Functions and Analysis

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Recent submissions

Any replacements are listed further down

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

Foundation of a General Theory of Functions of a Variable Complex Magnitude

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

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

Analytic Functions for Clifford Algebras

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

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

A Series Expansion for a Real Function

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

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

On the Expansion of Dirac Delta in Legendre Series

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

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

The Analytic and Numerical Proof of the Sophomore's Dream

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

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

New Developments in Clifford Fourier Transforms

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

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

Semicontinuous Filter Limits of Nets of Lattice Group-Valued Functions

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

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

Strong Uniform Continuity and Filter Exhaustiveness of Nets of Cone Metric Space-Valued Functions

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

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

Ascoli-Type Theorems in the Cone Metric Space Setting

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

[111] viXra:1405.0351 [pdf] submitted on 2014-05-28 15:44:07

Invariant Subspace Problem

Authors: Giuseppe Rauti
Comments: 1 Page.

Invariant Subspace Problem.
Category: Functions and Analysis

[110] viXra:1405.0350 [pdf] submitted on 2014-05-28 15:44:52

Invariant Subspace Problem II

Authors: Giuseppe Rauti
Comments: 1 Page.

Invariant Subspace Problem II.
Category: Functions and Analysis

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

Asymmetric Ascoli-Type Theorems and Filter Exhaustiveness

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

[108] viXra:1405.0269 [pdf] submitted on 2014-05-19 08:30:03

Hilbert's Xix Problem

Authors: Giuseppe Rauti
Comments: 1 Page.

De Giorgi's solution and Nash's solution of Hilbert's XIX problem.
Category: Functions and Analysis

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

The Analysis Techniques for Convexity: Convex Bodies (2)

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

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

The Analysis Techniques for Convexity: Convex Bodies (1)

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

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

On The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions

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

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

An Elementary Primer on Gaussian Integrals

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

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

Solid Angle of the Off-Axis Circle Sector

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

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

Recent Results on Modular Convergence Theorems, Rates of Approximation and Korovkin Theorems for Filter Convergence

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

[101] viXra:1403.0774 [pdf] submitted on 2014-03-23 08:46:22

Notes

Authors: Giuseppe Rauti
Comments: 9 Pages.

@@Notes.
Category: Functions and Analysis

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

Operator Exponentials for the Clifford Fourier Transform on Multivector Fields

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

Nikodym-Type Theorems for Lattice Group-Valued Measures with Respect to Filter Convergence

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

A Mathematical Analysis of Crowds

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

A Novel Method to Evaluate Large Amounts of Data on Chronic Neurological Diseases and It's Consequences.

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

Examples of Products of Distributions

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

Solving Problems of Mathematical Analysis by using Methods of Probability Theory

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

Demystification of the Geometric Fourier Transforms

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

Extending Fourier Transformations to Hamilton’s Quaternions and Clifford’s Geometric Algebras

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

The Quest for Conformal Geometric Algebra Fourier Transformations

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

A Hypothesis about Infinite Series

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

New Concepts of Neutrosophic Sets

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

The Concept of pq-Functions (Ii)

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

Some New Types of Filter Limit Theorems for Topological Group-Valued Measures

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

Rates of Approximation for General Sampling-Type Operators in the Setting of Filter Convergence

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

Triple Integrals and Trilinear Forms

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

A Short Proof of the Luarendeau Conjecture

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

A Conjecture About Functions

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

Fifth Stone of the Sun and the Qi Men Dun Jia Model

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

РЯДОВ ФУРЬЕ С ТОЧКИ СОПРЯЖЕНИЯ

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

Exact Solution of Ordinary Differential Equations, Including Bessel's

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

A Mathematical Conjecture on the Wavefunctions of Quantum Mechanics

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

Tutorial on Fourier Transformations and Wavelet Transformations in Cliord Geometric Algebra

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

The Clifford Fourier Transform in Real Clifford Algebras

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

Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions N = 2 (Mod 4) and N = 3 (Mod 4)

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

Uncertainty Principle for Clifford Geometric Algebras Cl(n,0), N = 3 (Mod 4) Based on Clifford Fourier Transform

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

Basic Multivector Calculus

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

Foundations of Multidimensional Wavelet Theory: The Quaternion Fourier Transform and its Generalizations

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

Geometric Calculus – Engineering Mathematics for the 21st Century

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

Vector Differential Calculus

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

Geometric Calculus for Engineers

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

Windowed Fourier Transform of Two-Dimensional Quaternionic Signals

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

Clifford Algebra Cl(3,0)-valued Wavelets and Uncertainty Inequality for Clifford Gabor Wavelet Transformation

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

Clifford Algebra Cl(3,0)-valued Wavelet Transformation, Clifford Wavelet Uncertainty Inequality and Clifford Gabor Wavelets

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

Two-Dimensional Clifford Windowed Fourier Transform

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

An Uncertainty Principle for Quaternion Fourier Transform

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

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl(3,0)

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

Rational Structure, General Solution and Naked Barred Galaxies

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

A Particular Solution Formula For Inhomogeneous Second Order Linear Ordinary Differential Equations

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

Creator's Standard Equation, General Solution and Naked Barred Galaxies

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

The Creator's Equation System Without Composite Functions

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

Numerical Solution of Nonlinear Sine-Gordon Equation with Local RBF-Based Finite Difference Collocation Method

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

The Creator's Equation

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

The First Derivative Proof of B^^x , X^^x and X^^f(x) by Differentiation Fundamental Limits Method.

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

3D Navier-Stokes Regularity

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

Products of Generalised Functions

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

Dynamical Systems Determinable by Discrete Samples

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

Invertible Dynamic Systems

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

Analysis: Theory and Practice

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

Gaussian Quadrature of the Integrals Int_(-Infty)^infty F(x) dx / Cosh(x).

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

Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes

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

Integral Mean Estimates for the Polar Derivative of a Polynomial

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

Discuss the Navier-Stokes Equation in Fluid (1)

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

Some Problems on Orthogonal Cartesian Spaces

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

Nonlinear Solitary Waves—the Klein Gordon Equations

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

Exact Solutions of Space Dependent Korteweg-de Vries Equation by the Extended Unified Method

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

P vs NP Graphed

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

The Secret Side of Reflexivity

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

The Answers to Two Millennium Prize Problems

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

Clay Navier-Stokes Problem Correctly Solved Cmi Offers Its Reply

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

The Mathematical Theory Of Turbulence Or Chaos

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

Q-Formulӕ

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

On a Method to Find the Roots of a Function that Satisfies the Bolzano's Theorem

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

The Relationship Between the Roots of the Complex Numbers and the Spirals

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

L’Hospital’s Rule

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

The First Digit Of 2^n

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

Matrix Exponential

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

Function of a Matrix

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

The Navier-Stokes Equations And Turbulence

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

Random Consensus in Nonlinear Systems Under Fixed Topology

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

Fractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering

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

Riemann's Sums in Improper Integrals

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

The Local Fractional Hilbert Transform in Fractal Space

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

Apparent Measure and Relative Dimension

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

On the Growth of Meromorphic Solutions of a type of Systems of Complex Algebraic Differential Equations

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

The Finite Yang-Laplace Transform in Fractal Space

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

The Local Fractional Stieltjes Transform in Fractal Space

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

A Series of Constants in the First Three Iterations of the Logistic Map

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

Local Fractional Mellin Transform in Fractal Space

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

Local Fractional Improper Integral in Fractal Space

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

Quickly Identifying the Presence of the Golden Ratio in the Logistic Map

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

A L-Topology of Banach space and Separability of Lipschitz dual space

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

A L*-Convergence of Sequence of Nonlinear Lipschitz Functionals and its Applications in Banach Spaces

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

An Extension Theorem of Nonlinear Lipschitz Functional and its Application in Banach Spaces

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

Volume of the Off-center Spherical Pyramidal Trunk

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

On the Connectivity in One-Dimensional Ad Hoc Wireless Networks with a Forbidden Zone

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

Mean Value Theorems for Local Fractional Integrals on Fractal Space

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

The Introduction of Twist (The Skew) in the Mathematics

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

The Calculus Relation Determination, with Whatever Precision, of Complete Elliptic Integral of the First Kind.

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

Is Zero to the Zero Power Equal to One?

Authors: Ron Bourgoin
Comments: 4 pages

Sometimes in physics we end up with a function that resembles f(x)=00, 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 00 in such cases equal to unity?
Category: Functions and Analysis

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

Summary of the Zeta Regularization Method Applied to the Calculation of Divergent Series Σn5 and Divergent Integrals ∫x5dx

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

Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test

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

A Conjecture About the Riemann XI-Function ξ(1/2 + Iz) and Functional Determinants

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

The Theory of Distributions Applied to Divergent Integrals of the Form (See Paper for Equation)

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

Product of Distributions and Zeta Regularization of Divergent Integrals ∫ Xm-Sdx and Fourier Transforms

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

Immediate Calculation of Some Poisson Type Integrals Using Supermathematics Circular ex-Centric Functions

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 W1 and W2 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

A Triple Inequality with Series and Improper Integrals

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

A Recurrence Method for Generalizing Known Scientific Results

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

Orthogonal Polynomials, Moment Problem and the Riemann XI-Function ξ(1/2 + Iz)

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

The Exact Analytic Solution of Blasius Equation

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

The Total Differential Integral of Calculus

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

Replacements of recent Submissions

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

Analytic Functions for Clifford Algebras

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

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

Analytic Functions for Clifford Algebras

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

[56] viXra:1404.0427 [pdf] replaced on 2014-04-25 04:52:22

Notes A

Authors: Giuseppe Rauti
Comments: 9 Pages.

Function Theory,Functional Analysis,Partial Differential Equations.
Category: Functions and Analysis

[55] viXra:1401.0041 [pdf] replaced on 2014-04-01 17:56:16

Complex Analysis

Authors: Giuseppe Rauti
Comments: 21 Pages.

Complex Analysis
Category: Functions and Analysis

[54] viXra:1401.0041 [pdf] replaced on 2014-03-31 08:58:21

Complex Analysis

Authors: Giuseppe Rauti
Comments: 20 Pages.

Complex Analysis.
Category: Functions and Analysis

[53] viXra:1401.0041 [pdf] replaced on 2014-01-22 05:23:04

Complex Analysis and Fourier Analysis Part I

Authors: Giuseppe Rauti
Comments: 12 Pages.

Complex Analysis and Fourier Analysis.
Category: Functions and Analysis

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

The Concept of pq-Functions

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

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

The Concept of pq-Functions

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

[50] viXra:1309.0031 [pdf] replaced on 2014-02-25 16:27:09

Lecture notes in Complex Analysis and Functional Analysis

Authors: Giuseppe Rauti
Comments: 121 Pages.

Lecture notes in Complex Analysis and Functional Analysis.
Category: Functions and Analysis

[49] viXra:1309.0031 [pdf] replaced on 2014-02-14 10:58:40

Functional Analysis: Part I

Authors: Giuseppe Rauti
Comments: 118 Pages.

Notes about functional analysis.
Category: Functions and Analysis

[48] viXra:1309.0031 [pdf] replaced on 2014-01-20 18:13:58

Functional Analysis: Part I

Authors: Giuseppe Rauti
Comments: 46 Pages.

Baire's Category Theorem; the Uniform Boundedness Principle; Banach-Steinhaus Theorem; Hahn-Banach Theorem; Banach-Schauder Theorem or Open Mapping Theorem; Banach Inverse Mapping Theorem; Closed Graph Theorem; Banach-Caccioppoli Fixed Point Theorem; Banach Spaces; Hilbert Spaces; Frechet-Riesz Representation Theorem; Stampacchia Theorem; Lax-Milgram Theorem; Lp Spaces; Weak and Weak Star Topology; Compactness and Convergence; Reflexive and Separable Spaces; Integral Equations; Fredholm Alternative Theorem; Banach Algebras; Dirichlet's Problem; Hilbert-Polya Conjecture. (Personal lecture notes in Functional Analysis).
Category: Functions and Analysis

[47] viXra:1309.0031 [pdf] replaced on 2014-01-03 16:32:38

Functional Analysis: Part I

Authors: Giuseppe Rauti
Comments: 30 Pages.

Baire's Category Theorem; the Uniform Boundedness Principle; Banach-Steinhaus Theorem; Hahn-Banach Theorem; Banach-Schauder Theorem or Open Mapping Theorem; Banach Inverse Mapping Theorem; Closed Graph Theorem; Banach-Caccioppoli Fixed Point Theorem; Banach Spaces; Hilbert Spaces; Frechet-Riesz Representation Theorem; Stampacchia Theorem; Lax-Milgram Theorem; Lp Spaces; Weak and Weak Star Topology; Compactness and Convergence; Reflexive and Separable Spaces; Integral Equations; Fredholm Alternative Theorem; Banach Algebras; Dirichlet's Problem; Hilbert-Polya Conjecture. (Personal lecture notes in Functional Analysis).
Category: Functions and Analysis

[46] viXra:1307.0162 [pdf] replaced on 2014-02-26 02:17:37

Analytic Number Theory

Authors: Giuseppe Rauti
Comments: 149 Pages.

Analytic Number Theory.
Category: Functions and Analysis

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

The Creator's Equation

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

Products of Generalised Functions

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

Products of Generalised Functions

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

Products of Generalised Functions

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

Products of Generalised Functions

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

Products of Generalised Functions

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

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

Heuristic Study of the Concept of pq-Radial Functions as a New Class of Potentials

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

Gaussian Quadrature of the Integrals Int_(-Infty)^infty F(x) dx / Cosh(x)

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

Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes

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

Discuss the Navier-Stokes Equation in Fluid (1)

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

Discuss the Navier-Stokes Equation in Fluid (1)

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

Some Problems on Orthogonal Cartesian Spaces

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

Nonlinear Solitary Waves—the Klein Gordon Equations

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

Nonlinear Solitary Waves—the Klein Gordon Equations

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

Clay Navier-Stokes Problem Corrrectly Solved Cmi Gives Its Reply

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

Clay Navier-Stokes Problem Correctly Solved Cmi Offers Its Reply

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

Clay Navier-Stokes Problem Correctly Solved - CMI Offers Its Reply

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

Clay Navier-Stokes Problem Correctly Solved - CMI Offers Its Reply

Authors: Jorma Jormakka
Comments: 8 Pages. Minor clarification

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

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

Clay Navier-Stokes Poblem Correctly Solved - CMI Offers Its Reply

Authors: Jorma Jormakka
Comments: 8 Pages. Some typos fixed.

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

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

The Mathematical Theory Of Turbulence Or Chaos

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

Q-Formulӕ

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

Q-Formulӕ

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

Function of a Matrix

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

The Navier-Stokes Equations And Turbulence

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

Entropy and the Individual Iterations of the Logistic Map

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

A L*-Convergence of Sequence of Nonlinear Lipschitz Functionals and its Applications in Banach Spaces

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

The Calculus Relation Determination, with Whatever Precision, of Complete Elliptic Integral of the First Kind.

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

Summary of the Zeta Regularization Method Applied to the Calculation of Divergent Series Σn5 and Divergent Integrals ∫x5dx

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

Summary of the Zeta Regularization Method Applied to the Calculation of Divergent Series Σn5 and Divergent Integrals ∫x5dx

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

Summary of the Zeta Regularization Method Applied to the Calculation of Divergent Series Σn5 and Divergent Integrals ∫x5dx

Authors: Jose Javier Garcia Moreta
Comments: 14 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

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

Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 14 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

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 18 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

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

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

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

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function ξ(1/2 + Iz)

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

A Conjecture About the Riemann XI-Function ξ(1/2 + Iz) and Functional Determinants

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

A Conjecture About the Riemann XI-Function ξ(1/2 + Iz) and Functional Determinants

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

A Conjecture About the Riemann XI-Function ξ(1/2 + Iz) and Functional Determinants

Authors: Jose Javier Garcia Moreta
Comments: 9 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

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

Product of Distributions and Zeta Regularization of Divergent Integrals ∫ Xm-Sdx and Fourier Transforms

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

Product of Distributions and Zeta Regularization of Divergent Integrals ∫ Xm-Sdx and Fourier Transforms

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

A Self-Recurrence Method for Generalizing Known Scientific Results

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