**Previous months:**

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2009 - 0903(1)

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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) - 1707(4) - 1708(3) - 1709(9) - 1710(5) - 1711(4) - 1712(7)

2018 - 1801(1) - 1802(6) - 1803(2) - 1804(2) - 1806(8) - 1807(7) - 1808(4)

Any replacements are listed farther down

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

**Authors:** Seong Won Cha

**Comments:** 22 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 5 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 15 pages.

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

**Category:** Functions and Analysis

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

**Authors:** Zaid Laadjal

**Comments:** Pages.

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

**Category:** Functions and Analysis

[276] **viXra:1807.0233 [pdf]**
*submitted on 2018-07-12 19:46:47*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove that the isotropic constant depends only on the dimension of the domain. Thus we prove a result that implies the sharp isotropic constant conjecture.

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 6 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Edigles Guedes

**Comments:** 4 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Viktor Strohm

**Comments:** 4 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Thinh D. Nguyen

**Comments:** 1 Page.

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

**Category:** Functions and Analysis

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

**Authors:** Hassine Saidane

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Thinh Nguyen

**Comments:** 17 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Tejas Chandrakant Thakare

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

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

**Category:** Functions and Analysis

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

**Authors:** Michael Parfenov

**Comments:** 18 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Claude Michael Cassano

**Comments:** 9 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Claude Michael Cassano

**Comments:** 18 Pages.

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

**Category:** Functions and Analysis

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

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

**Comments:** 19 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Andrej Liptaj

**Comments:** 10 Pages.

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

**Category:** Functions and Analysis

[263] **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

[262] **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

[261] **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

[260] **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

[259] **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

[258] **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

[257] **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

[256] **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

[255] **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

[254] **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

[253] **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

[252] **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

[251] **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

[250] **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

[249] **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

[248] **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

[247] **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

[246] **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

[245] **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

[244] **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

[243] **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

[242] **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

[241] **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

[240] **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

[239] **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

[238] **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

[237] **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

[236] **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

[235] **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

[234] **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

[233] **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

[232] **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

[231] **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

[230] **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

[229] **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

[228] **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

[227] **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

[226] **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

[225] **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

[224] **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

[223] **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

[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

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

**Authors:** Zaid Laadjal

**Comments:** 12 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Hassine Saidane

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Jonathan W. Tooker

**Comments:** 5 Pages. two figures

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

**Category:** Functions and Analysis

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

**Authors:** Jonathan W. Tooker

**Comments:** 4 Pages. two figures

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

**Category:** Functions and Analysis

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

**Authors:** Andrej Liptaj

**Comments:** 10 Pages.

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

**Category:** Functions and Analysis

[238] **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

[237] **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

[236] **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

[235] **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

[234] **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

[233] **viXra:1709.0113 [pdf]**
*replaced on 2018-07-22 15:18:50*

**Authors:** Johan Aspegren

**Comments:** 7 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

[232] **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

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

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

**Category:** Functions and Analysis

[230] **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

[229] **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

[228] **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

[227] **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

[226] **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

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 31 Pages. Published at WISE Journal, Volume 7, No. 1 (Spring, 2018), pp. 110-140.

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

[224] **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

[223] **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

[222] **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

[221] **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

[220] **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

[219] **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

[218] **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

[217] **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

[216] **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

[215] **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

[214] **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

[213] **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

[212] **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

[211] **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

[210] **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

[209] **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

[208] **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

[207] **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

[206] **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

[205] **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

[204] **viXra:1612.0238 [pdf]**
*replaced on 2018-04-26 17:21:49*

**Authors:** Daniel Thomas Hayes

**Comments:** 9 Pages.

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

**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