Pages. Now in Russian

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used as a cardinality measure for fine comparison between infinite countable sets in metric space which respects the intuitive notion of the set's size.

**Category:** Functions and Analysis

[205] **viXra:1701.0617 [pdf]**
*replaced on 2017-04-27 23:34:08*

**Authors:** Ilya Chernykh

**Comments:** 9 Pages.

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size.

**Category:** Functions and Analysis

[204] **viXra:1701.0617 [pdf]**
*replaced on 2017-02-01 10:10:22*

**Authors:** Ilya Chernykh

**Comments:** 8 Pages. In Russian language

We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size.

**Category:** Functions and Analysis

[203] **viXra:1612.0238 [pdf]**
*replaced on 2018-01-29 19:07:23*

**Authors:** Daniel Thomas Hayes

**Comments:** 11 Pages.

The problem on the existence and smoothness of the Navier-Stokes equations is solved.

**Category:** Functions and Analysis

[202] **viXra:1611.0056 [pdf]**
*replaced on 2016-11-10 10:15:26*

**Authors:** O. P. Ferreira, S. Z. Németh

**Comments:** 12 Pages.

The extended second order cones were introduced by S. Z. Németh and G. Zhang in [S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. Németh and G. Zhang in [S.Z. Németh and G. Zhang. Positive operators of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods.

**Category:** Functions and Analysis

[201] **viXra:1611.0002 [pdf]**
*replaced on 2017-01-15 23:03:42*

**Authors:** Kenneth C. Johnson

**Comments:** 22 Pages. [v8] Revised Appendix B

This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed.

**Category:** Functions and Analysis

[200] **viXra:1611.0002 [pdf]**
*replaced on 2016-12-21 02:06:40*

**Authors:** Kenneth C. Johnson

**Comments:** 22 Pages. [v7] Algorithm runtime is improved. This version should now be stable.

This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed.

**Category:** Functions and Analysis

[199] **viXra:1611.0002 [pdf]**
*replaced on 2016-12-19 03:02:40*

**Authors:** Kenneth C. Johnson

**Comments:** 22 Pages. [v6] update: Further improvments to error/tolerance analysis

This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed.

**Category:** Functions and Analysis

[198] **viXra:1611.0002 [pdf]**
*replaced on 2016-12-16 18:53:00*

**Authors:** Kenneth C. Johnson

**Comments:** 23 Pages. [v5] update: Improved error/tolerance analysis

**Category:** Functions and Analysis

[197] **viXra:1611.0002 [pdf]**
*replaced on 2016-12-13 20:10:43*

**Authors:** Kenneth C. Johnson

**Comments:** 22 Pages. [v4] update: robust error/tolerance analysis

**Category:** Functions and Analysis

[196] **viXra:1611.0002 [pdf]**
*replaced on 2016-11-30 17:18:29*

**Authors:** Kenneth C. Johnson

**Comments:** 16 Pages. v3 includes links to associated MATLAB and Mathematica code

**Category:** Functions and Analysis

[195] **viXra:1611.0002 [pdf]**
*replaced on 2016-11-29 23:00:31*

**Authors:** Kenneth C. Johnson

**Comments:** 16 Pages.

**Category:** Functions and Analysis

[194] **viXra:1609.0349 [pdf]**
*replaced on 2016-12-20 11:35:40*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 22 Pages. Published at WISE Journal, Volume 6, No. 2 (Summer, 2017), pp. 60-81.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and then, in special case, solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force, the Bernoulli’s law. A little description of the Lamb’s solution for Euler equations is done.

**Category:** Functions and Analysis

[193] **viXra:1609.0349 [pdf]**
*replaced on 2016-11-03 19:59:17*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 22 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done.

**Category:** Functions and Analysis

[192] **viXra:1609.0349 [pdf]**
*replaced on 2016-11-03 11:39:04*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 22 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done.

**Category:** Functions and Analysis

[191] **viXra:1609.0349 [pdf]**
*replaced on 2016-10-27 18:38:53*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done.

**Category:** Functions and Analysis

[190] **viXra:1609.0349 [pdf]**
*replaced on 2016-10-25 00:35:09*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

**Category:** Functions and Analysis

[189] **viXra:1609.0349 [pdf]**
*replaced on 2016-10-20 20:17:39*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

**Category:** Functions and Analysis

[188] **viXra:1609.0349 [pdf]**
*replaced on 2016-10-07 11:19:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 18 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force.

**Category:** Functions and Analysis

[187] **viXra:1609.0349 [pdf]**
*replaced on 2016-09-29 09:09:56*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force.

**Category:** Functions and Analysis

[186] **viXra:1609.0141 [pdf]**
*replaced on 2016-12-27 13:03:54*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1.

**Category:** Functions and Analysis

[185] **viXra:1609.0141 [pdf]**
*replaced on 2016-12-16 16:27:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1.

**Category:** Functions and Analysis

[184] **viXra:1609.0141 [pdf]**
*replaced on 2016-09-27 17:45:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

**Category:** Functions and Analysis

[183] **viXra:1609.0074 [pdf]**
*replaced on 2016-09-27 17:12:07*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

We present a solution for the Navier-Stokes equations for incompressible case with any smooth (C^∞) initial velocity given a pressure and external force in n = 3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, n≥1.

**Category:** Functions and Analysis

[182] **viXra:1609.0006 [pdf]**
*replaced on 2017-01-19 09:35:22*

**Authors:** Michael Parfenov

**Comments:** 46 Pages

The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory.

**Category:** Functions and Analysis

[181] **viXra:1609.0006 [pdf]**
*replaced on 2016-09-22 07:34:07*

**Authors:** Michael Parfenov

**Comments:** 46 Pages

Abstract: The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory.

**Category:** Functions and Analysis

[180] **viXra:1608.0432 [pdf]**
*replaced on 2016-09-01 12:40:33*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

We present two proofs of theorems on solutions of the Navier-Stokes equations for incompressible case with a conservative external force in n = 3 spatial dimensions. Without major difficulties, it can be adapted to any spatial dimension, n>=1.

**Category:** Functions and Analysis

[179] **viXra:1606.0344 [pdf]**
*replaced on 2016-09-07 12:56:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

§ 1: remembering the need of imposed the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. This section is historical only. § 2: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. § 3: non-uniqueness in time for incompressible and potential flows, if u≠0. § 4: a more general solution of Euler and Navier-Stokes equations for incompressible and irrotational (potential) flows, given the initial velocity. § 5: Solution for Euler and Navier-Stokes equations using Taylor’s series of powers of t around t=0.

**Category:** Functions and Analysis

[178] **viXra:1606.0344 [pdf]**
*replaced on 2016-08-18 14:56:52*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. This date is historical only. § 2: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. § 3: non-uniqueness in time for incompressible and potential flows, if u≠0. § 4: a more general solution of Euler and Navier-Stokes equations for incompressible and irrotational (potential) flows, given the initial velocity.

**Category:** Functions and Analysis

[177] **viXra:1606.0344 [pdf]**
*replaced on 2016-08-15 15:48:39*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Two last dates: non-uniqueness in time for incompressible and potential flows, if u≠0.

**Category:** Functions and Analysis

[176] **viXra:1606.0344 [pdf]**
*replaced on 2016-07-30 08:27:18*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Third date: non-uniqueness in time for incompressible and potential flows, if u≠0.

**Category:** Functions and Analysis

[175] **viXra:1606.0344 [pdf]**
*replaced on 2016-07-29 20:50:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Third date: non-uniqueness in time for incompressible and potential flows.

**Category:** Functions and Analysis

[174] **viXra:1606.0344 [pdf]**
*replaced on 2016-07-19 18:59:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 3 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential velocity and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only.

**Category:** Functions and Analysis

[173] **viXra:1606.0344 [pdf]**
*replaced on 2016-07-19 14:28:34*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 3 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential velocity and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. The first date is historical only.

**Category:** Functions and Analysis

[172] **viXra:1606.0344 [pdf]**
*replaced on 2016-07-03 20:13:18*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 2 Pages.

Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations.

**Category:** Functions and Analysis

[171] **viXra:1606.0344 [pdf]**
*replaced on 2016-06-30 18:11:07*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 2 Pages.

Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations.

**Category:** Functions and Analysis

[170] **viXra:1606.0324 [pdf]**
*replaced on 2017-01-20 16:47:57*

**Authors:** Furkan Semih Dundar

**Comments:** 6 Pages. v3. Clarifications have been made.

We provide a theory of $n$-scales previously called as $n$ dimensional time scales. In previous approaches to the theory of time scales, multi-dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. $n$-scales make the mathematical structure more flexible and appropriate to real world applications in physics and related fields. Here we define an $n$-scale as an arbitrary closed subset of $\mathbb R^n$. Modified forward and backward jump operators, $\Delta$-derivatives and multiple integrals on $n$-scales are defined.

**Category:** Functions and Analysis

[169] **viXra:1606.0324 [pdf]**
*replaced on 2016-07-02 18:45:26*

**Authors:** Furkan Semih Dundar

**Comments:** 5 Pages. v2 typos are fixed

We give a theory of $n$-scale previously called as $n$ dimensional time scale. In previous approaches to the theory of time scales, multi dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. Here we define an $n$-scale as an arbitrary closed subset of $\mathbb R^n$. Modified forward and backward jump operators, $\Delta$-derivatives and multiple integrals on $n$-scales are defined.

**Category:** Functions and Analysis

[168] **viXra:1606.0141 [pdf]**
*replaced on 2016-06-21 11:42:55*

**Authors:** Richard J. Mathar

**Comments:** 24 Pages. Added Section 4 in version 2.

The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x)K_1^v(x) dx
are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are
Modified Bessel Functions. There are essentially 15 types of the 4-fold product.
Partial integration of each of these types leads correlations between these integrals. The main result are
(forward) recurrences of the integrals with respect to the exponent j
of the power.

**Category:** Functions and Analysis

[167] **viXra:1605.0236 [pdf]**
*replaced on 2016-08-28 12:57:14*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions.

**Category:** Functions and Analysis

[166] **viXra:1605.0236 [pdf]**
*replaced on 2016-05-27 07:05:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions.

**Category:** Functions and Analysis

[165] **viXra:1604.0244 [pdf]**
*replaced on 2017-07-26 09:02:29*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[164] **viXra:1604.0244 [pdf]**
*replaced on 2017-04-03 08:56:40*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[163] **viXra:1604.0244 [pdf]**
*replaced on 2017-03-28 21:39:58*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[162] **viXra:1604.0244 [pdf]**
*replaced on 2017-03-27 09:28:38*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

**Category:** Functions and Analysis

[161] **viXra:1604.0244 [pdf]**
*replaced on 2017-03-17 22:05:04*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 13 Pages.

**Category:** Functions and Analysis

[160] **viXra:1604.0244 [pdf]**
*replaced on 2016-08-27 06:08:44*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 12 Pages. The equations 19 and 20 are not correct for n > 1

**Category:** Functions and Analysis

[159] **viXra:1604.0244 [pdf]**
*replaced on 2016-07-14 12:31:22*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

**Category:** Functions and Analysis

[158] **viXra:1604.0244 [pdf]**
*replaced on 2016-07-07 14:39:22*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[157] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-24 10:51:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[156] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-23 12:10:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[155] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-20 07:01:30*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

**Category:** Functions and Analysis

[154] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-12 22:01:12*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages. Sorry, this paper yet is not good.

We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[153] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-09 12:53:33*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** ERRATA: Page 4, between equations (12) and (13) include the word "square" before the word "module": square module.

We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Functions and Analysis

[152] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-05 18:00:27*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

**Category:** Functions and Analysis

[151] **viXra:1604.0244 [pdf]**
*replaced on 2016-06-02 09:33:17*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

[150] **viXra:1604.0244 [pdf]**
*replaced on 2016-05-31 09:37:15*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

[149] **viXra:1604.0244 [pdf]**
*replaced on 2016-05-12 11:26:02*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages. ERRATA: x,y,z in equation (7) are in Lagrangian description

**Category:** Functions and Analysis

[148] **viXra:1604.0244 [pdf]**
*replaced on 2016-05-09 13:52:01*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

[147] **viXra:1604.0244 [pdf]**
*replaced on 2016-05-06 20:29:57*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

**Category:** Functions and Analysis

[146] **viXra:1604.0244 [pdf]**
*replaced on 2016-04-27 08:05:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages. It´s necessay enter the factor 1/3 multiplying the second ni in the equations 1, 5, 6 and 7.

**Category:** Functions and Analysis

[145] **viXra:1604.0244 [pdf]**
*replaced on 2016-04-23 21:37:32*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages. In english.

**Category:** Functions and Analysis

[144] **viXra:1604.0244 [pdf]**
*replaced on 2016-04-23 04:30:18*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. Sorry. Unfortunately, x, y, z are not so free.

**Category:** Functions and Analysis

[143] **viXra:1604.0244 [pdf]**
*replaced on 2016-04-19 11:14:23*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. In english.

**Category:** Functions and Analysis

[142] **viXra:1604.0244 [pdf]**
*replaced on 2016-04-18 08:03:49*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. Equations (6) and (7) are wrongs, both versions 1 and 2. Nothing like they is possible.

**Category:** Functions and Analysis

[141] **viXra:1604.0235 [pdf]**
*replaced on 2016-06-12 04:36:26*

**Authors:** Danil Krotkov

**Comments:** 13 Pages.

We derive some new formulas, connecting some series with Möbius function with Sine Integral and Cosine Integral functions, give the formal proof for full version of Stirling's formula with remainder term in form of definite integral of elementary function; investigate the values of new Dirichlet series function at natural numbers >1 and its behavior at the pole at s=1, connecting it with elementary constants.

**Category:** Functions and Analysis

[140] **viXra:1604.0107 [pdf]**
*replaced on 2016-11-18 06:21:50*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[139] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-15 07:02:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages. In english.

We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[138] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-14 23:45:15*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[137] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-11 08:05:39*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages. In english.

We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[136] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-10 07:28:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 16 Pages. In English.

We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide.

**Category:** Functions and Analysis

[135] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-08 21:17:12*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Pages. Sorry, the case (B) is damaged. (2.34) and related are wrong. For now, it is safe the case (C).

**Category:** Functions and Analysis

[134] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-07 11:27:47*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 16 Pages. Surprisingly, we also solved the case (B) of the Millenium Problem on Navier-Stokes equations.

**Category:** Functions and Analysis

[133] **viXra:1604.0107 [pdf]**
*replaced on 2016-04-06 11:51:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 15 Pages.

**Category:** Functions and Analysis

[132] **viXra:1603.0250 [pdf]**
*replaced on 2016-06-22 06:52:29*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

A most deep solution to the fifteent problem of Smale, the Navier-Stokes equations in three spatial dimentions. The answer is negative.

**Category:** Functions and Analysis

[131] **viXra:1603.0250 [pdf]**
*replaced on 2016-03-26 11:24:42*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

A most deep solution to the fifteent problem of Smale, the Navier-Stokes equations in three spatial dimentions. The answer is negative.

**Category:** Functions and Analysis

[130] **viXra:1603.0250 [pdf]**
*replaced on 2016-03-17 18:42:56*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 19 Pages.

**Category:** Functions and Analysis

[129] **viXra:1603.0050 [pdf]**
*replaced on 2016-03-26 14:37:27*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. Three examples are given.

**Category:** Functions and Analysis

[128] **viXra:1603.0050 [pdf]**
*replaced on 2016-03-20 13:12:37*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. Three examples are given.

**Category:** Functions and Analysis

[127] **viXra:1603.0050 [pdf]**
*replaced on 2016-03-20 07:44:30*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[126] **viXra:1603.0050 [pdf]**
*replaced on 2016-03-15 10:39:26*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[125] **viXra:1603.0050 [pdf]**
*replaced on 2016-03-06 01:44:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[124] **viXra:1602.0246 [pdf]**
*replaced on 2017-02-03 06:23:46*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Article (english version) published in J. Phys. Math (2016) 7: 196. The digitation or conversion of this paper in J. Phys. Math. need revision and many typographical corrections. I'm sorry.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0(x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[123] **viXra:1602.0246 [pdf]**
*replaced on 2017-01-18 09:59:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Article (english version) published in J. Phys. Math (2016) 7: 196. The digitation or conversion of this paper in J. Phys. Math. need revision and many typographical corrections. I'm sorry.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0(x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[122] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-26 16:53:40*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Article (english version) published in J. Phys. Math (2016) 7: 196. The digitation or conversion of this paper in J. Phys. Math. need revision and many typographical corrections. I'm sorry.

**Category:** Functions and Analysis

[121] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-23 06:18:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Sections 1 to 4 and 8 to 10 translated

**Category:** Functions and Analysis

[120] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-21 12:57:07*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Sections 1 to 4 and 8 to 9 (Conclusion) translated. Equation (92) corrected.

**Category:** Functions and Analysis

[119] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-18 12:54:01*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. Sections 1 to 4 and 9 (Conclusion) translated.

**Category:** Functions and Analysis

[118] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-11 06:52:07*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. To me it has not been given the gift of English...

**Category:** Functions and Analysis

[117] **viXra:1602.0246 [pdf]**
*replaced on 2016-03-03 06:17:25*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. In section 10, replace velocity v by velocity u, vector I=(1,1,1) by J=(1,1,-2) and include f in (94). In next version the complete translation into English (I hope).

**Category:** Functions and Analysis

[116] **viXra:1602.0246 [pdf]**
*replaced on 2016-02-28 19:38:15*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 49 Pages. Páginas 14 e 38, onde se lê "sistema inexistente" leia-se "sistema impossível".

**Category:** Functions and Analysis

[115] **viXra:1602.0246 [pdf]**
*replaced on 2016-02-27 15:37:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 49 Pages. Need complete translation for english.

**Category:** Functions and Analysis

[114] **viXra:1602.0246 [pdf]**
*replaced on 2016-02-26 05:25:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Article in construction. Number of pages with total prepared for 2 idioms.

**Category:** Functions and Analysis

[113] **viXra:1602.0246 [pdf]**
*replaced on 2016-02-24 05:40:26*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Article in construction. Number of pages with total prepared for 2 idioms.

**Category:** Functions and Analysis

[112] **viXra:1602.0246 [pdf]**
*replaced on 2016-02-22 07:08:59*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Article in construction. Number of pages with total prepared for 2 idioms.

**Category:** Functions and Analysis

[111] **viXra:1601.0189 [pdf]**
*replaced on 2016-09-26 21:24:48*

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

The Newton's second law and third law has proven to be wrong, his mistake is the derivative operation the limitation resulting. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative algorithm. The Inverse Derivative is a kind of extension of the derivative computation, it is based on the derivative calculated generated, using most of the derivative formulas to converted adjusted. In the field of physics and mechanical engineering, Inverse Derivative may be a large number of employed, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[110] **viXra:1601.0189 [pdf]**
*replaced on 2016-06-20 20:19:06*

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

The Newton's second law and third law has proven to be wrong, his mistake is the derivative operation the limitation resulting. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative algorithm. The Inverse Derivative is a kind of extension of the derivative computation, it is based on the derivative calculated generated, using most of the derivative formulas to converted adjusted. In the field of physics and mechanical
engineering, Inverse Derivative may be a large number of employed, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[109] **viXra:1601.0189 [pdf]**
*replaced on 2016-04-17 21:27:29*

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

**Category:** Functions and Analysis

[108] **viXra:1601.0165 [pdf]**
*replaced on 2016-01-23 08:13:42*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. Submitted to special issue of Adv. in Appl. Cliff. Algs., 1 color figure.

In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over $\R^2$ with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.

**Category:** Functions and Analysis

[107] **viXra:1601.0129 [pdf]**
*replaced on 2018-01-02 16:48:29*

**Authors:** Wu Sheng-Ping

**Comments:** 1 Page.

I find the mistake in Cauchy's theorem of Complex variables.

**Category:** Functions and Analysis

[106] **viXra:1601.0039 [pdf]**
*replaced on 2016-09-26 21:16:25*

**Authors:** GuagSan Yu

**Comments:** 7 Pages. The Chinese

The Newton second law and the law of third have beened wrong by the proof, his mistake be caused by the localization of the derivative. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative computation method. The inverse derivative is a kind of extension of the derivative computation, it creation been according to derivative the computation, the adoption is through regulatory and conversion the computation formula of the mostly derivative. In the physics and mechanics engineering realm, the Inverse Derivative be may adopted in largeness, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[105] **viXra:1601.0039 [pdf]**
*replaced on 2016-06-20 20:27:03*

**Authors:** GuagSan Yu

**Comments:** 7 Pages. This paper is Chinese

The Newton second law and the law of third have beened wrong by the proof, his mistake be caused by the localization of the derivative. Therefore new derivative the computation method desiderate the creation, this is the Inverse Derivative computation method. The inverse derivative is a kind of extension of the derivative computation, it creation been according to derivative the computation, the adoption is through regulatory and conversion the computation formula of the mostly derivative. In the physics and mechanics engineering realm, the Inverse Derivative be may adopted in largeness, to replace original derivative computation method. Will produce the enormous and profound influence.

**Category:** Functions and Analysis

[104] **viXra:1601.0039 [pdf]**
*replaced on 2016-04-17 21:13:41*

**Authors:** GuagSan Yu

**Comments:** 7 Pages. This paper is Chinese

**Category:** Functions and Analysis

[103] **viXra:1601.0039 [pdf]**
*replaced on 2016-01-11 01:02:38*

**Authors:** GuagSan Yu

**Comments:** 7 Pages. This paper is Chinese

**Category:** Functions and Analysis