**Previous months:**

2007 - 0703(1)

2009 - 0903(1)

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

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

2012 - 1202(4) - 1203(8) - 1204(1) - 1205(1) - 1206(3) - 1207(2) - 1210(6) - 1211(1) - 1212(2)

2013 - 1301(4) - 1302(2) - 1303(2) - 1304(5) - 1305(8) - 1306(16) - 1307(5) - 1308(3) - 1309(1) - 1310(5) - 1311(2)

2014 - 1402(1) - 1403(5) - 1404(4) - 1405(1) - 1406(1) - 1407(3) - 1408(5) - 1409(1) - 1410(1) - 1411(1)

Any replacements are listed further down

[119] **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

[118] **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

[117] **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

[116] **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 lied the foundation of an aspect of complex analysis.--Typesetter)

**Category:** Functions and Analysis

[115] **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

[114] **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

[113] **viXra:1408.0059 [pdf]**
*submitted on 2014-08-10 06:08:31*

**Authors:** Enrico Masina

**Comments:** 3 Pages.

In this work I will provide a suitable method to have an expansion in Legendre series of the Dirac Delta distribution.
The work has its limits, anyway.

**Category:** Functions and Analysis

[112] **viXra:1408.0058 [pdf]**
*submitted on 2014-08-10 06:10:02*

**Authors:** Enrico Masina

**Comments:** 4 Pages.

In this work I'll prove the so called Sophomore's Dream, id est the calculus of \int from 0 to 1 of x^x dx.

**Category:** Functions and Analysis

[111] **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

[110] **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

[109] **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

[108] **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

[107] **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

[106] **viXra:1404.0132 [pdf]**
*submitted on 2014-04-16 03:36:02*

**Authors:** Cheng Tianren

**Comments:** 12 Pages.

we study the continuity, smoothing, and convergence properties of Steiner
symmetrization in higher dimension space. a stability version of the blaschke-santalo inequality and
the affine isoperimetric inequality for convex bodies is proved. the euler characteristic plays an
important role in many subjects of discrete and continuous mathematic. these distribution function
are defined here in terms of a distance function which is associated with a strictly convex gauge
body that contains the origin in its interior.

**Category:** Functions and Analysis

[105] **viXra:1404.0122 [pdf]**
*submitted on 2014-04-15 03:17:11*

**Authors:** Cheng Tianren

**Comments:** 21 Pages.

we show that every upper semicontinuous and invariant valuation on d-dimensional convex bodies is a linear combination of affine area,volume and characteristic. Motivated by the blaschke-santalo inequality, we define for a convex body in R^n. the volume of the polar body of a symmetric convex set K of R^d is investigate. The theory of curvature measures and Steiner formulae for parallel bodies of sets of positive reach in Euclidean space is generalized to space forms.

**Category:** Functions and Analysis

[104] **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

[103] **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

[102] **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

[101] **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

[100] **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

[99] **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

[98] **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

[97] **viXra:1402.0007 [pdf]**
*submitted on 2014-02-01 21:03:52*

**Authors:** Mustafa A. Khan

**Comments:** 4 Pages.

ABSTRACT:
Objective:
The purpose of this article is to express mathematically all the available information about any chronic neurological disease, such as relapsing-remitting multiple sclerosis, in such a way that all the known variables of the disease can be expressed as a function with time as one of the variables and thereby follow the disease in real time both generally and specifically in a given patient.
Methods:
The method consists of mapping all the known variables about any chronic neurological disease, such as relapsing-remitting multiple sclerosis on a (n+1) dimensional space with time being one of the variables and deriving from this certain functions that represent the disease generally and also specifically in any given patient.
Results:
The results of using this method is the derivation of functions with (n+1) variables. One of the functions will represent the entire disease generically, while other functions will represent the disease in any given individual patient.
Conclusions:
Using this method one can derive several conclusions. These include, finding any sub-types of a chronic neurological disease, help in the prognostication of the disease course in a given patient, stratification of the treatments for the disease, selection of a treatment that is the most useful for a given patient and finally remove the need for doing long and expensive head to head clinical trials of the different treatments for a disease.

**Category:** Functions and Analysis

[96] **viXra:1311.0185 [pdf]**
*submitted on 2013-11-28 04:31:36*

**Authors:** Vincenzo Nardozza

**Comments:** 7 Pages.

A new method for multiplying Distributions is proposed. The method is used to prove interesting equalities involving products among elements of D'.

**Category:** Functions and Analysis

[95] **viXra:1311.0147 [pdf]**
*submitted on 2013-11-20 12:51:39*

**Authors:** Yaremko O., Yaremko N.

**Comments:** 8 Pages.

The author proves statements from mathematical analysis by using methods of Probability theory. Inequalities were proved by means of geometric probability, the relationship of convex functions and random variables is grounded, the decomposi-tion theorems in the limit shape are proved with the help of the law of large numbers, the normal distribution is used to calculate the volume and surface of the n-dimensional unit ball, some integrals are calculated as corollaries.

**Category:** Functions and Analysis

[94] **viXra:1310.0255 [pdf]**
*submitted on 2013-10-29 20:39:08*

**Authors:** Roxana Bujack, Eckhard Hitzer, Gerik Scheuermann

**Comments:** 5 Pages. In T. Simos, G. Psihoyios and C. Tsitouras (eds.), Numerical Analysis and Applied Mathematics ICNAAM 2013, AIP Conf. Proc. 1558, pp. 525-528 (2013). DOI: 10.1063/1.4825543, with minor revisions.

As it will turn out in this paper, the recent hype about most of the Clifford Fourier transforms is not worth the pain.
Almost every one that has a real application is separable and these transforms can be decomposed into a sum of real valued
transforms with constant multivector factors. This fact makes their interpretation, their analysis and their implementation
almost trivial.

**Keywords:** geometric algebra, Clifford algebra, Fourier transform, trigonometric transform, convolution theorem.

**Category:** Functions and Analysis

[93] **viXra:1310.0249 [pdf]**
*submitted on 2013-10-29 03:30:11*

**Authors:** Eckhard Hitzer

**Comments:** 4 Pages. In T. Simos, G. Psihoyios and C. Tsitouras (eds.), Numerical Analysis and Applied Mathematics ICNAAM 2013, AIP Conf. Proc. 1558, pp. 529 -532 (2013). DOI: 10.1063/1.4825544. 2 figures.

We show how Fourier transformations can be extended to Hamilton’s algebra of quaternions. 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. Hamilton’s algebra of quaternions is only one example of the larger class
of Clifford’s geometric algebras, complete algebras encoding a vector space and all its subspace elements. We introduce how
Fourier transformations are extended to Clifford algebras and applied in electromagnetism, and in the processing of images,
color images, vector field and climate data.

**Keywords:** Clifford geometric algebra, quaternion Fourier transform, Clifford Fourier transform, Clifford Fourier-Mellin transform,
Mulitvector wavepackets, Spacetime Fourier transform.

AMS Subj. Class. 15A66, 42A38

**Category:** Functions and Analysis

[92] **viXra:1310.0248 [pdf]**
*submitted on 2013-10-29 03:33:41*

**Authors:** Eckhard Hitzer

**Comments:** 4 Pages. In T. Simos, G. Psihoyios and C. Tsitouras (eds.), Numerical Analysis and Applied Mathematics ICNAAM 2013, AIP Conf. Proc. 1558, pp. 30-33 (2013). DOI: 10.1063/1.4825413

Conformal geometric algebra is preferred in many applications. Clifford Fourier transforms (CFT) allow holistic signal processing of (multi) vector fields, different from marginal (channel wise) processing: Flow fields, color fields, electromagnetic
fields, ... The Clifford algebra sets (manifolds) of $\sqrt{-1}$ lead to continuous manifolds of CFTs. A frequently asked
question is: What does a Clifford Fourier transform of conformal geometric algebra look like? We try to give a first answer.

**Keywords:** Clifford geometric algebra, Clifford Fourier transform, conformal geometric algebra, horosphere.

AMS Subj. Class. 15A66, 42A38

**Category:** Functions and Analysis

[91] **viXra:1310.0176 [pdf]**
*submitted on 2013-10-20 10:06:59*

**Authors:** Sidharth Ghoshal

**Comments:** 11 Pages.

The goal of the following document is to highlight an idea for generating new infinite series besides the ones that the standard mauclarin approach produce:

**Category:** Functions and Analysis

[90] **viXra:1310.0080 [pdf]**
*submitted on 2013-10-12 19:32:01*

**Authors:** A.A.Salama, S.A.Albolwi, Mohmed Eisa

**Comments:** 8 Pages. New neutrosophic sets and possible Applications

In this paper we will introduce and study some types of neutrosophic sets. Finally, we extend the concept of intuitionistic fuzzy ideal [8] to the case of neutrosophic sets. We can use the new of neutrosophic notions in the following
applications: compiler, networks robots, codes and database.

**Category:** Functions and Analysis

[89] **viXra:1309.0073 [pdf]**
*submitted on 2013-09-10 15:25:19*

**Authors:** M. E. Hassani

**Comments:** 27 Pages; 1Table; 2 Figures; 4 References

In this article we study the concept of pq-functions which should regard as an extension of a prior work relative to pq-Radial Functions [1]. Here, our main aim is to generalize this concept to the field of complex numbers. As direct consequences, new kind of (partial) differential equations, polynomials, series and integrals are derived, and Joukowski function is generalized.

**Category:** Functions and Analysis

[88] **viXra:1308.0105 [pdf]**
*submitted on 2013-08-19 12:19:46*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 6 Pages.

Some new types of limit theorems for topological group-valued measures are proved in the context of filter convergence for suitable classes of
filters. We investigate some fundamental properties of topological group-valued measures.
We consider also Schur-type theorems,
using the sliding hump technique, and prove some
convergence theorems in the particular case of
positive measures. We deal with
the notion of uniform filter exhaustiveness,
by means of which we prove some theorems
on existence of the limit measure,
some other kinds of limit theorems and their
equivalence, using known results on existence of
countably additive restrictions of strongly
bounded measures.

**Category:** Functions and Analysis

[87] **viXra:1308.0096 [pdf]**
*submitted on 2013-08-19 04:32:59*

**Authors:** Antonio Boccuto, Xenofon Dimitriou

**Comments:** 5 Pages.

We investigate the order of
approximation of a real-valued function
by means of suitable families of sampling type
operators, which include both discrete
and integral ones. 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 deal with filter convergence, and our results are proper extensions of the classical ones.

**Category:** Functions and Analysis

[86] **viXra:1308.0017 [pdf]**
*submitted on 2013-08-03 18:40:35*

**Authors:** Quentin Hampus Dawkings

**Comments:** 2 Pages.

We derive a formula for triple integrating trilinear forms.

**Category:** Functions and Analysis

[85] **viXra:1307.0163 [pdf]**
*submitted on 2013-07-29 19:03:00*

**Authors:** Raymond Cote

**Comments:** 2 Pages.

We prove the Laurendeau Conjecture.

**Category:** Functions and Analysis

[84] **viXra:1307.0159 [pdf]**
*submitted on 2013-07-29 03:34:53*

**Authors:** Julien Laurendeau

**Comments:** 1 Page.

In this paper I will announce a conjecture invented by myself,hoping that the great mathematiciens will have the pleasure to try to prove it.

**Category:** Functions and Analysis

[83] **viXra:1307.0156 [pdf]**
*submitted on 2013-07-28 04:44:28*

**Authors:** John Frederick Sweeney

**Comments:** 45 Pages.

The Fifth Stone of the Sun, or the Aztec Calendar, bears similarities to the Qi Men Dun Jia Cosmic Board. In addition, the stone contains the natural logarithm e or Euler logarithm, as well as a series of related logarithms. The author has noted that matter begins with the natural logarithm e in a previous paper. The similarities and the high level mathematics lead towards the Clifford Algebra Cl (8) and the Exceptional Lie Algebra E8, which imply that Meso - American civilization enjoyed high - level mathematics.

**Category:** Functions and Analysis

[82] **viXra:1307.0155 [pdf]**
*submitted on 2013-07-27 11:20:41*

**Authors:** O.E. Yaremko

**Comments:** 7 Pages. Русский язык

Фурье формула для 2π-периодических функций с точки сопряжения изучаются. В случае периодической функции , точки сопряжения преобразование Фурье может быть упрощена для вычисления дискретного множества комплексных амплитуд, называемых Фурье-коэффициентов. Исследование основные свойства рядов Фурье.

**Category:** Functions and Analysis

[81] **viXra:1307.0063 [pdf]**
*submitted on 2013-07-12 09:47:39*

**Authors:** Claude Michael Cassano

**Comments:** 18 Pages.

The fact that there are N linearly independent solutions to a N-th order homogeneous linear ordinary differential equation suggests that linear transformations may yield solutions from solutions. This is, indeed, shown to be true by applying the technique to Bessel differential equations.

**Category:** Functions and Analysis

[80] **viXra:1306.0228 [pdf]**
*submitted on 2013-06-28 13:11:44*

**Authors:** HaengJin Choe

**Comments:** 4 Pages.

The uncertainty principle is one of the fundamental principles of quantum mechanics. While studying quantum mechanics recently, the author made an exciting mathematical discovery about the product of two expectation values. The author explains the discovery.

**Category:** Functions and Analysis

[79] **viXra:1306.0133 [pdf]**
*submitted on 2013-06-17 05:21:04*

**Authors:** Eckhard Hitzer

**Comments:** 45 Pages. 3 tables. In K. Tachibana (ed.) Tutorial on Fourier Transf. and Wavelet Transf. in Clifford Geometric Algebra, Lect. notes of the Int. Workshop for “Computational Science with Geometric Algebra” (FCSGA2007), Nagoya Univ., JP, Feb. 2007, pp. 65-87 (2007).

First, the basic concept multivector functions and their vector derivative
in geometric algebra (GA) is introduced. Second, beginning
with the Fourier transform on a scalar function we generalize to a
real Fourier transform on GA multivector-valued functions (f : R^3 -> Cl(3,0)). Third, we show a set of important properties of the Clifford
Fourier transform (CFT) on Cl(3,0) such as dierentiation properties,
and the Plancherel theorem. We round o the treatment of the CFT
(at the end of this tutorial) by applying the Clifford Fourier transform
properties for proving an uncertainty principle for Cl(3,0) multivector
functions.
For wavelets in GA it is shown how continuous Clifford Cl(3,0)-
valued admissible wavelets can be constructed using the similitude
group SIM(3), a subgroup of the ane group of R^3. We express the
admissibility condition in terms of the CFT and then derive a set of
important properties such as dilation, translation and rotation covariance,
a reproducing kernel, and show how to invert the Clifford wavelet
transform of multivector functions. We explain (at the end of this tutorial)
a generalized Clifford wavelet uncertainty principle. For scalar
admissibility constant it sets bounds of accuracy in multivector wavelet
signal and image processing. As concrete example we introduce
multivector Clifford Gabor wavelets, and describe important properties
such as the Clifford Gabor transform isometry, a reconstruction
formula, and (at the end of this tutorial) an uncertainty principle for
Clifford Gabor wavelets.
Keywords: vector derivative, multivector-valued function, Clifford
(geometric) algebra, Clifford Fourier transform, uncertainty principle,
similitude group, geometric algebra wavelet transform, geometric
algebra Gabor wavelets.

**Category:** Functions and Analysis

[78] **viXra:1306.0130 [pdf]**
*submitted on 2013-06-17 01:29:18*

**Authors:** Eckhard Hitzer

**Comments:** 21 Pages. 2 figures, 1 table. First published: Proc. of 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany, 04–06 July 2012.

We use the recent comprehensive research [17, 19] on the manifolds of
square roots of -1 in real Clifford’s geometric algebras Cl(p,q) in order to
construct the Clifford Fourier transform. Basically in the kernel of the complex
Fourier transform the imaginary unit j in C (complex numbers) is replaced by a square root
of -1 in Cl(p,q). The Clifford Fourier transform (CFT) thus obtained generalizes
previously known and applied CFTs [9, 13, 14], which replaced j in C
only by blades (usually pseudoscalars) squaring to -1. A major advantage
of real Clifford algebra CFTs is their completely real geometric interpretation.
We study (left and right) linearity of the CFT for constant multivector
coefficients in Cl(p,q), translation (x-shift) and modulation (w-shift) properties,
and signal dilations. We show an inversion theorem. We establish the
CFT of vector differentials, partial derivatives, vector derivatives and spatial
moments of the signal. We also derive Plancherel and Parseval identities as
well as a general convolution theorem.
Keywords: Clifford Fourier transform, Clifford algebra, signal processing,
square roots of -1.

**Category:** Functions and Analysis

[77] **viXra:1306.0127 [pdf]**
*submitted on 2013-06-17 01:59:58*

**Authors:** Eckhard Hitzer, Bahri Mawardi

**Comments:** 24 Pages. 2 tables. Adv. App. Cliff. Alg. Vol. 18, S3,4, pp. 715-736 (2008). DOI: 10.1007/s00006-008-0098-3.

First, the basic concepts of the multivector functions, vector differential
and vector derivative in geometric algebra are introduced. Second, we
dene a generalized real Fourier transform on Clifford multivector-valued functions
( f : R^n -> Cl(n,0), n = 2,3 (mod 4) ). Third, we show a set of important
properties of the Clifford Fourier transform on Cl(n,0), n = 2,3 (mod 4) such as
dierentiation properties, and the Plancherel theorem, independent of special
commutation properties. Fourth, we develop and utilize commutation properties
for giving explicit formulas for f x^m; f Nabla^m and for the Clifford convolution. Finally,
we apply Clifford Fourier transform properties for proving an uncertainty
principle for Cl(n,0), n = 2,3 (mod 4) multivector functions.
Keywords: Vector derivative, multivector-valued function, Clifford (geometric)
algebra, Clifford Fourier transform, uncertainty principle.

**Category:** Functions and Analysis

[76] **viXra:1306.0126 [pdf]**
*submitted on 2013-06-17 02:09:49*

**Authors:** Eckhard Hitzer, Bahri Mawardi

**Comments:** 10 Pages. 1 table. In T. Qian, M.I. Vai, X. Yusheng (eds.), Wavelet Analysis and Applications, Springer (SCI) Book Series Applied and Numerical Harmonic Analysis, Springer, pp. 45-54 (2006). DOI: 10.1007/978-3-7643-7778-6_6.

First, the basic concepts of the multivector functions, vector differential
and vector derivative in geometric algebra are introduced. Second,
we define a generalized real Fourier transform on Clifford multivector-valued functions (f : Rn -> Cl(n,0), n = 3 (mod 4)). Third, we introduce a set of important properties of the Clifford Fourier transform on Cl(n,0), n = 3 (mod 4) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving a directional uncertainty principle for Cl(n,0), n = 3 (mod 4) multivector functions.
Keywords. Vector derivative, multivector-valued function, Clifford (geometric) algebra, Clifford Fourier transform, uncertainty principle.
Mathematics Subject Classication (2000). Primary 15A66; Secondary 43A32.

**Category:** Functions and Analysis

[75] **viXra:1306.0124 [pdf]**
*submitted on 2013-06-17 02:53:25*

**Authors:** Eckhard Hitzer

**Comments:** 6 Pages. Proc. of 18th Intelligent Systems Symposium (FAN 2008), 23-24 Oct. 2008, Hiroshima, Japan, pp. 185 – 190 (2008).

We begin with introducing the generalization of real, complex, and quaternion numbers to hypercomplex
numbers, also known as Clifford numbers, or multivectors of geometric algebra. Multivectors encode everything from
vectors, rotations, scaling transformations, improper transformations (reflections, inversions), geometric objects (like
lines and spheres), spinors, and tensors, and the like. Multivector calculus allows to define functions mapping
multivectors to multivectors, differentiation, integration, function norms, multivector Fourier transformations and
wavelet transformations, filtering, windowing, etc. We give a basic introduction into this general mathematical
language, which has fascinating applications in physics, engineering, and computer science.

**Category:** Functions and Analysis

[74] **viXra:1306.0122 [pdf]**
*submitted on 2013-06-17 03:06:17*

**Authors:** Eckhard Hitzer

**Comments:** 3 Pages. E. Hitzer, Foundations of Multidimensional Wavelet Theory: The Quaternion Fourier Transf. and its Generalizations, Preprints of Meeting of the JSIAM, ISSN: 1345-3378, Tsukuba Univ., 16-18 Sep. 2006, Tsukuba, Japan, pp. 66,67.

Keywords: Multidimensional Wavelets, Quaternion Fourier Transform, Clifford geometric algebra

**Category:** Functions and Analysis

[73] **viXra:1306.0117 [pdf]**
*submitted on 2013-06-17 03:56:01*

**Authors:** Eckhard Hitzer

**Comments:** 12 Pages. 13 figures. Mem. Fac. Eng. Fukui Univ. 50(1), pp. 127-137 (2002).

This paper treats important questions at the interface of mathematics and the engineering sciences. It starts off with a quick quotation tour through 2300 years of mathematical history. At the beginning of the 21st century, technology has developed beyond every expectation. But do we also learn and practice an adequately modern form of mathematics? The paper argues that this role is very likely to be played by universal geometric calculus. The fundamental geometric product of vectors is introduced. This gives a quick-and-easy description of rotations as well as the ultimate geometric interpretation of the famous quaternions of Sir W.R. Hamilton. Then follows a one page review of the historical roots of geometric calculus. In order to exemplify the role of geometric calculus for the engineering sciences three representative examples are looked at in some detail: elasticity, image geometry and pose estimation. Next a current snapshot survey of geometric calculus software is provided. Finally the value of geometric calculus for teaching, research and development is commented.

**Category:** Functions and Analysis

[72] **viXra:1306.0116 [pdf]**
*submitted on 2013-06-17 04:00:42*

**Authors:** Eckhard Hitzer

**Comments:** 17 Pages. Mem. Fac. Eng. Fukui Univ. 50(1), pp. 109-125 (2002).

This paper treats the fundamentals of the vector differential calculus part of universal
geometric calculus. Geometric calculus simplifies and unifies the structure and notation of
mathematics for all of science and engineering, and for technological applications. In order to
make the treatment self-contained, I first compile all important geometric algebra relationships,
which are necessary for vector differential calculus. Then differentiation by vectors is introduced
and a host of major vector differential and vector derivative relationships is proven explicitly in a
very elementary step by step approach. The paper is thus intended to serve as reference material,
giving details, which are usually skipped in more advanced discussions of the subject matter.
Keywords: Geometric Calculus, Geometric Algebra, Clifford Algebra,
Vector Derivative, Vector Differential Calculus

**Category:** Functions and Analysis

[71] **viXra:1306.0114 [pdf]**
*submitted on 2013-06-17 04:13:56*

**Authors:** Eckhard Hitzer

**Comments:** 8 Pages. 7 figures. Proc. of the Pukyong National University - Fukui University International Symposium 2001 for Promotion of Research Cooperation, Pukyong National University, Busan, Korea, pp. 59-66 (2001).

This paper treats important questions at the interface of mathematics and the engineering sciences.
It starts off with a quick quotation tour through 2300 years of mathematical history. At the beginning
of the 21st century, technology has developed beyond every expectation. But do we also learn and
practice an adequately modern form of mathematics? The paper argues that this role is very likely to
be played by (universal) geometric calculus. The fundamental geometric product of vectors is
introduced. This gives a quick-and-easy description of rotations as well as the ultimate geometric
interpretation of the famous quaternions of Sir W.R. Hamilton. Then follows a one page review of the
historical roots of geometric calculus. In order to exemplify the role geometric calculus for the
engineering sciences three representative examples are looked at in some detail: elasticity, image
geometry and pose estimation. Finally the value of geometric calculus for teaching, research and
development and its worldwide impact are commented.

**Category:** Functions and Analysis

[70] **viXra:1306.0096 [pdf]**
*submitted on 2013-06-14 03:17:09*

**Authors:** B. Mawardi, E. Hitzer, R. Ashino, R. Vaillancourt

**Comments:** 20 Pages. Appl. Math. and Computation, 216, Iss. 8, pp. 2366-2379, 15 June 2010. 6 figures, 1 table.

In this paper, we generalize the classical windowed Fourier transform
(WFT) to quaternion-valued signals, called the quaternionic windowed
Fourier transform (QWFT).
Using the spectral
representation of the quaternionic Fourier transform (QFT), we derive
several important properties such as reconstruction formula,
reproducing kernel, isometry,
and orthogonality relation.
Taking the Gaussian function as window function we obtain quaternionic
Gabor filters which
play the role of coefficient functions when decomposing the signal in the
quaternionic Gabor
basis. We apply the QWFT properties and the (right-sided) QFT to establish
a Heisenberg type
uncertainty principle for the QWFT. Finally, we briefly introduce an
application of the QWFT to a linear time-varying system.
Keywords: quaternionic Fourier transform, quaternionic windowed Fourier
transform, signal processing, Heisenberg type uncertainty principle

**Category:** Functions and Analysis

[69] **viXra:1306.0095 [pdf]**
*submitted on 2013-06-14 03:21:42*

**Authors:** Mawardi Bahri, Eckhard Hitzer

**Comments:** 2 Pages. Preprints of Meeting of the Japan Society for Industrial and Applied Mathematics, ISSN: 1345-3378, Tsukuba University, 16-18 Sep. 2006, Tsukuba, Japan, pp. 64,65.

The purpose of this paper is to construct Clifford
algebra Cl(3,0)-valued wavelets using the similitude
group SIM(3) and then give a detailed explanation
of their properties using the Clifford Fourier
transform. Our approach can generalize complex
Gabor wavelets to multivectors called Clifford Gabor
wavelets. Finally, we describe some of their
important properties which we use to establish a
new uncertainty principle for the Clifford Gabor
wavelet transform.

**Category:** Functions and Analysis

[68] **viXra:1306.0094 [pdf]**
*submitted on 2013-06-14 03:35:04*

**Authors:** Mawardi Bahri, Eckhard Hitzer

**Comments:** 23 Pages. International Journal of Wavelets, Multiresolution and Information Processing, 5(6), pp. 997-1019 (2007). DOI: 10.1142/S0219691307002166, 2 tables.

In this paper, it is shown how continuous Clifford Cl(3,0)-valued admissible wavelets can
be constructed using the similitude group SIM(3), a subgroup of the affine group of R^3.
We express the admissibility condition in terms of a Cl(3,0) Clifford Fourier transform
and then derive a set of important properties such as dilation, translation and rotation
covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform of
multivector functions. We invent a generalized Clifford wavelet uncertainty principle. For
scalar admissibility constant it sets bounds of accuracy in multivector wavelet signal and
image processing. As concrete example we introduce multivector Clifford Gabor wavelets,
and describe important properties such as the Clifford Gabor transform isometry, a
reconstruction formula, and an uncertainty principle for Clifford Gabor wavelets.
Keywords: Similitude group, Clifford Fourier transform, Clifford wavelet transform, Clifford Gabor wavelets, uncertainty principle.

**Category:** Functions and Analysis

[67] **viXra:1306.0092 [pdf]**
*submitted on 2013-06-14 04:25:26*

**Authors:** Mawardi Bahri, Eckhard Hitzer, Sriwulan Adji

**Comments:** 15 Pages. in G. Scheuermann, E. Bayro-Corrochano (eds.), Geometric Algebra Computing, Springer, New York, 2010, pp. 93-106. 4 figures, 1 table.

Recently several generalizations to higher dimension of the classical
Fourier transform (FT) using Clifford geometric algebra have been introduced, including
the two-dimensional (2D) Clifford Fourier transform (CFT). Based on the
2D CFT, we establish the two-dimensional Clifford windowed Fourier transform
(CWFT). Using the spectral representation of the CFT, we derive several important
properties such as shift, modulation, a reproducing kernel, isometry and an orthogonality
relation. Finally, we discuss examples of the CWFT and compare the CFT
and the CWFT.

**Category:** Functions and Analysis

[66] **viXra:1306.0091 [pdf]**
*submitted on 2013-06-14 04:35:57*

**Authors:** Mawardi Bahri, Eckhard Hitzer, Akihisa Hayashi, Ryuichi Ashino

**Comments:** 20 Pages. Computer & Mathematics with Applications, 56, pp. 2398-2410 (2008). DOI: 10.1016/j.camwa.2008.05.032, 3 figures, 1 table.

We review the quaternionic Fourier transform (QFT). Using the properties of the QFT we establish an uncertainty principle for the right-sided QFT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternion signal minimizes the uncertainty.
Key words: Quaternion algebra, Quaternionic Fourier transform, Uncertainty principle, Gaussian quaternion signal, Hypercomplex functions
Math. Subj. Class.: 30G35, 42B10, 94A12, 11R52

**Category:** Functions and Analysis

[65] **viXra:1306.0089 [pdf]**
*submitted on 2013-06-14 04:44:58*

**Authors:** Bahri Mawardi, Eckhard Hitzer

**Comments:** 23 Pages. Advances in Applied Clifford Algebras, 16(1), pp. 41-61 (2006). DOI 10.1007/s00006-006-0003-x , 3 tables.

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions (f: R^3 -> Cl(3,0)). Third, we show a set of important properties of the Clifford Fourier transform on Cl(3,0) such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl(3,0) multivector functions.
Keywords: vector derivative, multivector-valued function, Clifford
(geometric) algebra, Clifford Fourier transform, uncertainty principle.

**Category:** Functions and Analysis

[64] **viXra:1305.0174 [pdf]**
*submitted on 2013-05-28 22:01:07*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 17 Pages. 2 Figures. Two theorems of rational structure are proved,

Rational structure in two dimension means that not only there exists an orthogonal net of curves in the plane but also, for each curve, the stellar density on one side of the curve is in constant ratio to the density on the other side of the curve. Such a curve is called a proportion curve or a Darwin curve. Such a distribution of matter is called a rational structure. Spiral galaxies are blended with dust and gas. Their longer wavelength (e.g. infrared) images present mainly the stellar distribution, which is called the naked galaxies. Jin He found many evidences that galaxies are rational stellar distribution. We list a few examples. Firstly, galaxy components (disks and bars) can be fitted with rational structure. Secondly, spiral arms can be fitted with Darwin curves. Thirdly, rational structure dictates New Universal Gravity which explains constant rotation curves simply and elegantly. This article presents the systematic theory of rational structure, its general solution and geometric meaning. A preliminary application to spiral galaxies is also discussed.

**Category:** Functions and Analysis

[63] **viXra:1305.0156 [pdf]**
*submitted on 2013-05-27 04:09:38*

**Authors:** Claude Michael Cassano

**Comments:** 5 Pages.

A formula for Particular solutions to any Linear Second Order Inhomogeneous Ordinary Diffrential equations is presented. For second order ODEs these make the methods of undetermined coefficients and variation of parameters obsolete.

**Category:** Functions and Analysis

[62] **viXra:1305.0147 [pdf]**
*submitted on 2013-05-23 19:33:33*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 14 Pages. 1 figure. Solve the beautiful equation and reveal the secret of galaxies

We have not found the general solution to the Creator's equation system. However, we have outlined the strategy for determining the solution. Firstly, we should study the stretch equation which is the first order linear and homogeneous partial differential equation, and find its all stretches which correspond to the given vector field (i.e., the gradient of the logarithmic stellar density). Our solution G(x,y), however, must be simultaneously the modulus of some analytic complex function. It is called the modulus stretch. Secondly, among all possible modulus stretches, we find the right solution (i.e., the orthogonal net of curves) which satisfies the Creator's standard equation.

**Category:** Functions and Analysis

[61] **viXra:1305.0094 [pdf]**
*submitted on 2013-05-15 22:17:49*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 11 Pages. 1 Figure. Nobel prizes come from the solutions of the Creator's equation.

Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors present the Creator's equation system without composite functions, the equation system being the necessary and sufficient condition for rational structure. However, we have not found its general solution. Please help us find the general solution.

**Category:** Functions and Analysis

[60] **viXra:1305.0082 [pdf]**
*submitted on 2013-05-14 00:58:36*

**Authors:** Yaqub Azari

**Comments:** 9 Pages.

This paper presents the local radial basis function based on finite difference (LRBF-FD) for the sine-Gordon equation. Advantages of the proposed method are that this method is mesh free unlike finite difference (FD) and finite element (FE) methods, and its coefficient matrix is sparse and well-conditioned as compared with the global RBF collocation method (GRBF). Numerical results show that the LRBF-FD method has good accuracy as compared with GRBF.

**Category:** Functions and Analysis

[59] **viXra:1305.0052 [pdf]**
*submitted on 2013-05-08 19:32:37*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 8 Pages. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

**Category:** Functions and Analysis

[58] **viXra:1305.0040 [pdf]**
*submitted on 2013-05-07 04:43:29*

**Authors:** Nasser Almismari

**Comments:** 12 Pages.

This paper is to find by proof the first derive of known tetration functions, fixed base iterated functions b^^x , general case for b^^f(x) and variable base with variable height iterated function x^^x. although the case of b^^x is already known by using the base change method but its derive function f(x) is still depend on the derive of f(x-1) which gives a shortcoming derivation. However, in the coming proofs, the resulted derivative functions are proved by applying differentiation elementary concepts step by step up to the final first derive ,but an unknown limit and a non-elementary product part of the resulted derivative function still needs study, Although I included approximation method for numerical solutions.

**Category:** Functions and Analysis

[57] **viXra:1305.0015 [pdf]**
*submitted on 2013-05-02 15:34:07*

**Authors:** Adam Chmaj

**Comments:** 4 Pages.

We solve the NS Millenium Prize Problem. This is
done with a modification of a pairing method used
earlier to establish the regularity of hyperdissipative NS equations.

**Category:** Functions and Analysis

[56] **viXra:1304.0158 [pdf]**
*submitted on 2013-04-28 13:23:26*

**Authors:** Vincenzo Nardozza

**Comments:** 15 Pages.

A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also derived.

**Category:** Functions and Analysis

[55] **viXra:1304.0138 [pdf]**
*submitted on 2013-04-24 17:00:58*

**Authors:** Ovidiu Ilie ŞANDRU, Luige VLĂDĂREANU, Alexandra ŞANDRU

**Comments:** 8 Pages.

In this paper we shall define and study an important class of dynamic systems which allow to effectively determine their mathematical model exclusively on experimental basis. The usefulness of these results of mathematical nature, obtained by extending the Whittaker-Shannon sampling theory, will be highlighted through an applied example from the field of optoelectronics.

**Category:** Functions and Analysis

[54] **viXra:1304.0137 [pdf]**
*submitted on 2013-04-24 17:02:49*

**Authors:** Ovidiu Ilie ŞANDRU, Luige VLĂDĂREANU, Alexandra ŞANDRU

**Comments:** 4 Pages.

In this paper we introduce the notion of invertible dynamic system, we indicate a very general
method to determine the inverse of such a system and we give evidence of the numerous applications of the
subclass of dynamic systems defined by this notion.

**Category:** Functions and Analysis

[53] **viXra:1304.0098 [pdf]**
*submitted on 2013-04-19 14:30:48*

**Authors:** Jesse Gilbert

**Comments:** 8 Pages.

[No abstract]

**Category:** Functions and Analysis

[52] **viXra:1303.0038 [pdf]**
*submitted on 2013-03-06 15:32:53*

**Authors:** Richard J. Mathar

**Comments:** 9 Pages.

The manuscript delivers
nodes and their weights for Gaussian quadratures with a ``non-classical''
weight in the integrand defined by a reciprocal hyperbolic cosine.
The associated monic orthogonal polynomials are constructed; their
coefficients
turn out to be simple multiples of the coefficients of the Meixner polynomials.
A final table shows the abscissae-weight pairs for up to 128 nodes.

**Category:** Functions and Analysis

[51] **viXra:1303.0013 [pdf]**
*submitted on 2013-03-03 09:07:05*

**Authors:** Richard J. Mathar

**Comments:** 14 Pages.

The manuscript provides tables of abscissae and weights
for Gauss-Laguerre integration on 64, 96 and 128 nodes,
and abscissae and weights
for Gauss-Hermite integration on 96 and 128 nodes.

**Category:** Functions and Analysis

[50] **viXra:1302.0138 [pdf]**
*submitted on 2013-02-20 21:39:58*

**Authors:** N. A. Rather, Suhail Gulzar

**Comments:** 8 Pages.

Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} \cite{d} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$
$$ n(|\alpha|-k)\left\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^r d\theta\right\}^{\frac{1}{r}}\leq\left\{ \int\limits_{0}^{2\pi}\left|1+ke^{i\theta}\right|^r d\theta\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_\alpha P(z)|. $$
\indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu},$ $1\leq\mu\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.

**Category:** Functions and Analysis

[49] **viXra:1302.0025 [pdf]**
*submitted on 2013-02-04 20:52:45*

**Authors:** Cheng Tianren

**Comments:** 30 Pages.

We prove the regularity of weak solutions of the navier-stokes equations for compressible,isentropic flow in three space dimension.We also prove the existence of a spatially periodic weak solution to the steady compressible navier-stokes equations for any specific heat ratio. Next we study the hyperbolic system of euler equations for isentropic,compressible fluid governed by a general law. We establish the vanishing viscosity limit of the navier-stokes equations to euler equations for one-dimensional compressible fluid flow.

**Category:** Functions and Analysis

[48] **viXra:1301.0181 [pdf]**
*submitted on 2013-01-29 20:47:08*

**Authors:** Cheng Tianren

**Comments:** 29 Pages.

we consider a special class of non-Archimedean Banach spaces, called Hilbertian,for which every one-dimensional linear subspaces has an orthogonal complement. We construct examples of hilbertian spaces over a non-spherically complete valued field without an orthogonal base.

**Category:** Functions and Analysis

[47] **viXra:1301.0169 [pdf]**
*submitted on 2013-01-27 19:46:56*

**Authors:** Cheng Tianren

**Comments:** 22 Pages.

We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.

**Category:** Functions and Analysis

[46] **viXra:1301.0036 [pdf]**
*submitted on 2013-01-07 01:37:06*

**Authors:** Hamdy I. Abdel-Gawad, Nasser S. Elazab, Mohamed Osman

**Comments:** 6 Pages. IOSR Journals

Abstract: Recently the unified method for finding traveling wave solutions of non-linear evolution equations
was proposed by one of the authors a. It was shown that, this method unifies all the methods being used to find
these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de
Vries (KdV) equation with space dependent coefficients. A new class of multiple-soliton or wave trains is
obtained.
Keywords: Exact solution, Extended unified method, Korteweg-deVries equation, variable coefficients

**Category:** Functions and Analysis

[45] **viXra:1301.0010 [pdf]**
*submitted on 2013-01-02 18:58:54*

**Authors:** Andrew Nassif

**Comments:** 2 Pages.

For many years lied a problem called the P vs NP. The question is to find the number of factorial possibilities to its orders. An example of this is finding the possibilities and comparison of improbabilities of picking 100 students out of 400 students. According to Lardner's theorem the number of known atoms in the universe is less then the number of combinations of possible orders and combinations of the answer to the P vs. NP problems. Finding the equation for the number of different orders a group of 400 people can be put into and subtracting 300 different people that couldn't get picked is equal to ((400!)-(100!*3)). My project is to represent this data through algorithms and different diagrams. When looking at my project you will know how I found a solution and the importance of it. My project will include all the required schematics, and graphs that coordinates with this answer. It will also acquire data showing different possibilities between P vs NP. As well as the combination where P can equal NP and N equals 1, or the possibilities where P doesn't equal NP and N isn't 1. P and NP is believed to stand for the number of possibilities and impossibilities.

**Category:** Functions and Analysis

[44] **viXra:1212.0168 [pdf]**
*submitted on 2012-12-31 08:22:43*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 3 Pages. (It's really more than 10 years old.)

It is proven that every complete, metrizable locally convex space (a.k.a. F-space)
is reflexive.
This in particular disproves an old conjecture that L^\infty was the dual of L^1.
It is shown that indeed, L^\infty contains a subspace of overcountable dimension
not contained in the dual of L^1.

**Category:** Functions and Analysis

[43] **viXra:1212.0137 [pdf]**
*submitted on 2012-12-23 13:38:03*

**Authors:** Andrew Nassif

**Comments:** 5 Pages.

For ten long years these two problems have not been solved after being offered a prize. Solving the Riemann hypothesis will bring dimensional analysis in mathematics and physics. Solving the P vs NP will increase our knowledge in programing and provide a wide expansion of mathematical understanding and industrilization.

**Category:** Functions and Analysis

[42] **viXra:1211.0055 [pdf]**
*submitted on 2012-11-11 06:36:23*

**Authors:** Jorma Jormakka

**Comments:** 8 Pages.

The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.

**Category:** Functions and Analysis

[41] **viXra:1210.0146 [pdf]**
*submitted on 2012-10-25 17:09:58*

**Authors:** Bertrand Wong

**Comments:** 16 Pages.

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would greatly affect the field of fluid mechanics. In this paper, which had been published in an international journal in 2010, a reasoned, practical approach towards resolving the issue is adopted and a practical, statistical kind of mathematical solution is proposed.

**Category:** Functions and Analysis

[40] **viXra:1210.0111 [pdf]**
*submitted on 2012-10-20 15:45:39*

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

**Comments:** 38 Pages.

This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.

**Category:** Functions and Analysis

[39] **viXra:1210.0069 [pdf]**
*submitted on 2012-10-13 04:32:55*

**Authors:** Imanol Pérez

**Comments:** 1 Page.

In this paper I introduce a new method to find the roots of a function with two known values a and b such that sgn(f(a)) = -sgn(f(b)).

**Category:** Functions and Analysis

[38] **viXra:1210.0068 [pdf]**
*submitted on 2012-10-13 04:38:34*

**Authors:** Imanol Pérez

**Comments:** 3 Pages.

This paper shows a relationship between spiral and the roots of complex numbers.

**Category:** Functions and Analysis

[37] **viXra:1210.0041 [pdf]**
*submitted on 2012-10-09 02:27:46*

**Authors:** Pierre-Yves Gaillard

**Comments:** 1 Page.

We give a short proof of l'Hospital's Rule.

**Category:** Functions and Analysis

[36] **viXra:1210.0007 [pdf]**
*submitted on 2012-10-01 22:45:24*

**Authors:** Ren Shiquan

**Comments:** 3 Pages. This is an undergraduate level assignment.

In this paper, we give a study on the probability of the first digit of 2^n. This is an undergraduate level assignment..

**Category:** Functions and Analysis

[35] **viXra:1207.0066 [pdf]**
*submitted on 2012-07-18 02:17:26*

**Authors:** Pierre-Yves Gaillard

**Comments:** 1 Page.

Let a be an element of a finite dimensional C-algebra with 1. Then there is a unique polynomial f_a such that f_a(a) = exp(a) and deg f_a < dim C[a]. We give an explicit formula for f_a.

**Category:** Functions and Analysis

[34] **viXra:1207.0046 [pdf]**
*submitted on 2012-07-12 01:17:41*

**Authors:** Pierre-Yves Gaillard

**Comments:** 1 Page.

Let a be a square matrix with complex entries and f a function holomorphic on an open subset U of the complex plane. It is well known that f can be evaluated on a if the spectrum of a is contained in U. We show that, for a fixed f, the resulting matrix depends holomorphically on a.

**Category:** Functions and Analysis

[33] **viXra:1206.0086 [pdf]**
*submitted on 2012-06-24 14:14:23*

**Authors:** Bertrand Wong

**Comments:** 2 Pages.

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

**Category:** Functions and Analysis

[32] **viXra:1206.0017 [pdf]**
*submitted on 2012-06-05 10:52:56*

**Authors:** Radha F. Gupta, Poom Kumam

**Comments:** 9 Pages.

This paper investigates the consensus problem in almost sure sense for uncertain multi-agent systems with noises and fixed topology. By
combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of
distributed stochastic type non-linear protocols. Numerical examples are given to illustrate the results.

**Category:** Functions and Analysis

[31] **viXra:1206.0005 [pdf]**
*submitted on 2012-06-02 21:56:55*

**Authors:** Xiong Wang

**Comments:** 6 Pages.

This paper discuss the longstanding problems of fractional calculus such as too
many definitions while lacking physical or geometrical meanings, and try to extend fractional
calculus to any dimension. First, some different definitions of fractional derivatives, such as the
Riemann-Liouville derivative, the Caputo derivative, Kolwankar's local derivative and Jumarie's modified
Riemann-Liouville derivative, are discussed and conclude that the very reason for introducing
fractional derivative is to study nondifferentiable functions. Then, a concise and essentially local
definition of fractional derivative for one dimension function is introduced and its geometrical
interpretation is given. Based on this simple definition, the fractional calculus is extended to
any dimension and the \emph{Fractional Geometric Calculus} is proposed. Geometric algebra provided
an powerful mathematical framework in which the most advanced concepts modern physic, such
as quantum mechanics, relativity, electromagnetism, etc., can be expressed in this framework
graciously. At the other hand, recent developments in nonlinear science and complex system
suggest that scaling, fractal structures, and nondifferentiable functions occur much more
naturally and abundantly in formulations of physical theories. In this paper, the extended
framework namely the Fractional Geometric Calculus is proposed naturally, which aims
to give a unifying language for mathematics, physics and science of complexity of the 21st century.

**Category:** Functions and Analysis

[30] **viXra:1205.0078 [pdf]**
*submitted on 2012-05-19 15:27:40*

**Authors:** Hilário Fernandes de Araújo Júnior

**Comments:** 2 Pages. Article part 1, Copyright© 19 may 2012

In this article, is exposed two sum representations for integrals in which the integration interval is infinite.

**Category:** Functions and Analysis

[29] **viXra:1204.0091 [pdf]**
*submitted on 2012-04-26 07:16:21*

**Authors:** Guang-Sheng Chen

**Comments:** 6 Pages.

In this paper, we establish local fractional Hilbert transform in fractal space, consider
some properties of local fractional Hilbert Transforms.

**Category:** Functions and Analysis

[28] **viXra:1203.0078 [pdf]**
*submitted on 2012-03-20 07:02:58*

**Authors:** Faycal Ben Adda

**Comments:** 24 Pages. A short version of this paper was published in Journal Européen des Systèmes Automatisés, Fractional order systems, 42, p733-746, 2008.

In this paper, we introduce a concept of
"apparent" measure in R^n and we define a concept of relative dimension (of real order) with it, which depends on the geometry
of the object to measure and on the distance which separates it
from an observer. At the end we discuss the relative dimension of the Cantor set. This measure enables us to provide a geometric
interpretation of the Riemann-Liouville's integral of order alpha between 0 and 1.

**Category:** Functions and Analysis

[27] **viXra:1203.0065 [pdf]**
*submitted on 2012-03-16 20:48:14*

**Authors:** Xiao-meng Li, Xianfeng Su

**Comments:** 4 Pages.

This paper is concerned with the growth of meromorphic solutions of a class of systems of complex algebraic differentialequations. A general estimate the growth order of solutions of the systems of differential equation is obtained by Zalacman Lemma. We also take an example to show that the result is right.

**Category:** Functions and Analysis

[26] **viXra:1203.0038 [pdf]**
*submitted on 2012-03-11 08:37:50*

**Authors:** Guang-Sheng Chen

**Comments:** 4 Pages.

In this paper, we establish finte Yang-Laplace Transform on fractal space, considered
some properties of finte Yang-Laplace Transform.

**Category:** Functions and Analysis

[25] **viXra:1203.0037 [pdf]**
*submitted on 2012-03-11 08:40:07*

**Authors:** Guang-Sheng Chen

**Comments:** 4 Pages.

This paper deals with the theory of the local fractional Stieltjes transform. We derive
the Stieltjes transform. This is followed by several examples and the basic operational properties
of Stieltjes transforms.

**Category:** Functions and Analysis

[24] **viXra:1203.0035 [pdf]**
*submitted on 2012-03-11 03:35:12*

**Authors:** S Halayka

**Comments:** 9 Pages. Lots of figures.

A rough analysis of the first three iterations of the logistic map $x^\prime = rx(1-x)$ produces a series of special constants.
The three constants are $1$, the inverse of the golden ratio, and Catalan's constant.

**Category:** Functions and Analysis

[23] **viXra:1203.0030 [pdf]**
*submitted on 2012-03-08 22:45:49*

**Authors:** Guang-Sheng Chen

**Comments:** 12 Pages.

This paper deals with the theory and applications of the local fractional Mellin
transform of the real order α . We define the local fractional Mellin transform and its inverse
transform. This is followed by several examples and the basic operational properties of local
fractional Mellin transform. We discuss applications of local fractional Mellin transforms to local
fractional boundary value problems.

**Category:** Functions and Analysis

[22] **viXra:1203.0029 [pdf]**
*submitted on 2012-03-08 22:49:14*

**Authors:** Guang-Sheng Chen

**Comments:** 8 Pages.

In this paper we study Local fractional improper integrals on fractal space. By
some mean value theorems for Local fractional integrals, we prove an analogue of the classical
Dirichlet-Abel test for Local fractional improper integrals.

**Category:** Functions and Analysis

[21] **viXra:1203.0023 [pdf]**
*submitted on 2012-03-07 02:27:32*

**Authors:** S Halayka

**Comments:** 2 Pages.

A brief visual demonstration of the presence of the golden ratio in the logistic map is given.

**Category:** Functions and Analysis

[20] **viXra:1202.0071 [pdf]**
*submitted on 2012-02-21 22:27:58*

**Authors:** Choe Ryong Gil

**Comments:** 23 pages

In this paper we have introduced a new topology and a convergence in Banach space, which would be called a L-topology and a L-convergence. It is similar to the weak topology and weak convergence, but there are some essential differences. For example, the L-topology is stronger than weak topology, but weaker than the strong one. On the basis of the notion, we have considered the problem on the separability and reflexibility of Lipschitz (Lip-) dual space. Furthermore, we have introduced a new topology of Lip-dual space, which is similar to the weak* (W*-) topology of linear dual of Banach space and would be called an L*-topology, and we have considered the problems on the metrizability of L*-topology and on the L*-separability of Lip-dual space, too.

**Category:** Functions and Analysis

[19] **viXra:1202.0069 [pdf]**
*submitted on 2012-02-20 20:24:29*

**Authors:** Choe Ryong Gil

**Comments:** 18 pages

In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.

**Category:** Functions and Analysis

[18] **viXra:1202.0060 [pdf]**
*submitted on 2012-02-19 02:03:52*

**Authors:** Choe Ryong Gil

**Comments:** 17 pages

In this paper we have obtained a new theorem that a nonlinear Lipschitz (Lip-) functional defined on the closed subset of Banach spaces can be extended to the whole space with Lip-continuity and maintenance of Lip-constant, which would be called an extension theorem (ET). This theorem is a generalization to the Lip-functional of the famous Hahn-Banach theorem on the bounded linear functional. By the ET, we have completely solved the open problem on the relation of the invertibility between the Lip-operator and its Lip-dual operator.

**Category:** Functions and Analysis

[17] **viXra:1202.0015 [pdf]**
*submitted on 2012-02-06 15:20:56*

**Authors:** Richard J. Mathar

**Comments:** 12 Pages. Includes complete C++ source listing.

The volume inside intersecting spheres may be computed by a standard method which computes
a surface integral over all visible sections of the spheres. If the visible sections are divided in simple
zonal sections, the individual contribution by each zone follows from basic analysis. We implement
this within a semi-numerical program which marks the zones individually as visible or invisible.

**Category:** Functions and Analysis

[16] **viXra:1112.0044 [pdf]**
*submitted on 2011-12-15 09:36:44*

**Authors:** Xiaodong Hu, Evgeniy Grechnikov

**Comments:** 11 Pages.

This paper investigates the connectivity in one-dimensional ad hoc wireless networks with a forbidden zone. We derive the probability of the wireless networks which are composed of exactly m
clusters by means of the methods of combinatorics and probability. The probability of connectivity, i.e. $m = 1$, can be obtained as a special case. Further, we explain how the transmission range of node affects the connectivity of the wireless network.

**Category:** Functions and Analysis

[15] **viXra:1110.0075 [pdf]**
*submitted on 30 Oct 2011*

**Authors:** Guang-Sheng Chen

**Comments:** 6 pages.

In this paper, by some properties of Local fractional integral, we establish the
generalized Mean value theorems for Local Fractional Integral.

**Category:** Functions and Analysis

[14] **viXra:1106.0056 [pdf]**
*submitted on 27 Jun 2011*

**Authors:** Mircea Selariu

**Comments:** 10 pages.

The article define a mathematic entity called twist, which generates, in this way, notion of straight line.
Straight line becom thus a twist of eccentricity e = 0, and broken line (zigzag line) is a twist of s = ± 1.

**Category:** Functions and Analysis

[13] **viXra:1106.0055 [pdf]**
*submitted on 26 Jun 2011*

**Authors:** Mircea Selariu

**Comments:** 10 pages.

These papers show a calculus relation ( 50 ) of complete elliptic integral K(k) with minimum 9
precise decimals and the possibility to obtain a more precisely relation.. It results by application
Landen's method, of geometrical-arithmetical average, not for obtain a numerical value but to
obtain a compute algebraically relation after 5 steps of a geometrical transformation, called
"CENTERED PROCESS".

**Category:** Functions and Analysis

[12] **viXra:1106.0014 [pdf]**
*submitted on 9 Jun 2011*

**Authors:** Ron Bourgoin

**Comments:** 4 pages

Sometimes in physics we end up with a function that resembles
f(x)=0^{0}, where for example we have a radius that goes to zero and
an exponent goes to zero in k/r n , where k is a constant. Is 0^{0} in
such cases equal to unity?

**Category:** Functions and Analysis

[11] **viXra:1009.0047 [pdf]**
*submitted on 13 Sep 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 13 pages.

We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[10] **viXra:1008.0025 [pdf]**
*submitted on 9 Aug 2010*

**Authors:** Elemér E Rosinger

**Comments:** 166 pages

It is shown how the infinity of differential algebras of generalized
functions is naturally subjected to a basic dichotomic singularity test
regarding their significantly different abilities to deal with large classes
of singularities. In this respect, a review is presented of the way
singularities are dealt with in four of the infinitely many types of
differential algebras of generalized functions. These four algebras, in the
order they were introduced in the literature are : the nowhere dense,
Colombeau, space-time foam, and local ones. And so far, the first
three of them turned out to be the ones most frequently used in a
variety of applications. The issue of singularities is naturally not a
simple one. Consequently, there are different points of view, as well as
occasional misunderstandings. In order to set aside, and preferably,
avoid such misunderstandings, two fundamentally important issues
related to singularities are pursued. Namely, 1) how large are the sets
of singularity points of various generalized functions, and 2) how are
such generalized functions allowed to behave in the neighbourhood of
their point of singularity. Following such a two fold clarification on
singularities, it is further pointed out that, once one represents
generalized functions - thus as well a large class of usual singular functions
- as elements of suitable differential algebras of generalized functions,
one of the main advantages is the resulting freedom to perform
globally arbitrary algebraic and differential operations on such functions,
simply as if they did not have any singularities at all. With the same
freedom from singularities, one can perform globally operations such
as limits, series, and so on, which involve infinitely many generalized
functions. The property of a space of generalized functions of being
a flabby sheaf proves to be essential in being able to deal with large
classes of singularities. The first and third type of the mentioned
differential algebras of generalized functions are flabby sheaves, while the
second type fails to be so. The fourth type has not yet been studied
in this regard.

**Category:** Functions and Analysis

[9] **viXra:1007.0005 [pdf]**
*submitted on 5 Jul 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 7 pages.

We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)

**Category:** Functions and Analysis

[8] **viXra:1005.0075 [pdf]**
*submitted on 19 May 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 9 pages

In this paper we review some results on the regularization of divergent integrals of
the form ... (see paper for full abstract)

**Category:** Functions and Analysis

[7] **viXra:1005.0071 [pdf]**
*submitted on 17 May 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 9 pages

Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of dDirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[6] **viXra:1004.0053 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache, Mircea Eugen Șelariu

**Comments:** 10 pages

This article presents two methods, in parallel, of solving more complex integrals, among
which is the Poisson's integral, in order to emphasize the obvious advantages of a new method
of integration, which uses the supermathematics circular ex-centric functions.
We will specially analyze the possibilities of easy passing/changing of the supermathematics
circular ex-centric functions of a centric variable α to the same functions of ex-centric variable &theta.
The angle α is the angle at the center point O(0,0), which represents the centric variable and θ
is the angle at the ex-center E(k,ε), representing the ex-centric variable. These are the angles
from which the points W_{1} and W_{2} are visible on the unity circle - resulted from the intersection
of the unity/trigonometric circle with the revolving straight line d around the ex-centric
E(k,&epsilon) - from O and from E, respectively.

**Category:** Functions and Analysis

[5] **viXra:1004.0014 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

As a consequence of the Integral Test we find a triple inequality which bounds up and
down both a series with respect to its corresponding improper integral, and reciprocally
an improper integral with respect to its corresponding series.

**Category:** Functions and Analysis

[4] **viXra:1003.0166 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

A great number of articles widen known scientific results (theorems, inequalities,
math/physics/chemical etc. propositions, formulas), and this is due to a simple procedure,
of which it is good to say a few words

**Category:** Functions and Analysis

[3] **viXra:1003.0105 [pdf]**
*submitted on 10 Mar 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 12 Pages.

In this paper we study a set of orthogonal Polynomials with respect a certain given measure related to
the Taylor series expansion of the Xi-function , this paper is based on a previous conjecture by Carlon and
Gaston related to the fact that Riemann Hypothesis (with simple zeros) is equivalent to the limit for a
certain set of orthogonal Polynomials, we study the 'Hamburger moment problem' for even 'n' and 0 for n
odd here the moments are related to the power series expansion of Xi-function , we also give the
integral representation for the generating function , in terms of the Laplace transform of , and
in the end of the paper we study the connection of our orthogonal polynomial set with the Kernel ,
through all the paper we will use the simplified notation (see paper for abstract with equations)

**Category:** Functions and Analysis

[2] **viXra:0903.0007 [pdf]**
*submitted on 28 Mar 2009*

**Authors:** Chun-Xuan Jiang

**Comments:** recovered from sciprint.org

We find Blasius function to satisfy the boundary condition f(∞) = 1 and obtain the exact analytic soultion of Blasius equation.

**Category:** Functions and Analysis

[1] **viXra:0703.0011 [pdf]**
*submitted on 10 Mar 2007*

**Authors:** Gerardo Alvarado

**Comments:** recovered from sciprint.org

I deduce a series which satisfies the fundamental theorem of calculus without dependence on an
explicit function. I prove Taylor's theorem and show that it is closely related. I deduce a series for the
logarithm function and from this series deduce the power series representation of the logarithm function
along with the interval of convergence. I also solve an ordinary differential equation.

**Category:** Functions and Analysis

[49] **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

[48] **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

[47] **viXra:1309.0073 [pdf]**
*replaced on 2014-08-22 05:45:37*

**Authors:** Mohamed E. Hassani

**Comments:** 26 Pages

In this article we study the concept of pq-functions which should regard as an extension of prior work relating to pq-radial functions [1]. Here, our main aim is to generalize this concept to the field of complex numbers. As direct consequences, new kind of (partial) differential equations, polynomials, series and integrals are derived, and Joukowski function is generalized.

**Category:** Functions and Analysis

[46] **viXra:1309.0073 [pdf]**
*replaced on 2013-12-01 04:57:28*

**Authors:** M. E. Hassani

**Comments:** 27 Pages.

In this article we study the concept of pq-functions which should regard as an extension of a prior work relative to pq-Radial Functions [1]. Here, our main aim is to generalize this concept to the field of complex numbers. As direct consequences, new kind of (partial) differential equations, polynomials, series and integrals are derived, and Joukowski function is generalized.

**Category:** Functions and Analysis

[45] **viXra:1305.0052 [pdf]**
*replaced on 2013-05-15 22:21:50*

**Authors:** Jin He, Xiaoli Yang

**Comments:** 8 Pages. 1 Figure. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.

**Category:** Functions and Analysis

[44] **viXra:1304.0158 [pdf]**
*replaced on 2014-01-31 10:13:46*

**Authors:** Vincenzo Nardozza

**Comments:** 16 Pages.

An elementary algebra of products of distributions is constructed. The constructed algebra is equivalent, although less general, of the full Colombeau algebra of generalised functions. However, the loss of generality is compensated by the fact that the new algebra of generalised functions is very convenient for practical calculations. An equivalent relation, among elements of the above algebra, is proposed and a linear space of generalised functions is constructed as a partition space of the elementary algebra with respect to the equivalent relation. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also given.

**Category:** Functions and Analysis

[43] **viXra:1304.0158 [pdf]**
*replaced on 2013-12-25 10:07:31*

**Authors:** Vincenzo Nardozza

**Comments:** 15 Pages.

An elementary algebra of products of distributions is constructed. An equivalent relation between products of distributions is given and a space of generalised functions is constructed as a partition space of the elementary algebra with respect to the equivalent relation. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also derived.

**Category:** Functions and Analysis

[42] **viXra:1304.0158 [pdf]**
*replaced on 2013-11-23 09:17:36*

**Authors:** Vincenzo Nardozza

**Comments:** 14 Pages.

A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also derived.

**Category:** Functions and Analysis

[41] **viXra:1304.0158 [pdf]**
*replaced on 2013-11-17 18:37:28*

**Authors:** Vincenzo Nardozza

**Comments:** 13 Pages.

A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also derived.

**Category:** Functions and Analysis

[40] **viXra:1304.0158 [pdf]**
*replaced on 2013-05-01 10:41:48*

**Authors:** Vincenzo Nardozza

**Comments:** 14 Pages.

**Category:** Functions and Analysis

[39] **viXra:1304.0151 [pdf]**
*replaced on 2013-04-28 14:30:03*

**Authors:** M.E.Hassani

**Comments:** 20 Pages; 7 References

The main purpose of the present paper is the heuristic study of the structure, properties and consequences of new class of potential functions results from the concept of pq-Radial functions which are fundamental family of solutions of second order pq-PDE.

**Category:** Functions and Analysis

[38] **viXra:1303.0038 [pdf]**
*replaced on 2013-03-21 04:33:40*

**Authors:** Richard J. Mathar

**Comments:** 10 Pages. Corrected 2 digits in eq. (11). Added remark 2, eqs. (22) and (24), and 2 references

The manuscript delivers
nodes and their weights for Gaussian quadratures with a "non-classical"
weight in the integrand defined by a reciprocal hyperbolic cosine.
The associated monic orthogonal polynomials are constructed; their
coefficients
are simple multiples of the coefficients of Hahn polynomials.
A final table shows the abscissae-weight pairs for up to 128 nodes.

**Category:** Functions and Analysis

[37] **viXra:1303.0013 [pdf]**
*replaced on 2013-10-02 14:02:55*

**Authors:** Richard J. Mathar

**Comments:** 27 Pages. Tables extended to Hermite quadratures for x^m*exp(-x^2), m=2, 4, 6 or 8.

The manuscript provides tables of abscissae and weights
for Gauss-Laguerre integration on 64, 96 and 128 nodes,
and abscissae and weights
for Gauss-Hermite integration on 96 and 128 nodes.

**Category:** Functions and Analysis

[36] **viXra:1302.0025 [pdf]**
*replaced on 2014-03-09 20:57:38*

**Authors:** Cheng Tianren

**Comments:** 30 Pages.

We prove the regularity of weak solutions of the navier-stokes equations for compressible,isentropic flow in three space dimension.We also prove the existence of a spatially periodic weak solution to the steady compressible navier-stokes equations for any specific heat ratio. Next we study the hyperbolic system of euler equations for isentropic,compressible fluid governed by a general law. We establish the vanishing viscosity limit of the navier-stokes equations to euler equations for one-dimensional compressible fluid flow.

**Category:** Functions and Analysis

[35] **viXra:1302.0025 [pdf]**
*replaced on 2013-02-22 03:22:36*

**Authors:** Cheng Tianren

**Comments:** 30 Pages.

We prove the regularity of weak solutions of the navier-stokes equations for compressible,isentropic flow in three space dimension.We also prove the existence of a spatially periodic weak solution to the steady compressible navier-stokes equations for any specific heat ratio. Next we study the hyperbolic system of euler equations for isentropic,compressible fluid governed by a general law. We establish the vanishing viscosity limit of the navier-stokes equations to euler equations for one-dimensional compressible fluid flow.

**Category:** Functions and Analysis

[34] **viXra:1301.0181 [pdf]**
*replaced on 2013-02-04 06:51:44*

**Authors:** Cheng Tianren

**Comments:** 29 Pages.

we consider a special class of non-Archimedean Banach spaces, called Hilbertian,for which every one-dimensional linear subspaces has an orthogonal complement. We construct examples of hilbertian spaces over a non-spherically complete valued field without an orthogonal base.

**Category:** Functions and Analysis

[33] **viXra:1301.0169 [pdf]**
*replaced on 2013-02-07 20:26:07*

**Authors:** Cheng Tianren

**Comments:** 22 Pages.

We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.

**Category:** Functions and Analysis

[32] **viXra:1301.0169 [pdf]**
*replaced on 2013-02-05 19:38:21*

**Authors:** Cheng Tianren

**Comments:** 22 Pages.

We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.

**Category:** Functions and Analysis

[31] **viXra:1211.0055 [pdf]**
*replaced on 2013-03-06 23:49:55*

**Authors:** Jorma Jormakka

**Comments:** Corrected the number of pages and added a small clarifying comment to the text.

The Clay Navier-Stokes proble is correctly solved. The answer from CMI is included. The article discusses why and how the Clay Navier-Stokes problem should be corrected.

**Category:** Functions and Analysis

[30] **viXra:1211.0055 [pdf]**
*replaced on 2012-11-16 01:16:50*

**Authors:** Jorma Jormakka

**Comments:** 11 Pages. Addded a letter from Bombieri after the reply from CMI

The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.

**Category:** Functions and Analysis

[29] **viXra:1211.0055 [pdf]**
*replaced on 2012-11-13 02:20:09*

**Authors:** Jorma Jormakka

**Comments:** 8 Pages. A minor change

The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.

**Category:** Functions and Analysis

[28] **viXra:1211.0055 [pdf]**
*replaced on 2012-11-12 08:56:11*

**Authors:** Jorma Jormakka

**Comments:** 8 Pages. Minor clarification

**Category:** Functions and Analysis

[27] **viXra:1211.0055 [pdf]**
*replaced on 2012-11-11 23:01:43*

**Authors:** Jorma Jormakka

**Comments:** 8 Pages. Some typos fixed.

**Category:** Functions and Analysis

[26] **viXra:1210.0146 [pdf]**
*replaced on 2014-06-09 10:10:42*

**Authors:** Bertrand Wong

**Comments:** 16 Pages.

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the
three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are relatively simplelooking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would greatly affect the field of fluid mechanics.
In this paper, which had been published in the International Journal of Nonlinear Science, Vol. 10 (2010) No. 3, pp.264-278, a reasoned, practical approach towards resolving the issue is adopted and a practical, statistical kind of mathematical solution is proposed.

**Category:** Functions and Analysis

[25] **viXra:1210.0111 [pdf]**
*replaced on 2013-04-18 16:34:34*

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

**Comments:** 76 Pages.

This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.

**Category:** Functions and Analysis

[24] **viXra:1210.0111 [pdf]**
*replaced on 2012-11-20 14:35:42*

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

**Comments:** 71 Pages.

This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.

**Category:** Functions and Analysis

[23] **viXra:1207.0046 [pdf]**
*replaced on 2013-04-08 09:05:21*

**Authors:** Pierre-Yves Gaillard

**Comments:** 1 Page.

Let a be a square matrix with complex entries and f a function holomorphic on an open subset U of the complex plane. It is well known that f can be evaluated on a if the spectrum of a is contained in U. We show that, for a fixed f, the resulting matrix depends holomorphically on a.

**Category:** Functions and Analysis

[22] **viXra:1206.0086 [pdf]**
*replaced on 2014-06-09 10:32:56*

**Authors:** Bertrand Wong

**Comments:** 2 Pages.

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

**Category:** Functions and Analysis

[21] **viXra:1203.0035 [pdf]**
*replaced on 2012-03-12 15:27:40*

**Authors:** S Halayka

**Comments:** 7 Pages. Changed title, reduced clutter, submitted to journal.

It is presumed a priori that there is an entropy-area relationship inherent to the iterations of the logistic map. Several
interesting results are produced.

**Category:** Functions and Analysis

[20] **viXra:1202.0069 [pdf]**
*replaced on 2012-02-23 03:28:33*

**Authors:** Choe Ryong Gil, Kim Myong Il

**Comments:** 17 pages

In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.

**Category:** Functions and Analysis

[19] **viXra:1106.0055 [pdf]**
*replaced on 27 Jun 2011*

**Authors:** Mircea Selariu

**Comments:** 10 pages. v1 in Romanian, v2 in English.

These papers show a calculus relation ( 50 ) of complete elliptic integral K(k) with minimum 9
precise decimals and the possibility to obtain a more precisely relation.. It results by application
Landen's method, of geometrical-arithmetical average, not for obtain a numerical value but to
obtain a compute algebraically relation after 5 steps of a geometrical transformation, called
"CENTERED PROCESS".

**Category:** Functions and Analysis

[18] **viXra:1009.0047 [pdf]**
*replaced on 23 Feb 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 19 pages.

We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[17] **viXra:1009.0047 [pdf]**
*replaced on 11 Feb 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 18 pages.

We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[16] **viXra:1009.0047 [pdf]**
*replaced on 8 Nov 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 14 pages.

**Category:** Functions and Analysis

[15] **viXra:1008.0025 [pdf]**
*replaced on 12 Aug 2010*

**Authors:** Elemér E Rosinger

**Comments:** 184 pages

It is shown how the infinity of differential algebras of generalized
functions is naturally subjected to a basic dichotomic singularity test
regarding their significantly different abilities to deal with large classes
of singularities. In this respect, a review is presented of the way
singularities are dealt with in four of the infinitely many types of
differential algebras of generalized functions. These four algebras, in the
order they were introduced in the literature are : the nowhere dense,
Colombeau, space-time foam, and local ones. And so far, the first
three of them turned out to be the ones most frequently used in a
variety of applications. The issue of singularities is naturally not a
simple one. Consequently, there are different points of view, as well as
occasional misunderstandings. In order to set aside, and preferably,
avoid such misunderstandings, two fundamentally important issues
related to singularities are pursued. Namely, 1) how large are the sets
of singularity points of various generalized functions, and 2) how are
such generalized functions allowed to behave in the neighbourhood of
their point of singularity. Following such a two fold clarification on
singularities, it is further pointed out that, once one represents
generalized functions - thus as well a large class of usual singular functions
- as elements of suitable differential algebras of generalized functions,
one of the main advantages is the resulting freedom to perform
globally arbitrary algebraic and differential operations on such functions,
simply as if they did not have any singularities at all. With the same
freedom from singularities, one can perform globally operations such
as limits, series, and so on, which involve infinitely many generalized
functions. The property of a space of generalized functions of being
a flabby sheaf proves to be essential in being able to deal with large
classes of singularities. The first and third type of the mentioned
differential algebras of generalized functions are flabby sheaves, while the
second type fails to be so. The fourth type has not yet been studied
in this regard.

**Category:** Functions and Analysis

[14] **viXra:1007.0005 [pdf]**
*replaced on 28 Nov 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 23 Pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant
for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) ,
this potential V is related to the half-integral of the logarithmic derivative for the Riemann
Xi-function, through the paper
we will assume that the reduced Planck constant is defined in units where and that the mass is

**Category:** Functions and Analysis

[13] **viXra:1007.0005 [pdf]**
*replaced on 13 Nov 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 20 pages.

We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1

**Category:** Functions and Analysis

[12] **viXra:1007.0005 [pdf]**
*replaced on 3 Nov 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 19 pages.

We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1

**Category:** Functions and Analysis

[11] **viXra:1007.0005 [pdf]**
*replaced on 4 Oct 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 16 pages.

We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1

**Category:** Functions and Analysis

[10] **viXra:1007.0005 [pdf]**
*replaced on 28 Jun 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 14 pages.

**Category:** Functions and Analysis

[9] **viXra:1007.0005 [pdf]**
*replaced on 2 May 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 18 pages.

**Category:** Functions and Analysis

[8] **viXra:1007.0005 [pdf]**
*replaced on 5 Apr 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 16 pages.

**Category:** Functions and Analysis

[7] **viXra:1007.0005 [pdf]**
*replaced on 10 Mar 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 15 pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one dimension () for a real-valued function V(x) ,
this potential V is related to the half-integral of the logarithmic derivative for the Riemann
Xi-function, through
the paper we will assume that the reduced Planck constant is defined in units where h-bar = 1 and that the mass is 2m = 1

**Category:** Functions and Analysis

[6] **viXra:1007.0005 [pdf]**
*replaced on 18 Nov 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 13 pages.

We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)

**Category:** Functions and Analysis

[5] **viXra:1007.0005 [pdf]**
*replaced on 3 Aug 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 10 pages.

We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)

**Category:** Functions and Analysis

[4] **viXra:1007.0005 [pdf]**
*replaced on 27 Jul 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 9 pages.

**Category:** Functions and Analysis

[3] **viXra:1005.0071 [pdf]**
*replaced on 20 Jun 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 13 pages

Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of Dirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[2] **viXra:1005.0071 [pdf]**
*replaced on 15 Jan 2011*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 13 pages

Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of Dirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)

**Category:** Functions and Analysis

[1] **viXra:1003.0166 [pdf]**
*replaced on 20 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 7 pages

A great number of articles widen known scientific results (theorems, inequalities,
math/physics/chemical etc. propositions, formulas), and this is due to a simple procedure,
of which it is good to say a few words:
Let suppose that we want to generalizes a known mathematical proposition P(a) ,
where a is a constant, to the proposition P(n) , where n is a variable which belongs to
subset of N .
To prove that P is true for n by recurrence means the following: the first step is
trivial, since it is about the known result P(a) (and thus it was already verified before by
other mathematicians!). To pass from P(n) to P(n + 1) , one uses too P(a) : therefore one
widens a proposition by using the proposition itself, in other words the found
generalization will be paradoxically proved with the help of the particular case from
which one started!
We present below the generalizations of Hölder, Minkovski, and respectively
Tchebychev inequalities.

**Category:** Functions and Analysis