**Previous months:**

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Any replacements are listed further down

[192] **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

[191] **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

[190] **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

[189] **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

[188] **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

[187] **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

[186] **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

[185] **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

[184] **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

[183] **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

[182] **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

[181] **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

[180] **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

[179] **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

[178] **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

[177] **viXra:1608.0090 [pdf]**
*submitted on 2016-08-08 23:29:14*

**Authors:** Stephen Crowley

**Comments:** 4 Pages. I've been super busy. I will update this with corrections soon. Dr Kawhira has indicated I typed the hypothesis wrong, I do understand it but was typing too fast and loose and this isn't my dayjob...

An assertion about the superaattractiveness of the fixed-points of the Newton map of the Hardy Z
function is shown to be equivalent to the Riemann Hypothesis.

**Category:** Functions and Analysis

[176] **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

[175] **viXra:1608.0015 [pdf]**
*submitted on 2016-08-02 05:05:37*

**Authors:** Kolosov Petro

**Comments:** 9 Pages. -

The main aim of this paper to establish the relations between forward, backward and central finite and divided differences (that is discrete analog of the derivative) and partial and ordinary high-order derivatives of the polynomials.

**Category:** Functions and Analysis

[174] **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

[173] **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

[172] **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

[171] **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

[170] **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

[169] **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

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

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 13 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Claude Michael Cassano

**Comments:** 4 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Danil Krotkov

**Comments:** 12 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Bing He

**Comments:** 16 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. 1 figure

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

**Category:** Functions and Analysis

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

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 12 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 19 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Oleh Kyrhan

**Comments:** 6 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Eckhard Hitzer

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

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

**Category:** Functions and Analysis

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

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 7 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Eckhard Hitzer

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

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Eckhard Hitzer

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

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

**Category:** Functions and Analysis

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

**Authors:** Wu Sheng-Ping

**Comments:** 1 Page.

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

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

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

**Category:** Functions and Analysis

[149] **viXra:1512.0253 [pdf]**
*submitted on 2015-12-07 03:05:14*

**Authors:** Viktor Strohm

**Comments:** 5 Pages.

In this article we consider a differential equation for 2nd-order curve of the motion of a parametric pendulum in zero gravity.

**Category:** Functions and Analysis

[148] **viXra:1511.0302 [pdf]**
*submitted on 2015-11-30 20:55:26*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. Published in Advances in Applied Clifford Algebras, pp. 1-16, Online First: 02 Nov. 2015, DOI: 10.1007/s00006-015-0620-3..

So far quaternion Fourier transforms have been mainly defined over
$\mathbb{R}^2$ as signal domain space. But it seems natural to define
a quaternion Fourier transform for quaternion valued signals over quaternion
domains. This quaternion domain Fourier transform (QDFT)
transforms quaternion valued signals (for example electromagnetic scalarvector
potentials, color data, space-time data, etc.) defined over a quaternion
domain (space-time or other 4D domains) from a quaternion
position space to a quaternion frequency space. The QDFT uses the
full potential provided by hypercomplex algebra in higher dimensions
and may moreover be useful for solving quaternion partial differential
equations or functional equations, and in crystallographic texture analysis.
We define the QDFT and analyze its main properties, including
quaternion dilation, modulation and shift properties, Plancherel and
Parseval identities, covariance under orthogonal transformations, transformations
of coordinate polynomials and differential operator polynomials,
transformations of derivative and Dirac derivative operators, as
well as signal width related to band width uncertainty relationships.

**Category:** Functions and Analysis

[147] **viXra:1511.0247 [pdf]**
*submitted on 2015-11-25 11:42:54*

**Authors:** Sinisa Bubonja

**Comments:** 23 Pages.

We are interested in finding sums of some divergent series using the general method for summing divergent series discovered in our previous work and symbolic mathematical computation program Mathematica. We make a comparison to other five summation methods implemented in Mathematica and show that our method is the stronger method than methods of Abel, Borel, Cesaro, Dirichlet and Euler.

**Category:** Functions and Analysis

[146] **viXra:1511.0246 [pdf]**
*submitted on 2015-11-25 07:46:15*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 93 Pages. In French.

It is a course of numerical analysis and applied mathematics for the students of the geomatic and topography option of the ESAT School, Tunis, Tunisia.

**Category:** Functions and Analysis

[145] **viXra:1511.0077 [pdf]**
*submitted on 2015-11-10 07:45:51*

**Authors:** Richard J. Mathar

**Comments:** Few text; the majority of pages show pairs of abscissae and weights.

We tabulate the abscissae and associated weights for numerical integration of
integrals with kernels which contain a power of x
and Modified Bessel Functions K_nu(x).
The first family of integrals contains the factor x^m K_nu(x)
with integer indices nu=0 or 1 and integer powers nu <= m <= 3.
The second family of integrals contains the factor x^m K_nu(x)K_nu'(x)
with
integer indices 0 <= nu, nu' <= 1
and integer powers nu+nu' <= m <= 3.

**Category:** Functions and Analysis

[144] **viXra:1511.0071 [pdf]**
*submitted on 2015-11-09 06:26:54*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 2 Pages. In portuguese.

A critique to Otelbaev solution for the sixth millenium problem (Navier-Stokes equation). Begins with a dialogue with the teacher Odilon in the IME-USP library.

**Category:** Functions and Analysis

[143] **viXra:1510.0506 [pdf]**
*submitted on 2015-10-30 10:22:42*

**Authors:** Alex Patterson

**Comments:** 42 Pages. Special Thanks to Julian C. Boyd, December 25, 1931 – April 5, 2005

Abstract: In ideal circumstances, we know exactly what to do. But there are no ideal circumstances; but then we are confusing and conflating two things if you look: When we speak of ideal circumstances, we are philosophers and thinking of our version of a Platonic ideal, not our value ideals or our ideal values [a common mistake that does not justify nominalism – it is the nominalist’s backdoor malware into the reductive Platonism], such as freedom of speech, right bear arms, habeas corpus, each one of us has those which are most important to us, sometimes many at the same time. These are not however social value-propositions. They derive of natural law in the Constitution. However, certain Platonic ideal circumstances as to our individually important value ideals can be created in real life, to the degree that we are able to invent. That is why we have Critique. It takes notice of the differences and the possibilities offered by the differences, which would be less without noticing the differences. However, in in that sense where ideal circumstances can be created [actually, are present], we have created them in our everyday lives, they are our individual lives, exist within the network of constitutive grounds for law and institutions. In strict terms of modality, these value-propositions become axiomatic with the given system of our everyday lives that.
And so, with a leap of faith to living our everyday lives practically, theoretical matters, axiomatically classical matters, classical axiomatic matters, a serious mind would see that you that in the debates in physics over cosmology, we literally have geodesics flying out through time from centrifugal non-global hyperbolic Cauchy topology like Frisbees, each one of which you can play an infinite number of games on. You don’t need or have a CTC, so there is no causal blockade in the hyperbolic Cauchy topology of strong cause and effect. Modus ponens as explained above is the only mechanical mechanism is the mode for cause and effect that man has, and its antecedence-precedence minimalism yet strong presence is enough; and it is minimum. With modus ponens what it has been since remote antiquities, to say anything otherwise would be to ask help from God to unravel all of the physics up until today’s counterarguments in favor of a vt time-lapse based arithmetic time line with counterintuitive information transfer and strangely in relation to the later, if that goes badly, the need for constructivism’s interference in mathematics, resulting in the need to use MP to take the physics apart. The solution itself takes the piled data apart. We can’t read God’s mind. But Gödel left us this. Otherwise we need God’s help, and I have written the PRF.POSTV [proof-positive] against that possibility. All that we retain from that would be the nominalist’s eased conventional respect for Occam in expostulation and discussions of formal matters such as sets.
In this [theoretical] critique of theory, new terms and tools are introduced for the Gödel material and literature. It would be precisely named, “Critique of The Theory.” Not Gödel’s theory, if he had one, but ‘The Theory’ as an object. It’s revealing that no other concept can be spoken of in that way.
Picture trees in a forest, and rocks displaced by gravity and falling through forest, hitting one tree, then another, until they stop. Hubble shows to astronomers which super-structures are found, how big, and how far back in time. It is consistent with Mach’s principle if you rotate yourself before you look into the forest at 90 degrees and find a way of suspending yourself and turnig.
Asymptotes are used within vertical and horizontal graphs to justify a future that need not be seen as a future in the sense of grammatical future-tense [which is a dubious tense in “philosophical grammar”] but as a potential part in such systems themselves that we deal with with respect to incompleteness.
The thesis is that we can approach incompleteness by using [theoretical] reasoning and available tools that are allowed in theoretical reasoning and in reasoning as we think of it in reasoning through situations, and in the sense of pure reasoning itself; and, in order to critique the very meaning of incompleteness itself. As well, as the theoretical reasoning possible inside the tools themselves, should they have domains. We will not always a domain for ourselves. We will not always be here; we will die, and epochal turns in history amounting to new antiquities will occur.
One of the most famous axioms of this type was invented by Vico, an axiom he discovered by thinking things through and studying.
In the long course that rumor has run from the beginning of the world it has been the perennial source of all the exaggerated opinions which have hitherto been held concerning remote antiquities unknown to us, by virtue of that property of the human mind noted by Tacitus in his Life of Agricola, where he says that everything unknown is taken for something great (omne ignotum fro magnifico est). – Gianbattista Vico, La Scienze nouva, The New Science

**Category:** Functions and Analysis

[142] **viXra:1510.0494 [pdf]**
*submitted on 2015-10-28 20:16:06*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 8 Pages.

The present work purports to the generation of a signal based chaos based on Ramanujan’s Mock Theta Functions. Specifically, the variable in these functions is viewed as an additively coupled sum of sinusoidal signals, with competing frequencies. Thus, by adapting the seven third order mock theta functions into signals, the derivatives are computed and used to form the corresponding iterative maps, which are studied using phase portraits. It is seen that the phase portraits of three of the seven forms exhibit rich, ornamental patterns, characteristic of chaos. Using these, the bifurcation diagrams are plotted, and the chaotic behavior is quantitatively characterized using Lyapunov Exponent and Kolmogorov Entropy. It is seen that the nature of chaos in the mock theta form based signals indeed depend on the frequency ratio of the driving signals, thus pertaining to a case of signal based chaos, which has the key advantage of easy tunability, which forms the novelty of the present work.

**Category:** Functions and Analysis

[141] **viXra:1510.0492 [pdf]**
*submitted on 2015-10-28 20:18:43*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 5 Pages.

The present work pertains to the formulation and generation of signal based frequency controlled chaos based on Einstein Functions. Specifically, the variable in the four Einstein functions is taken as an additively coupled sum of sinusoids with competing frequencies. By adapting the four Einstein functions into signals, the time derivatives are computed and used to form the iterative maps and phase portraits. It is seen that the four phase portraits display to varying degrees, ornamental patterns, characteristic of quasiperiodicity and chaos. Using these, bifurcation diagrams are plotted, and the chaotic signal is quantitatively characterized using largest Lyapunov Exponents. It is seen that the nature of chaos in the generated signals depend on the frequency ratio of the driving signals, thus pertaining to a case of signal based chaos, which has the key advantage of easy tunability, forming the novelty of the present work.

**Category:** Functions and Analysis

[140] **viXra:1510.0447 [pdf]**
*submitted on 2015-10-28 09:39:32*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 4 Pages.

We investigate some properties of
unconditional convergence of series taking
values in lattice groups.
We give some matrix and Schur-type
theorems in the filter convergence context for lattice group-valued measures, and deduce
an interchange theorem for series.

**Category:** Functions and Analysis

[139] **viXra:1510.0426 [pdf]**
*submitted on 2015-10-28 02:22:32*

**Authors:** Claude Michael Cassano

**Comments:** 5 Pages.

Several theorems are established to ease exact solution of homogeneous linear ordinary differential equations.

**Category:** Functions and Analysis

[138] **viXra:1510.0398 [pdf]**
*submitted on 2015-10-26 08:34:31*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 6 Pages.

The present paper pertains to the formulation and generation of a signal based frequency controlled chaos based on the Heart Curves, an assortment of parametric functions so named due to their ability to generate cardoid (heart-shaped) curves. Specifically, the variable in two parametric functions is taken as an additively coupled sum of sinusoids with competing frequencies. By adapting the x and y components of these functions into signals, the time derivatives are computed and used to form the iterative maps and phase portraits. It is seen that the four phase portraits display to varying degrees, ornamental patterns, characteristic of quasiperiodicity and chaos. Using these, bifurcation diagrams are plotted in order to investigate the chaotic behavior. It is seen that the nature of chaos in the generated signals depend on the frequency ratio of the driving signals, thus pertaining to a case of signal based chaos, which has the key advantage of easy tunability, forming the novelty of the present work.

**Category:** Functions and Analysis

[137] **viXra:1510.0386 [pdf]**
*submitted on 2015-10-25 08:11:52*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 6 Pages.

The present work purports to a signal based generation of chaos, thereby offering a radically innovative solution to the issues of tunability caused in conventional system based chaos generation circuits. Specifically, the Ramanujan Theta Function is seen to represent the output signal of a coupled nonlinear system with two driving signals. Using this concept, the iterative map of the Ramanujan Theta Function is developed using the frequency ratio of driving signals as the control parameter, and the ratio dependent dynamics are studied using the bifurcation plot. The proposed system is implemented in hardware using FPGA and the presence of chaos is validated qualitatively using phase portraits and quantitatively using Lyapunov Exponents, whose trend indeed agree with the one observed in the bifurcation plot. The innovative perspective of signal based chaos proposed using the Ramanujan Theta Function enables easy tunability in chaotic generation systems and this forms the novelty of the present work.

**Category:** Functions and Analysis

[136] **viXra:1510.0364 [pdf]**
*submitted on 2015-10-23 12:56:09*

**Authors:** Danil Krotkov

**Comments:** 4 Pages.

Generalized Ramanujan identity

**Category:** Functions and Analysis

[135] **viXra:1510.0361 [pdf]**
*submitted on 2015-10-23 09:16:29*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 8 Pages.

The present work purports to the formulation and characterization of signal based chaos based on Bessel Functions. Specifically, the variable in these functions is viewed as an additively coupled sum of sinusoidal signals, with competing frequencies. By adapting the regular and modified Bessel Functions of the first and second kinds into signals, the derivatives are computed and used to form the corresponding iterative maps, which are studied using phase portraits. It is seen that the phase portraits of the regular and modified Bessel functions of the first kind exhibit rich, ornamental patterns, characteristic of quasiperiodicity and chaos. Using these, the bifurcation diagrams are plotted, and the chaotic behavior is quantitatively characterized using largest Lyapunov Exponents. It is seen that the nature of chaos in the generated signals indeed depend on the frequency ratio of the driving signals, thus pertaining to a case of signal based chaos, which has the key advantage of easy tunability, which forms the novelty of the present work.

**Category:** Functions and Analysis

[134] **viXra:1510.0098 [pdf]**
*submitted on 2015-10-12 08:22:11*

**Authors:** Danil Krotkov

**Comments:** Pages.

Generalized Identity of Johann Bernoulli

**Category:** Functions and Analysis

[133] **viXra:1509.0286 [pdf]**
*submitted on 2015-09-30 00:40:38*

**Authors:** Kenneth C. Johnson

**Comments:** 5 Pages. See also http://www.mathworks.com/matlabcentral/fileexchange/50413-generalized-matrix-exponential

This paper reports work-in-progress on the solution of first-order, linear, homogeneous differential equation systems, with non-constant coefficients, by generalization of the Padé approximant method for exponential matrices.

**Category:** Functions and Analysis

[132] **viXra:1509.0193 [pdf]**
*submitted on 2015-09-21 05:27:01*

**Authors:** Alex Patterson

**Comments:** 39 Pages. Special thanks for Michael J. Burns, of Hacking Physics

Abstract: in ideal circumstances, we know exactly what to do. But there are no ideal circumstances. That is why we have Critique. The object of critique matters not.
However, this paper is also very concerned not only with critique as such but with a possible intersection, that has arisen out of such critique, of a heretofore un-combined areas of mathematical induction, Gödelian Relativity, and their Algebras within Bayesian Networks. It is very rough attempt, but uses all the right tools, should such an Intersection be theoretically viable in the short term, at least.
Further steps are made into Ethics, which the author believes to be the basis of any elegance for that matter, of any excellent piece of intellectual effort.
This paper uses its own peculiar lettering system for each paragraph. I don’t like it: But the way I write it is the only way I can figure to impose some kind of order for the content. Seemingly arcane references will be made to philosophers such as Vico and Kant, to Heine and Goethe, to Wittgenstein and B.F. Skinner, J.L. Austin, Turing, Thomas Paine, G.H. von Wright, Marx, the “Availability Heuristic,” Quantum Logic, More Classical Systems of Axiomatic Problems, Epistemology, etc., et. al. Towards the end a Narrative will be given that attempts to address some concerns of the present world in its ethical, moral, and political makeup. There will be no cut-off at that point, but the change will be clearly visible, it is hoped that there will be a “fade to gray" effect. The Reader’s forbearance and endurance to “strange thoughts”(but which are informed by the author’s education) is solicited as the paper proceeds: Even a crank might not be a crank, as a liar may not be a liar. Such are our weak paradoxes.
Therefore, this:
1.New terms and tools are introduced for quantification that creates a more synthetic (logical, reasonable, coherent) intervention and inter-weaving into these (those) now classical problems of the assumptions and the successes in the Gödel material and literature.
2.Asymptotes are used within vertical and horizontal graphs to justify a future that need not be seen as a future in the sense of grammatical future-tense (it is considered by the best philosophical grammarians to be a tense that doesn’t exist), but as a potential part in such systems themselves that we deal with respect to incompleteness.
3.The thesis is that we can approach incompleteness by using theoretical reasoning and available tools that are allowed in theoretical reasoning to critique the very theory of incompleteness itself. That is the essential Abstract Thesis.
4.It will be seen that a real attempt is attempted. It is quite a ride. I’ve looked at everything I can think of so far. In doing Gödel you can’t tell a hypothesis from a position. So it is necessary to be very analytical, use Occam’s razor wherever you can but still brave the undesired feat of blunting the blade. This is not Quine, and there is no Neurath’s Figure to resort to as Quine does, and in his area, must do, much to his discredit, as great as he may be.
This critique (work, paper) works from Gödel’s incompleteness to Gödel relativity. It does so in with the help of formal logic and inductive thinking, and a lot of hard analysis and questioning of various facts, the interrogation of assumptions, so that the critique can end up where it belongs: in the ethical domain of inquiry, not an easy category of thought to express, but which will be further versioned in subsequent editions, or entirely new works.

**Category:** Functions and Analysis

[131] **viXra:1508.0214 [pdf]**
*submitted on 2015-08-27 06:58:21*

**Authors:** Alex Patterson

**Comments:** 5 Pages. Special thanks to many

Abstract
This short paper is a simple attempt to see whether the Lorentz Gamma Factor, when restrained by Wittgenstein’s TLP (Tractatus Logico Philosophicus) 3.333 proposition on a Function containing a function which he supposes rather imperatively that it excises Russell’s Paradox in Principia Mathematica, may be ‘a rationalist/rational under-study’ on that very of excisement, and vice versa. Further, potentially moot but nonetheless by Vico’s Dictum attention is given to what the consequences of finalizing the Under-Study might entail, be consequent on, pertain to, maybe even elucidate, etc.
Such under-studies are abundant in the classics, for example Vico to Petrarch. That’s sufficient and needs more of in the author’s view.

**Category:** Functions and Analysis

[130] **viXra:1508.0204 [pdf]**
*submitted on 2015-08-25 14:15:46*

**Authors:** Kunle Adegoke, Olawanle Layeni

**Comments:** 22 Pages.

We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both finite and infinite sums. The high points of this paper are perhaps the discovery of several previously unknown infinite summation results involving {\em non-linear} generalized harmonic number terms and the derivation of interesting alternating summation formulas involving these numbers.

**Category:** Functions and Analysis

[129] **viXra:1508.0152 [pdf]**
*submitted on 2015-08-19 11:23:23*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. example (13) is wrong. very sorry. i´m sad. will be corrected.

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

**Category:** Functions and Analysis

[128] **viXra:1508.0096 [pdf]**
*submitted on 2015-08-13 08:16:42*

**Authors:** Han Geurdes

**Comments:** 5 Pages.

In this paper it is demonstrated that the Navier Stokes equation has a smooth type A nontrivial exact solution combining two radial solutions inside and outside the unit sphere.b

**Category:** Functions and Analysis

[127] **viXra:1508.0083 [pdf]**
*submitted on 2015-08-11 10:24:30*

**Authors:** Carlos Armando De Castro

**Comments:** 7 pages, one column, 8 figures

In this paper it is shown a simple approximation of the function exp(x) for positive values of x, deduced from the implicit Euler numerical solution of first order lineal differential equations (ODE). The results show that the approximation has an error of less than 10% for exp(x) when x < 0.35 and for exp(-x) when x < 0.5, which is acceptable for many engineering applications, and helps facilitate the analysis of some systems without the use of computers.
Keywords: approximation, Euler implicit method, exponential function, first order ODE.

**Category:** Functions and Analysis

[126] **viXra:1507.0225 [pdf]**
*submitted on 2015-07-31 07:39:39*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 1 Page.

The qualitative equality between n = 2 and n = 3 spatial dimensions.

**Category:** Functions and Analysis

[125] **viXra:1507.0063 [pdf]**
*submitted on 2015-07-09 14:51:16*

**Authors:** Sidharth Ghoshal

**Comments:** 3 Pages.

A technique for optimizing a more general class of linear operator expressions than those covered by the Euler Lagrange Equations

**Category:** Functions and Analysis

[124] **viXra:1507.0051 [pdf]**
*submitted on 2015-07-08 08:47:05*

**Authors:** Danil Krotkov

**Comments:** 2 Pages.

Generalized identity of Johann Bernoulli

**Category:** Functions and Analysis

[123] **viXra:1507.0022 [pdf]**
*submitted on 2015-07-03 12:26:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 2 Pages.

The solution of the fifteenth problem of Smale, the Navier-Stokes equations in three spatial dimensions.

**Category:** Functions and Analysis

[122] **viXra:1506.0019 [pdf]**
*submitted on 2015-06-03 16:14:12*

**Authors:** Carlos Oscar Rodríguez Leal

**Comments:** Language: Spanish. Number of pages: 25

In this paper several existing numerical methods of solution of ordinary
differential equations of first order with initial conditions are modified , so
that they can be generalized as methods of solution of ordinary differential
equations of order n.

**Category:** Functions and Analysis

[121] **viXra:1505.0083 [pdf]**
*submitted on 2015-05-11 08:09:42*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 18 Pages. I'm working on a better proof.

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[120] **viXra:1503.0177 [pdf]**
*submitted on 2015-03-23 23:22:06*

**Authors:** Sidharth Ghoshal

**Comments:** 10 Pages.

We write a solution for solving all equations of the type
a(x)f(x+h) + b(x)f(x) + c(x)f(x)^2 = 0
And use this to create a variety of formula for infinite nested radicals

**Category:** Functions and Analysis

[119] **viXra:1503.0047 [pdf]**
*submitted on 2015-03-07 05:44:06*

**Authors:** Edigles Guedes

**Comments:** 3 pages.

In this paper, we demonstrate some integral representations for harmonic number and digamma function.

**Category:** Functions and Analysis

[118] **viXra:1503.0032 [pdf]**
*submitted on 2015-03-04 20:15:34*

**Authors:** Edigles Guedes

**Comments:** 8 pages.

In present article, we developed infinite series representations for inverse sine function and other functions. Our main goal is to get the hypergeometric representation for Catalan constant and hyperbolic sine function; and new integral
representation for inverse sine function.

**Category:** Functions and Analysis

[117] **viXra:1502.0210 [pdf]**
*submitted on 2015-02-22 15:39:24*

**Authors:** Edigles Guedes

**Comments:** 2 pages.

I prove the expansion in infinite seiries for the multiplication between sine, hyperbolic sine and exponential functions, that do not exist in the mathematical literature.

**Category:** Functions and Analysis

[116] **viXra:1502.0202 [pdf]**
*submitted on 2015-02-22 10:45:18*

**Authors:** Edigles Guedes

**Comments:** 3 pages.

I prove some accelerations of the infinite series for the hyperbolic sine, hyperbolic cosine, Struve and Bessel function of the first kind.

**Category:** Functions and Analysis

[115] **viXra:1502.0201 [pdf]**
*submitted on 2015-02-22 11:09:50*

**Authors:** Edigles Guedes

**Comments:** 3 pages.

I proved some accelerations of the infinite series for the sine and cosine functions.

**Category:** Functions and Analysis

[114] **viXra:1502.0152 [pdf]**
*submitted on 2015-02-17 23:04:56*

**Authors:** Sidharth Ghoshal

**Comments:** 8 Pages.

This highlights some of my findings and derivation in the theory of arbitrary step size finite differences. Most are fairly simple

**Category:** Functions and Analysis

[113] **viXra:1502.0074 [pdf]**
*submitted on 2015-02-10 07:53:24*

**Authors:** Sinivsa Bubonja

**Comments:** 20 Pages. Serbian language

In this work I am going to mention historical development of divergent series theory, and to give a number of different examples, as some of the methods for their summing. After that, I am going to introduce the general method, which I discovered, for summing divergent series, which we can also consider as a method for computing limits of divergent sequences and functions in divergent points, In this case, limits of sequences of their partials sums. Through the exercises, I am going to apply this method on given examples and prove its validity. Then I'm going to apply the method to compute the value of some divergent integrals.

**Category:** Functions and Analysis

[112] **viXra:1412.0001 [pdf]**
*submitted on 2014-12-01 01:59:59*

**Authors:** Pith Xie

**Comments:** 29 Pages.

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions.

**Category:** Functions and Analysis

[111] **viXra:1411.0560 [pdf]**
*submitted on 2014-11-25 09:47:20*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 8 Pages.

We consider (strong uniform) continuity of the
limit of a pointwise convergent net of lattice
group-valued functions, (strong weak)
exhaustiveness and (strong)
alpha-convergence with respect to a pair
of filters, which in the setting of nets are
more natural than the corresponding notions
formulated with respect to a single filter. Some
comparison results are given between such concepts, in connection with suitable properties of filters. Moreover, some modes of filter
(strong uniform) continuity for lattice
group-valued functions are investigated, giving
some characterization. As an application, we get
some Ascoli-type theorem in an abstract setting,
extending earlier results to the context of filter
alpha-convergence.

**Category:** Functions and Analysis

[110] **viXra:1411.0548 [pdf]**
*submitted on 2014-11-23 10:10:13*

**Authors:** Edigles Guedes

**Comments:** 9 pages.

I proved some infinity series power and integral representations for error function
and relations for special functions.

**Category:** Functions and Analysis

[109] **viXra:1411.0228 [pdf]**
*submitted on 2014-11-17 18:51:33*

**Authors:** Edigles Guedes

**Comments:** 8 pages.

I prove two expansions of infinite series and some integral representations for Bessel function of the first kind.

**Category:** Functions and Analysis

[108] **viXra:1410.0109 [pdf]**
*submitted on 2014-10-19 11:39:09*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 4 Pages.

We investigate some properties of
lattice group-valued positive k-subadditive set functions, and in particular we give some
comparisons between regularity and continuity from above. Moreover we prove different kinds of limit theorems in the non-additive case with respect to filter convergence, in which it is supposed that the involved filter is diagonal.

**Category:** Functions and Analysis

[107] **viXra:1409.0125 [pdf]**
*submitted on 2014-09-16 10:33:17*

**Authors:** Jianwen Huang, Shouquan Chen, Jiaojiao Liu

**Comments:** 16 Pages.

In this paper, we derive the extreme value
distributions of independent identically distributed random
variables with mixed distributions of two and finite components,
which include generalized logistic, asymmetric Laplace and
asymmetric normal distributions.

**Category:** Functions and Analysis

[106] **viXra:1408.0132 [pdf]**
*submitted on 2014-08-20 05:53:16*

**Authors:** Bernhard Riemann

**Comments:** 38 Pages.

N/A (This is the LaTeXed version of Riemann's 1851 thesis, which laid the foundation of an aspect of complex analysis.--Typesetter)

**Category:** Functions and Analysis

[105] **viXra:1408.0084 [pdf]**
*submitted on 2014-08-14 05:38:48*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 9 Pages.

Cauchy Theory is applied and extended to n-dimensional functions in Clifford algebras, showing the existence of integrals that do not exist in Euclidean spaces. It celebrates the depth of Cauchy's lecture, held on the 22nd of August, 1814, so 200 years ago, in times of bitter warfare. (I should like to recommend reading about his life, e.g. in Wikipedia.)

**Category:** Functions and Analysis

[104] **viXra:1408.0081 [pdf]**
*submitted on 2014-08-13 15:39:49*

**Authors:** Alessio Mangoni

**Comments:** 4 Pages.

We show that a class C-infinite real function can be written as an n-summation of terms involving its derivative. For many functions, under certain conditions, this summation can become a particular series expansion.

**Category:** Functions and Analysis

[103] **viXra:1407.0169 [pdf]**
*submitted on 2014-07-21 15:55:09*

**Authors:** Eckhard Hitzer

**Comments:** 7 Pages. in N. E. Mastorakis, P. M. Pardalos, R. P. Agarwal, L. Kocinac (eds.), Adv. in Appl. and Pure Math., Proc. of the 2014 Int. Conf. on Pure Math., Appl. Math., Comp. Methods (PMAMCM 2014), Santorini, Greece, July 2014, Math. & Comp. in Sci. & Eng., Vol. 29.

We show how real and complex Fourier transforms
are extended to W.R. Hamilton's algebra of quaternions and to
W.K. Clifford’s geometric algebras. This was initially motivated by
applications in nuclear magnetic resonance and electric engineering.
Followed by an ever wider range of applications in color image and
signal processing. Clifford's geometric algebras are complete algebras,
algebraically encoding a vector space and all its subspace elements.
Applications include electromagnetism, and the processing of images,
color images, vector field and climate data. Further developments of
Clifford Fourier Transforms include operator exponential representations,
and extensions to wider classes of integral transforms, like
Clifford algebra versions of linear canonical transforms and wavelets.

**Category:** Functions and Analysis

[102] **viXra:1407.0054 [pdf]**
*submitted on 2014-07-07 18:13:59*

**Authors:** Antonio Boccuto, X. Dimitriou

**Comments:** 4 Pages.

Some conditions for semicontinuity of the limit function of a pointwise convergent net of lattice group-valued functions with respect to filter convergence are given. In this framework we consider some kinds of filter exhaustiveness.

**Category:** Functions and Analysis

[101] **viXra:1407.0044 [pdf]**
*submitted on 2014-07-05 12:22:07*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 3 Pages.

We give necessary and sufficient conditions for (strong uniform) continuity of the limit of a pointwise convergent net of cone metric space-valued functions. In this framework we consider several types of convergence in the filter context and some kinds of filter exhaustiveness.

**Category:** Functions and Analysis

[100] **viXra:1406.0024 [pdf]**
*submitted on 2014-06-04 12:51:50*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 7 Pages.

We give some necessary and sufficient conditions for (global) continuity of the limit of a pointwise convergent net of cone metric space-valued functions, defined on a Hausdorff topological space, in terms of weak filter exhaustiveness. In this framework, we prove some Ascoli-type theorems, considering also possibly asymmetric and extended real-valued distance functions.

**Category:** Functions and Analysis

[99] **viXra:1405.0322 [pdf]**
*submitted on 2014-05-26 09:56:23*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 4 Pages.

We prove an Ascoli-type theorem, giving a necessary and sufficient condition for forward compactness of sets of functions, defined and with values in asymmetric metric spaces.

**Category:** Functions and Analysis

[98] **viXra:1404.0072 [pdf]**
*submitted on 2014-04-10 02:19:40*

**Authors:** Xiong Wang

**Comments:** 7 Pages.

In the recent paper {\it Communications in Nonlinear Science and Numerical Simulation.
Vol.18. No.11. (2013) 2945-2948}, it was demonstrated that a violation of the Leibniz rule is a characteristic property
of derivatives of non-integer orders.
It was proved that all fractional derivatives ${\cal D}^{\alpha}$,
which satisfy the Leibniz rule
${\cal D}^{\alpha}(fg)=({\cal D}^{\alpha}f) \, g + f \, ({\cal D}^{\alpha}g)$,
should have the integer order $\alpha=1$, i.e.
fractional derivatives of non-integer orders cannot satisfy the Leibniz rule. However, it should be noted that this result is only for differentiable functions.
We argue that the very reason for introducing
fractional derivative is to study non-differentiable functions.
In this note, we try to clarify and summarize the Leibniz rule for both differentiable and non-differentiable functions. The Leibniz rule holds for differentiable functions with classical integer order derivative. Similarly the Leibniz rule still holds for non-differentiable functions with a concise and essentially local
definition of fractional derivative. This could give a more unified picture and understanding for Leibniz rule and the geometrical interpretation for both integer order and fractional derivative.

**Category:** Functions and Analysis

[97] **viXra:1404.0026 [pdf]**
*submitted on 2014-04-03 22:36:37*

**Authors:** William O. Straub

**Comments:** 9 Pages.

Gaussian integrals appear frequently in mathematics and physics, especially probability, statistics and quantum mechanics. One of the truly odd things about these integrals is that they cannot be evaluated in closed form over finite limits but are generally exactly integrable over +/- infinity. Yet their evaluation is still often difficult, particularly multi-dimensional integrals and those involving quadratics, vectors and matrices in the exponential. An added complication is that Gaussian integrals can involve ordinary real or complex variables as well as the less familiar Grassmann variables, which are important in the description of fermions. In this elementary primer
we present some of the more common Gaussian integrals of both types, along with methods for their evaluation.

**Category:** Functions and Analysis

[96] **viXra:1403.0977 [pdf]**
*submitted on 2014-03-31 12:36:51*

**Authors:** Richard J. Mathar

**Comments:** 6 Pages.

The solid angle of a circular sector specified by
circle radius, angle of the sector, and distance of the circle plane to the
observer is calculated in terms of various trigonometric and cyclometric
functions. This generalizes previous results for the full circle
that have appeared in the literature.

**Category:** Functions and Analysis

[95] **viXra:1403.0951 [pdf]**
*submitted on 2014-03-27 10:50:26*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 9 Pages.

We give a survey on recent results
about the problem of approximating a real-valued function by means of suitable families of sampling type operators, which include both discrete and integral ones, and about
the order of approximation, and abstract
Korovkin-type theorems with respect to
different types of test functions,
in the context of filter convergence.
We give a unified approach, by
means of which it is possible to consider several
kinds of classical operators, for instance Urysohn integral operators, in particular Mellin-type convolution integrals, and generalized sampling series.
We obtain proper extensions of classical results.

**Category:** Functions and Analysis

[94] **viXra:1403.0310 [pdf]**
*submitted on 2014-03-20 00:14:06*

**Authors:** David Eelbode, Eckhard Hitzer

**Comments:** Submitted to Publications of Research Institute for Mathematical Sciences (PRIMS), March 2014, 18 pages.

This paper briefly reviews the notion of Clifford's geometric algebras and vector to multivector functions; as well as the field of Clifford analysis (function theory of the Dirac operator). In Clifford Fourier transformations (CFT) on multivector signals the complex unit $i\in \mathbb{C}$ is replaced by a multivector square root of $-1$, which may be a pseudoscalar in the simplest case. For these transforms we derive, via a multivector function representation in terms of monogenic polynomials, the operator representation of the CFTs by exponentiating the Hamilton operator of a harmonic oscillator.

**Category:** Functions and Analysis

[93] **viXra:1403.0304 [pdf]**
*submitted on 2014-03-19 18:07:58*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 3 Pages.

We present some new convergence and
boundedness theorem with respect to
filter convergence for lattice
group-valued measures, whose techniques
are based on sliding hump
arguments.

**Category:** Functions and Analysis

[92] **viXra:1403.0262 [pdf]**
*submitted on 2014-03-14 21:51:47*

**Authors:** Shreyak Chakraborty

**Comments:** 8 Pages.

Crowds are generally analyzed in the regime of sociology- where
they are studied and classified on the basis of crowd psychology.
This analysis arises from the study of collective behavior and treats
crowds as dependent on psychology of humans in the crowd. In this
introductory paper we show a generalized treatment of crowds as a
set of living objects: called members of the crowd. We classify
crowds based on various parameters and study some general and
specific characteristics of crowd of humans and study the response
of a simple crowd to an external situation or stimulus by deriving the
solution of the generalized crowd equation. We also define some
terminology regarding the mathematical description of crowds and
hence arrive at some useful conjectures.

**Category:** Functions and Analysis

[182] **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

[181] **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

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

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

**Authors:** Kenneth C. Johnson

**Comments:** 16 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 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

[179] **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

[178] **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

[177] **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

[176] **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

[175] **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

[174] **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

[173] **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

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 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

[171] **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

[170] **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

[169] **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

[168] **viXra:1608.0090 [pdf]**
*replaced on 2016-11-06 13:44:32*

**Authors:** Stephen Crowley

**Comments:** 11 Pages.

There is shown to exist a unique solution to the LeClaire-França exact equation for the n-th Riemann zero via the construction of Cauchy sequences whose accumulation points are the zeros of the Hardy Z function thus proving the Riemann hypothesis.

**Category:** Functions and Analysis

[167] **viXra:1608.0090 [pdf]**
*replaced on 2016-10-15 03:20:39*

**Authors:** Stephen Crowley

**Comments:** 8 Pages. under construction

An assertion about the super-attractiveness of the fixed-points of the Newton map of the Hardy Z
function and their immediate basins is shown to be equivalent to the Riemann Hypothesis.

**Category:** Functions and Analysis

[166] **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

[165] **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

[164] **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

[163] **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

[162] **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

[161] **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

[160] **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

[159] **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

[158] **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

[157] **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

[156] **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

[155] **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

[154] **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

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 12 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Danil Krotkov

**Comments:** 13 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 17 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 15 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 19 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 53 Pages. in english and portuguese

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** Valdir Monteiro dos Santos Godoi

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

**Comments:** 8 Pages.

**Category:** Functions and Analysis

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

**Authors:** Eckhard Hitzer

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

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

**Comments:** 7 Pages. The Chinese

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

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

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

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

**Category:** Functions and Analysis

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

**Authors:** GuagSan Yu

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

**Category:** Functions and Analysis

[98] **viXra:1510.0475 [pdf]**
*replaced on 2016-02-26 19:52:02*

**Authors:** Stephen Crowley

**Comments:** 8 Pages. adding references and a few more limit/residue calculations

The integral of the logarithmic derivative of the Hardy Z function is calculated and used as a basis for the construction of an entire function χ(t)=-χ(1-t) which shares roots at exactly the same points of the Riemann zeta function, but with additional (complex) roots which interlace the roots of ζ and also having real roots at the positive odd integers, 0, and the negative even integers.

**Category:** Functions and Analysis

[97] **viXra:1510.0364 [pdf]**
*replaced on 2016-04-14 21:15:05*

**Authors:** Danil Krotkov

**Comments:** 4 Pages.

Generalized Ramanujan identity

**Category:** Functions and Analysis

[96] **viXra:1509.0286 [pdf]**
*replaced on 2016-04-22 13:45:55*

**Authors:** Kenneth C. Johnson

**Comments:** 7 Pages. 4/22/2016 update: added 8-th order approximant (Eq 22)

This paper reports work-in-progress on the solution of first-order, linear, homogeneous differential equation systems, with non-constant coefficients, by generalization of the Padé-approximant method for exponential matrices.

**Category:** Functions and Analysis

[95] **viXra:1508.0152 [pdf]**
*replaced on 2016-01-24 06:06:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages. See viXra:1601.0312: "Breakdown of Navier-Stokes Solutions – Unbounded Energy for t > 0"

Considerations on (and solution to) the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[94] **viXra:1508.0152 [pdf]**
*replaced on 2016-01-22 16:28:38*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages. Soon a new article organizing what I wrote here.

Considerations on (and solution to) the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[93] **viXra:1508.0152 [pdf]**
*replaced on 2016-01-18 05:17:25*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages. Soon a new article organizing what I wrote here.

Considerations on (and solution to) the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[92] **viXra:1508.0152 [pdf]**
*replaced on 2016-01-12 10:00:46*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 14 Pages. soon a new article organizing what I wrote here.

**Category:** Functions and Analysis

[91] **viXra:1508.0152 [pdf]**
*replaced on 2016-01-11 12:00:44*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 13 Pages.

**Category:** Functions and Analysis

[90] **viXra:1508.0152 [pdf]**
*replaced on 2015-12-16 08:40:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 13 Pages.

Considerations on the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[89] **viXra:1508.0152 [pdf]**
*replaced on 2015-12-14 08:31:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 12 Pages. Considerations on the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

Considerations on the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[88] **viXra:1508.0152 [pdf]**
*replaced on 2015-12-10 13:04:25*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 12 Pages.

Considerations on the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy.

**Category:** Functions and Analysis

[87] **viXra:1508.0152 [pdf]**
*replaced on 2015-11-13 04:36:56*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages. Experts understand that it is also possible the case \alpha = 0, and thus the zero-order partial derivative of a function g is equal to g. Therefore the samples u^0 = c and u^0 = x not obeyed (4) for \alpha = 0.

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

**Category:** Functions and Analysis

[86] **viXra:1508.0152 [pdf]**
*replaced on 2015-10-15 11:27:17*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages. The important problem solution, according to their respective description.

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

**Category:** Functions and Analysis

[85] **viXra:1508.0152 [pdf]**
*replaced on 2015-09-03 11:59:04*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages. soon a better English translation of this article

**Category:** Functions and Analysis

[84] **viXra:1508.0152 [pdf]**
*replaced on 2015-08-22 19:52:54*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

**Category:** Functions and Analysis

[83] **viXra:1508.0152 [pdf]**
*replaced on 2015-08-21 10:26:14*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages. I hope to translate this article soon

**Category:** Functions and Analysis

[82] **viXra:1508.0096 [pdf]**
*replaced on 2015-09-03 08:41:14*

**Authors:** Han Geurdes

**Comments:** 5 Pages.

It is demonstrated that the Navier Stokes equation has a smooth nontrivial exact solution in (3+1). All the required characteristics of a type B solution are verified.

**Category:** Functions and Analysis

[81] **viXra:1508.0096 [pdf]**
*replaced on 2015-08-18 04:34:42*

**Authors:** Han Geurdes

**Comments:** 9 Pages.

In this paper it is demonstrated, via construction, that the NS equation has a smooth nontrivial exact solution in (2+1). The smoothness is accomplished by connecting disconnectedp pieces off solution over a vanishing small interval. We show by example that the algebra of connecting coefficients is consistent.

**Category:** Functions and Analysis

[80] **viXra:1508.0096 [pdf]**
*replaced on 2015-08-14 05:01:09*

**Authors:** Han Geurdes

**Comments:** 4 Pages.

In this paper it is demonstrated that the Navier Stokes equation has a smooth nontrivial exact solution. The solution is a heuristic and is the smoothly glueing together of positive and negative domain solutions.

**Category:** Functions and Analysis

[79] **viXra:1507.0022 [pdf]**
*replaced on 2015-09-11 10:48:17*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

The solution of the fifteenth problem of Smale, the Navier-Stokes equations in three spatial dimensions.

**Category:** Functions and Analysis

[78] **viXra:1507.0022 [pdf]**
*replaced on 2015-07-18 19:13:45*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

The solution of the fifteenth problem of Smale, the Navier-Stokes equations in three spatial dimensions.

**Category:** Functions and Analysis

[77] **viXra:1507.0022 [pdf]**
*replaced on 2015-07-16 12:33:02*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Pages. Unfortunately I need to send another version. I hope the last. I need write that grad q(x) = 0 has only constant solution. Anxiety is (very) enemy of perfection. Sorry.

**Category:** Functions and Analysis

[76] **viXra:1507.0022 [pdf]**
*replaced on 2015-07-15 12:54:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

**Category:** Functions and Analysis

[75] **viXra:1507.0022 [pdf]**
*replaced on 2015-07-14 11:13:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. English and portuguese

**Category:** Functions and Analysis

[74] **viXra:1507.0022 [pdf]**
*replaced on 2015-07-05 12:29:37*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. English and portuguese version.

**Category:** Functions and Analysis

[73] **viXra:1506.0019 [pdf]**
*replaced on 2015-06-06 01:44:51*

**Authors:** Carlos Oscar Rodriguez Leal

**Comments:** Pages: 25. Languaje. Spanish.

In this paper several existing numerical methods of solution of ordinary differential equations of first order with initial conditions are modified, so that they can be generalized as methods of solution of ordinary differential equations of order n, according to the theory.

**Category:** Functions and Analysis

[72] **viXra:1505.0083 [pdf]**
*replaced on 2016-02-12 05:16:48*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 27 Pages.

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. First we study these cases at t=0 and then at t≥0.

**Category:** Functions and Analysis

[71] **viXra:1505.0083 [pdf]**
*replaced on 2016-01-26 05:30:49*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 27 Pages.

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. First we study these cases at t=0 and then at t≥0.

**Category:** Functions and Analysis

[70] **viXra:1505.0083 [pdf]**
*replaced on 2015-12-18 07:52:04*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages. soon a new paper about Navier-Stokes equations.

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute. First we study these cases at t=0 and then at t≥0.

**Category:** Functions and Analysis

[69] **viXra:1505.0083 [pdf]**
*replaced on 2015-12-17 10:20:25*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 27 Pages. Long article, in portuguese, many versions, corresponds to the development of my ideas on the Breakdown of Navier-Stokes solutions.

**Category:** Functions and Analysis

[68] **viXra:1505.0083 [pdf]**
*replaced on 2015-08-18 10:17:36*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages. >> This paper will change soon. << Long article, in portuguese, many versions, corresponds to the development of my ideas on the Breakdown of the N-S solutions.

**Category:** Functions and Analysis

[67] **viXra:1505.0083 [pdf]**
*replaced on 2015-08-16 13:45:13*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[66] **viXra:1505.0083 [pdf]**
*replaced on 2015-08-16 10:26:14*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[65] **viXra:1505.0083 [pdf]**
*replaced on 2015-08-12 08:34:24*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages. We're almost there!

**Category:** Functions and Analysis

[64] **viXra:1505.0083 [pdf]**
*replaced on 2015-07-27 06:01:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[63] **viXra:1505.0083 [pdf]**
*replaced on 2015-07-14 06:20:02*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 26 Pages.

**Category:** Functions and Analysis

[62] **viXra:1505.0083 [pdf]**
*replaced on 2015-06-11 11:45:00*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Pages. Por um equívoco interpretei errado o significado do termo "divergence-free". Isto será acertado em breve.

**Category:** Functions and Analysis

[61] **viXra:1505.0083 [pdf]**
*replaced on 2015-06-10 05:44:13*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** see Note A on page 24

**Category:** Functions and Analysis

[60] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-25 08:44:30*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 25 Pages.

**Category:** Functions and Analysis

[59] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-23 11:33:52*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 24 Pages.

**Category:** Functions and Analysis

[58] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-19 07:26:47*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** Pages. The examples like exp(-x) are unhappy because diverge at x negative. When like the exp (-x^2) are better.

**Category:** Functions and Analysis

[57] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-18 06:19:12*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 24 Pages.

**Category:** Functions and Analysis

[56] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-17 06:46:51*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 24 Pages.

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[55] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-14 11:40:26*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 22 Pages. Eureka! (Arquimedes) It´s!

We have proved that there are initial velocities u^0 (x) and forces F(x,t) such that there is no solution to the Navier-Stokes equations, which corresponds to the cases (C) and (D) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Functions and Analysis

[54] **viXra:1505.0083 [pdf]**
*replaced on 2015-05-12 07:22:14*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 18 Pages. In next version a proof for all t

**Category:** Functions and Analysis

[53] **viXra:1503.0047 [pdf]**
*replaced on 2015-03-08 08:39:49*

**Authors:** Edigles Guedes

**Comments:** 3 pages

In this paper, we demonstrate some integral representations for harmonic number
and digamma function.

**Category:** Functions and Analysis

[52] **viXra:1502.0074 [pdf]**
*replaced on 2015-02-14 07:41:19*

**Authors:** Sinisa Bubonja

**Comments:** 20 Pages. Serbian language, Title of paper in translation from Serbian to English changed, Corrected typos, Revised definitions and added example in section 4, results unchanged

In this work I am going to mention historical development of divergent series theory, and to give a number of different examples, as some of the methods for their summing. After that, I am going to introduce the general method, which I discovered, for summing divergent series, which we can also consider as a method for computing limits of divergent sequences and functions in divergent points, In this case, limits of sequences of their partials sums. Through the exercises, I am going to apply this method on given examples and prove its validity. Then I'm going to apply the method to compute the value of some divergent integrals.

**Category:** Functions and Analysis

[51] **viXra:1412.0001 [pdf]**
*replaced on 2015-12-28 21:16:51*

**Authors:** Pith Xie

**Comments:** 28 Pages.

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions and pose some conjectures. Real operators can give new equations and inequalities so as to precisely describe the relation of mathematical objects or scientific objects.

**Category:** Functions and Analysis

[50] **viXra:1412.0001 [pdf]**
*replaced on 2015-09-18 03:36:17*

**Authors:** Pith Xie

**Comments:** 28 pages

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions and pose some conjectures.

**Category:** Functions and Analysis

[49] **viXra:1412.0001 [pdf]**
*replaced on 2015-06-06 20:50:20*

**Authors:** Pith Xie

**Comments:** 29 Pages.

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions.

**Category:** Functions and Analysis

[48] **viXra:1412.0001 [pdf]**
*replaced on 2014-12-13 20:30:23*

**Authors:** Pith Xie

**Comments:** 29 Pages.

The reference [2] contructs the Operator axioms to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions.

**Category:** Functions and Analysis

[47] **viXra:1411.0228 [pdf]**
*replaced on 2014-11-23 09:56:15*

**Authors:** Edigles Guedes

**Comments:** 7 pages

I prove one expansion of infinite serie and some integral representations for Bessel function of the first kind.

**Category:** Functions and Analysis

[46] **viXra:1408.0084 [pdf]**
*replaced on 2014-08-21 06:48:24*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 9 Pages. minor corrections

Cauchy Theory is applied and extended to n-dimensional functions in Clifford algebras, showing the existence of integrals that do not exist in Euclidean spaces. It celebrates the depth of Cauchy's lecture, held on the 22nd of August, 1814, so 200 years ago, in times of bitter warfare. (I should like to recommend reading about his life, e.g. in Wikipedia.)

**Category:** Functions and Analysis

[45] **viXra:1408.0084 [pdf]**
*replaced on 2014-08-15 05:53:17*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 9 Pages. The inroductory Clifford algebra proposition was hopelessly misspelled. Sorry. I hope you got its meaning still.

Cauchy Theory is applied and extended to n-dimensional functions in Clifford algebras, showing the existence of integrals that do not exist in Euclidean spaces. It celebrates the depth of Cauchy's lecture, held on the 22nd of August, 1814, so 200 years ago, in times of bitter warfare. (I should like to recommend reading about his life, e.g. in Wikipedia.)

**Category:** Functions and Analysis