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2012 - 1202(4) - 1203(8) - 1204(1) - 1205(1) - 1206(3) - 1207(2) - 1210(6) - 1211(1) - 1212(2)
2013 - 1301(4) - 1302(3) - 1303(2) - 1304(5) - 1305(6)
Any replacements are listed further down
[64] viXra:1305.0135 [pdf] submitted on 2013-05-23 01:32:35
Authors: S. Sambasivarao, And P. Jagadeeshwar
Comments: 3 Pages.
In this paper we present an explicit formula to
find a square root of certain class of matrices using Cayley-Hamiliton Theorem. Our formula extend and generalize the corresponding approach for 2-by-2 matrix. Also, some examples are given in support of our method.
Category: Functions and Analysis
[63] viXra:1305.0094 [pdf] submitted on 2013-05-15 22:17:49
Authors: Jin He, Xiaoli Yang
Comments: 11 Pages. 1 Figure. Nobel prizes come from the solutions of the Creator's equation.
Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors present the Creator's equation system without composite functions, the equation system being the necessary and sufficient condition for rational structure. However, we have not found its general solution. Please help us find the general solution.
Category: Functions and Analysis
[62] viXra:1305.0082 [pdf] submitted on 2013-05-14 00:58:36
Authors: Yaqub Azari
Comments: 9 Pages.
This paper presents the local radial basis function based on finite difference (LRBF-FD) for the sine-Gordon equation. Advantages of the proposed method are that this method is mesh free unlike finite difference (FD) and finite element (FE) methods, and its coefficient matrix is sparse and well-conditioned as compared with the global RBF collocation method (GRBF). Numerical results show that the LRBF-FD method has good accuracy as compared with GRBF.
Category: Functions and Analysis
[61] viXra:1305.0052 [pdf] submitted on 2013-05-08 19:32:37
Authors: Jin He, Xiaoli Yang
Comments: 8 Pages. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.
Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.
Category: Functions and Analysis
[60] viXra:1305.0040 [pdf] submitted on 2013-05-07 04:43:29
Authors: Nasser Almismari
Comments: 12 Pages.
This paper is to find by proof the first derive of known tetration functions, fixed base iterated functions b^^x , general case for b^^f(x) and variable base with variable height iterated function x^^x. although the case of b^^x is already known by using the base change method but its derive function f(x) is still depend on the derive of f(x-1) which gives a shortcoming derivation. However, in the coming proofs, the resulted derivative functions are proved by applying differentiation elementary concepts step by step up to the final first derive ,but an unknown limit and a non-elementary product part of the resulted derivative function still needs study, Although I included approximation method for numerical solutions.
Category: Functions and Analysis
[59] viXra:1305.0015 [pdf] submitted on 2013-05-02 15:34:07
Authors: Adam Chmaj
Comments: 4 Pages.
We solve the NS Millenium Prize Problem. This is
done with a modification of a pairing method used
earlier to establish the regularity of hyperdissipative NS equations.
Category: Functions and Analysis
[58] viXra:1304.0158 [pdf] submitted on 2013-04-28 13:23:26
Authors: Vincenzo Nardozza
Comments: 15 Pages.
A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A way of multiplying the defined generalised functions with polynomials is also derived.
Category: Functions and Analysis
[57] viXra:1304.0138 [pdf] submitted on 2013-04-24 17:00:58
Authors: Ovidiu Ilie ŞANDRU, Luige VLĂDĂREANU, Alexandra ŞANDRU
Comments: 8 Pages.
In this paper we shall define and study an important class of dynamic systems which allow to effectively determine their mathematical model exclusively on experimental basis. The usefulness of these results of mathematical nature, obtained by extending the Whittaker-Shannon sampling theory, will be highlighted through an applied example from the field of optoelectronics.
Category: Functions and Analysis
[56] viXra:1304.0137 [pdf] submitted on 2013-04-24 17:02:49
Authors: Ovidiu Ilie ŞANDRU, Luige VLĂDĂREANU, Alexandra ŞANDRU
Comments: 4 Pages.
In this paper we introduce the notion of invertible dynamic system, we indicate a very general
method to determine the inverse of such a system and we give evidence of the numerous applications of the
subclass of dynamic systems defined by this notion.
Category: Functions and Analysis
[55] viXra:1304.0098 [pdf] submitted on 2013-04-19 14:30:48
Authors: Jesse Gilbert
Comments: 8 Pages.
[No abstract]
Category: Functions and Analysis
[54] viXra:1303.0038 [pdf] submitted on 2013-03-06 15:32:53
Authors: Richard J. Mathar
Comments: 9 Pages.
The manuscript delivers
nodes and their weights for Gaussian quadratures with a ``non-classical''
weight in the integrand defined by a reciprocal hyperbolic cosine.
The associated monic orthogonal polynomials are constructed; their
coefficients
turn out to be simple multiples of the coefficients of the Meixner polynomials.
A final table shows the abscissae-weight pairs for up to 128 nodes.
Category: Functions and Analysis
[53] viXra:1303.0013 [pdf] submitted on 2013-03-03 09:07:05
Authors: Richard J. Mathar
Comments: 14 Pages.
The manuscript provides tables of abscissae and weights
for Gauss-Laguerre integration on 64, 96 and 128 nodes,
and abscissae and weights
for Gauss-Hermite integration on 96 and 128 nodes.
Category: Functions and Analysis
[52] viXra:1302.0138 [pdf] submitted on 2013-02-20 21:39:58
Authors: N. A. Rather, Suhail Gulzar
Comments: 8 Pages.
Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} \cite{d} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$
$$ n(|\alpha|-k)\left\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^r d\theta\right\}^{\frac{1}{r}}\leq\left\{ \int\limits_{0}^{2\pi}\left|1+ke^{i\theta}\right|^r d\theta\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_\alpha P(z)|. $$
\indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu},$ $1\leq\mu\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.
Category: Functions and Analysis
[51] viXra:1302.0067 [pdf] submitted on 2013-02-11 13:00:45
Authors: Masina Enrico
Comments: 7 Pages.
I want to analyze and solve this PDE, that has some implication in so many fields like applied math, quantum physics, general relativity and so on.
Category: Functions and Analysis
[50] viXra:1302.0025 [pdf] submitted on 2013-02-04 20:52:45
Authors: Cheng Tianren
Comments: 30 Pages.
We prove the regularity of weak solutions of the navier-stokes equations for compressible,isentropic flow in three space dimension.We also prove the existence of a spatially periodic weak solution to the steady compressible navier-stokes equations for any specific heat ratio. Next we study the hyperbolic system of euler equations for isentropic,compressible fluid governed by a general law. We establish the vanishing viscosity limit of the navier-stokes equations to euler equations for one-dimensional compressible fluid flow.
Category: Functions and Analysis
[49] viXra:1301.0181 [pdf] submitted on 2013-01-29 20:47:08
Authors: Cheng Tianren
Comments: 29 Pages.
we consider a special class of non-Archimedean Banach spaces, called Hilbertian,for which every one-dimensional linear subspaces has an orthogonal complement. We construct examples of hilbertian spaces over a non-spherically complete valued field without an orthogonal base.
Category: Functions and Analysis
[48] viXra:1301.0169 [pdf] submitted on 2013-01-27 19:46:56
Authors: Cheng Tianren
Comments: 22 Pages.
We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.
Category: Functions and Analysis
[47] viXra:1301.0036 [pdf] submitted on 2013-01-07 01:37:06
Authors: Hamdy I. Abdel-Gawad, Nasser S. Elazab, Mohamed Osman
Comments: 6 Pages. IOSR Journals
Abstract: Recently the unified method for finding traveling wave solutions of non-linear evolution equations
was proposed by one of the authors a. It was shown that, this method unifies all the methods being used to find
these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de
Vries (KdV) equation with space dependent coefficients. A new class of multiple-soliton or wave trains is
obtained.
Keywords: Exact solution, Extended unified method, Korteweg-deVries equation, variable coefficients
Category: Functions and Analysis
[46] viXra:1301.0010 [pdf] submitted on 2013-01-02 18:58:54
Authors: Andrew Nassif
Comments: 2 Pages.
For many years lied a problem called the P vs NP. The question is to find the number of factorial possibilities to its orders. An example of this is finding the possibilities and comparison of improbabilities of picking 100 students out of 400 students. According to Lardner's theorem the number of known atoms in the universe is less then the number of combinations of possible orders and combinations of the answer to the P vs. NP problems. Finding the equation for the number of different orders a group of 400 people can be put into and subtracting 300 different people that couldn't get picked is equal to ((400!)-(100!*3)). My project is to represent this data through algorithms and different diagrams. When looking at my project you will know how I found a solution and the importance of it. My project will include all the required schematics, and graphs that coordinates with this answer. It will also acquire data showing different possibilities between P vs NP. As well as the combination where P can equal NP and N equals 1, or the possibilities where P doesn't equal NP and N isn't 1. P and NP is believed to stand for the number of possibilities and impossibilities.
Category: Functions and Analysis
[45] viXra:1212.0168 [pdf] submitted on 2012-12-31 08:22:43
Authors: Hans Detlef Hüttenbach
Comments: 3 Pages. (It's really more than 10 years old.)
It is proven that every complete, metrizable locally convex space (a.k.a. F-space)
is reflexive.
This in particular disproves an old conjecture that L^\infty was the dual of L^1.
It is shown that indeed, L^\infty contains a subspace of overcountable dimension
not contained in the dual of L^1.
Category: Functions and Analysis
[44] viXra:1212.0137 [pdf] submitted on 2012-12-23 13:38:03
Authors: Andrew Nassif
Comments: 5 Pages.
For ten long years these two problems have not been solved after being offered a prize. Solving the Riemann hypothesis will bring dimensional analysis in mathematics and physics. Solving the P vs NP will increase our knowledge in programing and provide a wide expansion of mathematical understanding and industrilization.
Category: Functions and Analysis
[43] viXra:1211.0055 [pdf] submitted on 2012-11-11 06:36:23
Authors: Jorma Jormakka
Comments: 8 Pages.
The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.
Category: Functions and Analysis
[42] viXra:1210.0146 [pdf] submitted on 2012-10-25 17:09:58
Authors: Bertrand Wong
Comments: 16 Pages.
The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. However, the three-dimensional Navier-Stokes equations for modelling turbulence misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would greatly affect the field of fluid mechanics. In this paper, which had been published in an international journal in 2010, a reasoned, practical approach towards resolving the issue is adopted and a practical, statistical kind of mathematical solution is proposed.
Category: Functions and Analysis
[41] viXra:1210.0111 [pdf] submitted on 2012-10-20 15:45:39
Authors: J.A.J. van Leunen
Comments: 38 Pages.
This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.
Category: Functions and Analysis
[40] viXra:1210.0069 [pdf] submitted on 2012-10-13 04:32:55
Authors: Imanol Pérez
Comments: 1 Page.
In this paper I introduce a new method to find the roots of a function with two known values a and b such that sgn(f(a)) = -sgn(f(b)).
Category: Functions and Analysis
[39] viXra:1210.0068 [pdf] submitted on 2012-10-13 04:38:34
Authors: Imanol Pérez
Comments: 3 Pages.
This paper shows a relationship between spiral and the roots of complex numbers.
Category: Functions and Analysis
[38] viXra:1210.0041 [pdf] submitted on 2012-10-09 02:27:46
Authors: Pierre-Yves Gaillard
Comments: 1 Page.
We give a short proof of l'Hospital's Rule.
Category: Functions and Analysis
[37] viXra:1210.0007 [pdf] submitted on 2012-10-01 22:45:24
Authors: Ren Shiquan
Comments: 3 Pages. This is an undergraduate level assignment.
In this paper, we give a study on the probability of the first digit of 2^n. This is an undergraduate level assignment..
Category: Functions and Analysis
[36] viXra:1207.0066 [pdf] submitted on 2012-07-18 02:17:26
Authors: Pierre-Yves Gaillard
Comments: 1 Page.
Let a be an element of a finite dimensional C-algebra with 1. Then there is a unique polynomial f_a such that f_a(a) = exp(a) and deg f_a < dim C[a]. We give an explicit formula for f_a.
Category: Functions and Analysis
[35] viXra:1207.0046 [pdf] submitted on 2012-07-12 01:17:41
Authors: Pierre-Yves Gaillard
Comments: 1 Page.
Let a be a square matrix with complex entries and f a function holomorphic on an open subset U of the complex plane. It is well known that f can be evaluated on a if the spectrum of a is contained in U. We show that, for a fixed f, the resulting matrix depends holomorphically on a.
Category: Functions and Analysis
[34] viXra:1206.0086 [pdf] submitted on 2012-06-24 14:14:23
Authors: Bertrand Wong
Comments: 2 Pages.
The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon.
Category: Functions and Analysis
[33] viXra:1206.0017 [pdf] submitted on 2012-06-05 10:52:56
Authors: Radha F. Gupta, Poom Kumam
Comments: 9 Pages.
This paper investigates the consensus problem in almost sure sense for uncertain multi-agent systems with noises and fixed topology. By
combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of
distributed stochastic type non-linear protocols. Numerical examples are given to illustrate the results.
Category: Functions and Analysis
[32] viXra:1206.0005 [pdf] submitted on 2012-06-02 21:56:55
Authors: Xiong Wang
Comments: 6 Pages.
This paper discuss the longstanding problems of fractional calculus such as too
many definitions while lacking physical or geometrical meanings, and try to extend fractional
calculus to any dimension. First, some different definitions of fractional derivatives, such as the
Riemann-Liouville derivative, the Caputo derivative, Kolwankar's local derivative and Jumarie's modified
Riemann-Liouville derivative, are discussed and conclude that the very reason for introducing
fractional derivative is to study nondifferentiable functions. Then, a concise and essentially local
definition of fractional derivative for one dimension function is introduced and its geometrical
interpretation is given. Based on this simple definition, the fractional calculus is extended to
any dimension and the \emph{Fractional Geometric Calculus} is proposed. Geometric algebra provided
an powerful mathematical framework in which the most advanced concepts modern physic, such
as quantum mechanics, relativity, electromagnetism, etc., can be expressed in this framework
graciously. At the other hand, recent developments in nonlinear science and complex system
suggest that scaling, fractal structures, and nondifferentiable functions occur much more
naturally and abundantly in formulations of physical theories. In this paper, the extended
framework namely the Fractional Geometric Calculus is proposed naturally, which aims
to give a unifying language for mathematics, physics and science of complexity of the 21st century.
Category: Functions and Analysis
[31] viXra:1205.0078 [pdf] submitted on 2012-05-19 15:27:40
Authors: Hilário Fernandes de Araújo Júnior
Comments: 2 Pages. Article part 1, Copyright© 19 may 2012
In this article, is exposed two sum representations for integrals in which the integration interval is infinite.
Category: Functions and Analysis
[30] viXra:1204.0091 [pdf] submitted on 2012-04-26 07:16:21
Authors: Guang-Sheng Chen
Comments: 6 Pages.
In this paper, we establish local fractional Hilbert transform in fractal space, consider
some properties of local fractional Hilbert Transforms.
Category: Functions and Analysis
[29] viXra:1203.0078 [pdf] submitted on 2012-03-20 07:02:58
Authors: Faycal Ben Adda
Comments: 24 Pages. A short version of this paper was published in Journal Européen des Systèmes Automatisés, Fractional order systems, 42, p733-746, 2008.
In this paper, we introduce a concept of
"apparent" measure in R^n and we define a concept of relative dimension (of real order) with it, which depends on the geometry
of the object to measure and on the distance which separates it
from an observer. At the end we discuss the relative dimension of the Cantor set. This measure enables us to provide a geometric
interpretation of the Riemann-Liouville's integral of order alpha between 0 and 1.
Category: Functions and Analysis
[28] viXra:1203.0065 [pdf] submitted on 2012-03-16 20:48:14
Authors: Xiao-meng Li, Xianfeng Su
Comments: 4 Pages.
This paper is concerned with the growth of meromorphic solutions of a class of systems of complex algebraic differentialequations. A general estimate the growth order of solutions of the systems of differential equation is obtained by Zalacman Lemma. We also take an example to show that the result is right.
Category: Functions and Analysis
[27] viXra:1203.0038 [pdf] submitted on 2012-03-11 08:37:50
Authors: Guang-Sheng Chen
Comments: 4 Pages.
In this paper, we establish finte Yang-Laplace Transform on fractal space, considered
some properties of finte Yang-Laplace Transform.
Category: Functions and Analysis
[26] viXra:1203.0037 [pdf] submitted on 2012-03-11 08:40:07
Authors: Guang-Sheng Chen
Comments: 4 Pages.
This paper deals with the theory of the local fractional Stieltjes transform. We derive
the Stieltjes transform. This is followed by several examples and the basic operational properties
of Stieltjes transforms.
Category: Functions and Analysis
[25] viXra:1203.0035 [pdf] submitted on 2012-03-11 03:35:12
Authors: S Halayka
Comments: 9 Pages. Lots of figures.
A rough analysis of the first three iterations of the logistic map $x^\prime = rx(1-x)$ produces a series of special constants.
The three constants are $1$, the inverse of the golden ratio, and Catalan's constant.
Category: Functions and Analysis
[24] viXra:1203.0030 [pdf] submitted on 2012-03-08 22:45:49
Authors: Guang-Sheng Chen
Comments: 12 Pages.
This paper deals with the theory and applications of the local fractional Mellin
transform of the real order α . We define the local fractional Mellin transform and its inverse
transform. This is followed by several examples and the basic operational properties of local
fractional Mellin transform. We discuss applications of local fractional Mellin transforms to local
fractional boundary value problems.
Category: Functions and Analysis
[23] viXra:1203.0029 [pdf] submitted on 2012-03-08 22:49:14
Authors: Guang-Sheng Chen
Comments: 8 Pages.
In this paper we study Local fractional improper integrals on fractal space. By
some mean value theorems for Local fractional integrals, we prove an analogue of the classical
Dirichlet-Abel test for Local fractional improper integrals.
Category: Functions and Analysis
[22] viXra:1203.0023 [pdf] submitted on 2012-03-07 02:27:32
Authors: S Halayka
Comments: 2 Pages.
A brief visual demonstration of the presence of the golden ratio in the logistic map is given.
Category: Functions and Analysis
[21] viXra:1202.0071 [pdf] submitted on 2012-02-21 22:27:58
Authors: Choe Ryong Gil
Comments: 23 pages
In this paper we have introduced a new topology and a convergence in Banach space, which would be called a L-topology and a L-convergence. It is similar to the weak topology and weak convergence, but there are some essential differences. For example, the L-topology is stronger than weak topology, but weaker than the strong one. On the basis of the notion, we have considered the problem on the separability and reflexibility of Lipschitz (Lip-) dual space. Furthermore, we have introduced a new topology of Lip-dual space, which is similar to the weak* (W*-) topology of linear dual of Banach space and would be called an L*-topology, and we have considered the problems on the metrizability of L*-topology and on the L*-separability of Lip-dual space, too.
Category: Functions and Analysis
[20] viXra:1202.0069 [pdf] submitted on 2012-02-20 20:24:29
Authors: Choe Ryong Gil
Comments: 18 pages
In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.
Category: Functions and Analysis
[19] viXra:1202.0060 [pdf] submitted on 2012-02-19 02:03:52
Authors: Choe Ryong Gil
Comments: 17 pages
In this paper we have obtained a new theorem that a nonlinear Lipschitz (Lip-) functional defined on the closed subset of Banach spaces can be extended to the whole space with Lip-continuity and maintenance of Lip-constant, which would be called an extension theorem (ET). This theorem is a generalization to the Lip-functional of the famous Hahn-Banach theorem on the bounded linear functional. By the ET, we have completely solved the open problem on the relation of the invertibility between the Lip-operator and its Lip-dual operator.
Category: Functions and Analysis
[18] viXra:1202.0015 [pdf] submitted on 2012-02-06 15:20:56
Authors: Richard J. Mathar
Comments: 12 Pages. Includes complete C++ source listing.
The volume inside intersecting spheres may be computed by a standard method which computes
a surface integral over all visible sections of the spheres. If the visible sections are divided in simple
zonal sections, the individual contribution by each zone follows from basic analysis. We implement
this within a semi-numerical program which marks the zones individually as visible or invisible.
Category: Functions and Analysis
[17] viXra:1112.0044 [pdf] submitted on 2011-12-15 09:36:44
Authors: Xiaodong Hu, Evgeniy Grechnikov
Comments: 11 Pages.
This paper investigates the connectivity in one-dimensional ad hoc wireless networks with a forbidden zone. We derive the probability of the wireless networks which are composed of exactly m
clusters by means of the methods of combinatorics and probability. The probability of connectivity, i.e. $m = 1$, can be obtained as a special case. Further, we explain how the transmission range of node affects the connectivity of the wireless network.
Category: Functions and Analysis
[16] viXra:1111.0105 [pdf] submitted on 28 Nov 2011
Authors: Jose Javier Garcia Moreta
Comments: 23 Pages.
We give a possible interpretation of the Xi-function of Riemann as the Functional determinant
for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) ,
this potential V is related to the half-integral of the logarithmic derivative for the Riemann
Xi-function, through the paper
we will assume that the reduced Planck constant is defined in units where and that the mass is
Category: Functions and Analysis
[15] viXra:1110.0075 [pdf] submitted on 30 Oct 2011
Authors: Guang-Sheng Chen
Comments: 6 pages.
In this paper, by some properties of Local fractional integral, we establish the
generalized Mean value theorems for Local Fractional Integral.
Category: Functions and Analysis
[14] viXra:1106.0056 [pdf] submitted on 27 Jun 2011
Authors: Mircea Selariu
Comments: 10 pages.
The article define a mathematic entity called twist, which generates, in this way, notion of straight line.
Straight line becom thus a twist of eccentricity e = 0, and broken line (zigzag line) is a twist of s = ± 1.
Category: Functions and Analysis
[13] viXra:1106.0055 [pdf] submitted on 26 Jun 2011
Authors: Mircea Selariu
Comments: 10 pages.
These papers show a calculus relation ( 50 ) of complete elliptic integral K(k) with minimum 9
precise decimals and the possibility to obtain a more precisely relation.. It results by application
Landen's method, of geometrical-arithmetical average, not for obtain a numerical value but to
obtain a compute algebraically relation after 5 steps of a geometrical transformation, called
"CENTERED PROCESS".
Category: Functions and Analysis
[12] viXra:1106.0014 [pdf] submitted on 9 Jun 2011
Authors: Ron Bourgoin
Comments: 4 pages
Sometimes in physics we end up with a function that resembles
f(x)=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
Authors: Jose Javier Garcia Moreta
Comments: 13 pages.
We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[10] viXra:1008.0025 [pdf] submitted on 9 Aug 2010
Authors: Elemér E Rosinger
Comments: 166 pages
It is shown how the infinity of differential algebras of generalized
functions is naturally subjected to a basic dichotomic singularity test
regarding their significantly different abilities to deal with large classes
of singularities. In this respect, a review is presented of the way
singularities are dealt with in four of the infinitely many types of
differential algebras of generalized functions. These four algebras, in the
order they were introduced in the literature are : the nowhere dense,
Colombeau, space-time foam, and local ones. And so far, the first
three of them turned out to be the ones most frequently used in a
variety of applications. The issue of singularities is naturally not a
simple one. Consequently, there are different points of view, as well as
occasional misunderstandings. In order to set aside, and preferably,
avoid such misunderstandings, two fundamentally important issues
related to singularities are pursued. Namely, 1) how large are the sets
of singularity points of various generalized functions, and 2) how are
such generalized functions allowed to behave in the neighbourhood of
their point of singularity. Following such a two fold clarification on
singularities, it is further pointed out that, once one represents
generalized functions - thus as well a large class of usual singular functions
- as elements of suitable differential algebras of generalized functions,
one of the main advantages is the resulting freedom to perform
globally arbitrary algebraic and differential operations on such functions,
simply as if they did not have any singularities at all. With the same
freedom from singularities, one can perform globally operations such
as limits, series, and so on, which involve infinitely many generalized
functions. The property of a space of generalized functions of being
a flabby sheaf proves to be essential in being able to deal with large
classes of singularities. The first and third type of the mentioned
differential algebras of generalized functions are flabby sheaves, while the
second type fails to be so. The fourth type has not yet been studied
in this regard.
Category: Functions and Analysis
[9] viXra:1007.0005 [pdf] submitted on 5 Jul 2010
Authors: Jose Javier Garcia Moreta
Comments: 7 pages.
We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)
Category: Functions and Analysis
[8] viXra:1005.0075 [pdf] submitted on 19 May 2010
Authors: Jose Javier Garcia Moreta
Comments: 9 pages
In this paper we review some results on the regularization of divergent integrals of
the form ... (see paper for full abstract)
Category: Functions and Analysis
[7] viXra:1005.0071 [pdf] submitted on 17 May 2010
Authors: Jose Javier Garcia Moreta
Comments: 9 pages
Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of dDirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[6] viXra:1004.0053 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache, Mircea Eugen Șelariu
Comments: 10 pages
This article presents two methods, in parallel, of solving more complex integrals, among
which is the Poisson's integral, in order to emphasize the obvious advantages of a new method
of integration, which uses the supermathematics circular ex-centric functions.
We will specially analyze the possibilities of easy passing/changing of the supermathematics
circular ex-centric functions of a centric variable α to the same functions of ex-centric variable &theta.
The angle α is the angle at the center point O(0,0), which represents the centric variable and θ
is the angle at the ex-center E(k,ε), representing the ex-centric variable. These are the angles
from which the points 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
Authors: Florentin Smarandache
Comments: 4 pages
As a consequence of the Integral Test we find a triple inequality which bounds up and
down both a series with respect to its corresponding improper integral, and reciprocally
an improper integral with respect to its corresponding series.
Category: Functions and Analysis
[4] viXra:1003.0166 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
A great number of articles widen known scientific results (theorems, inequalities,
math/physics/chemical etc. propositions, formulas), and this is due to a simple procedure,
of which it is good to say a few words
Category: Functions and Analysis
[3] viXra:1003.0105 [pdf] submitted on 10 Mar 2010
Authors: Jose Javier Garcia Moreta
Comments: 12 Pages.
In this paper we study a set of orthogonal Polynomials with respect a certain given measure related to
the Taylor series expansion of the Xi-function , this paper is based on a previous conjecture by Carlon and
Gaston related to the fact that Riemann Hypothesis (with simple zeros) is equivalent to the limit for a
certain set of orthogonal Polynomials, we study the 'Hamburger moment problem' for even 'n' and 0 for n
odd here the moments are related to the power series expansion of Xi-function , we also give the
integral representation for the generating function , in terms of the Laplace transform of , and
in the end of the paper we study the connection of our orthogonal polynomial set with the Kernel ,
through all the paper we will use the simplified notation (see paper for abstract with equations)
Category: Functions and Analysis
[2] viXra:0903.0007 [pdf] submitted on 28 Mar 2009
Authors: Chun-Xuan Jiang
Comments: recovered from sciprint.org
We find Blasius function to satisfy the boundary condition f(∞) = 1 and obtain the exact analytic soultion of Blasius equation.
Category: Functions and Analysis
[1] viXra:0703.0011 [pdf] submitted on 10 Mar 2007
Authors: Gerardo Alvarado
Comments: recovered from sciprint.org
I deduce a series which satisfies the fundamental theorem of calculus without dependence on an
explicit function. I prove Taylor's theorem and show that it is closely related. I deduce a series for the
logarithm function and from this series deduce the power series representation of the logarithm function
along with the interval of convergence. I also solve an ordinary differential equation.
Category: Functions and Analysis
[36] viXra:1305.0052 [pdf] replaced on 2013-05-15 22:21:50
Authors: Jin He, Xiaoli Yang
Comments: 8 Pages. 1 Figure. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.
Is the sum of rational structures also a rational structure? It is called the Creator's big question for humans. Numerical calculation suggests that it is approximately rational for the fitted parameter values of barred spiral galaxies. However, we need mathematical justification. The authors are very old and are not experts in mathematics. Please help us humans to resolve the question.
Category: Functions and Analysis
[35] viXra:1304.0158 [pdf] replaced on 2013-05-01 10:41:48
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
[34] viXra:1304.0151 [pdf] replaced on 2013-04-28 14:30:03
Authors: M.E.Hassani
Comments: 20 Pages; 7 References
The main purpose of the present paper is the heuristic study of the structure, properties and consequences of new class of potential functions results from the concept of pq-Radial functions which are fundamental family of solutions of second order pq-PDE.
Category: Functions and Analysis
[33] viXra:1303.0038 [pdf] replaced on 2013-03-21 04:33:40
Authors: Richard J. Mathar
Comments: 10 Pages. Corrected 2 digits in eq. (11). Added remark 2, eqs. (22) and (24), and 2 references
The manuscript delivers
nodes and their weights for Gaussian quadratures with a "non-classical"
weight in the integrand defined by a reciprocal hyperbolic cosine.
The associated monic orthogonal polynomials are constructed; their
coefficients
are simple multiples of the coefficients of Hahn polynomials.
A final table shows the abscissae-weight pairs for up to 128 nodes.
Category: Functions and Analysis
[32] viXra:1302.0025 [pdf] replaced on 2013-02-22 03:22:36
Authors: Cheng Tianren
Comments: 30 Pages.
We prove the regularity of weak solutions of the navier-stokes equations for compressible,isentropic flow in three space dimension.We also prove the existence of a spatially periodic weak solution to the steady compressible navier-stokes equations for any specific heat ratio. Next we study the hyperbolic system of euler equations for isentropic,compressible fluid governed by a general law. We establish the vanishing viscosity limit of the navier-stokes equations to euler equations for one-dimensional compressible fluid flow.
Category: Functions and Analysis
[31] viXra:1301.0181 [pdf] replaced on 2013-02-04 06:51:44
Authors: Cheng Tianren
Comments: 29 Pages.
we consider a special class of non-Archimedean Banach spaces, called Hilbertian,for which every one-dimensional linear subspaces has an orthogonal complement. We construct examples of hilbertian spaces over a non-spherically complete valued field without an orthogonal base.
Category: Functions and Analysis
[30] viXra:1301.0169 [pdf] replaced on 2013-02-07 20:26:07
Authors: Cheng Tianren
Comments: 22 Pages.
We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.
Category: Functions and Analysis
[29] viXra:1301.0169 [pdf] replaced on 2013-02-05 19:38:21
Authors: Cheng Tianren
Comments: 22 Pages.
We consider the problem of invariant nonlinear wave equations in any dimension. we show that the classical finite-difference scheme conserves the positive-definite discrete analog of the energy. We also show that, under certain generic assumptions, each solution converges to the two-dimensional set when the dimension .Another problem we proved is about the spectral stability of solitary wave solutions to the dirac equation in any dimension.
Category: Functions and Analysis
[28] viXra:1211.0055 [pdf] replaced on 2013-03-06 23:49:55
Authors: Jorma Jormakka
Comments: Corrected the number of pages and added a small clarifying comment to the text.
The Clay Navier-Stokes proble is correctly solved. The answer from CMI is included. The article discusses why and how the Clay Navier-Stokes problem should be corrected.
Category: Functions and Analysis
[27] viXra:1211.0055 [pdf] replaced on 2012-11-16 01:16:50
Authors: Jorma Jormakka
Comments: 11 Pages. Addded a letter from Bombieri after the reply from CMI
The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.
Category: Functions and Analysis
[26] viXra:1211.0055 [pdf] replaced on 2012-11-13 02:20:09
Authors: Jorma Jormakka
Comments: 8 Pages. A minor change
The Clay Navier-Stokes problem is correctly solved. The answer from CMI is
included. The article discusses why and how the Clay Navier-Stokes problem
should be corrected.
Category: Functions and Analysis
[25] viXra:1211.0055 [pdf] replaced on 2012-11-12 08:56:11
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
[24] viXra:1211.0055 [pdf] replaced on 2012-11-11 23:01:43
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
[23] viXra:1210.0111 [pdf] replaced on 2013-04-18 16:34:34
Authors: J.A.J. van Leunen
Comments: 76 Pages.
This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.
Category: Functions and Analysis
[22] viXra:1210.0111 [pdf] replaced on 2012-11-20 14:35:42
Authors: J.A.J. van Leunen
Comments: 71 Pages.
This is a compilation of formula of quaternionic algebra and quaternionic differentials
Two types of quaternionic differentiation exist.
Flat differentiation uses the quaternionic nabla and ignores the curvature of the parameter space.
Full differentiation uses the distance function ℘(x) that defines the curvature of the parameter space.
The text focuses at applications in quantum mechanics, in electrodynamics and in fluid dynamics.
Category: Functions and Analysis
[21] viXra:1207.0046 [pdf] replaced on 2013-04-08 09:05:21
Authors: Pierre-Yves Gaillard
Comments: 1 Page.
Let a be a square matrix with complex entries and f a function holomorphic on an open subset U of the complex plane. It is well known that f can be evaluated on a if the spectrum of a is contained in U. We show that, for a fixed f, the resulting matrix depends holomorphically on a.
Category: Functions and Analysis
[20] viXra:1203.0035 [pdf] replaced on 2012-03-12 15:27:40
Authors: S Halayka
Comments: 7 Pages. Changed title, reduced clutter, submitted to journal.
It is presumed a priori that there is an entropy-area relationship inherent to the iterations of the logistic map. Several
interesting results are produced.
Category: Functions and Analysis
[19] viXra:1202.0069 [pdf] replaced on 2012-02-23 03:28:33
Authors: Choe Ryong Gil, Kim Myong Il
Comments: 17 pages
In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.
Category: Functions and Analysis
[18] viXra:1106.0055 [pdf] replaced on 27 Jun 2011
Authors: Mircea Selariu
Comments: 10 pages. v1 in Romanian, v2 in English.
These papers show a calculus relation ( 50 ) of complete elliptic integral K(k) with minimum 9
precise decimals and the possibility to obtain a more precisely relation.. It results by application
Landen's method, of geometrical-arithmetical average, not for obtain a numerical value but to
obtain a compute algebraically relation after 5 steps of a geometrical transformation, called
"CENTERED PROCESS".
Category: Functions and Analysis
[17] viXra:1009.0047 [pdf] replaced on 23 Feb 2011
Authors: Jose Javier Garcia Moreta
Comments: 19 pages.
We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[16] viXra:1009.0047 [pdf] replaced on 11 Feb 2011
Authors: Jose Javier Garcia Moreta
Comments: 18 pages.
We study a generalization of the zeta regularization method applied to the
case of the regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[15] viXra:1009.0047 [pdf] replaced on 8 Nov 2010
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
[14] viXra:1008.0025 [pdf] replaced on 12 Aug 2010
Authors: Elemér E Rosinger
Comments: 184 pages
It is shown how the infinity of differential algebras of generalized
functions is naturally subjected to a basic dichotomic singularity test
regarding their significantly different abilities to deal with large classes
of singularities. In this respect, a review is presented of the way
singularities are dealt with in four of the infinitely many types of
differential algebras of generalized functions. These four algebras, in the
order they were introduced in the literature are : the nowhere dense,
Colombeau, space-time foam, and local ones. And so far, the first
three of them turned out to be the ones most frequently used in a
variety of applications. The issue of singularities is naturally not a
simple one. Consequently, there are different points of view, as well as
occasional misunderstandings. In order to set aside, and preferably,
avoid such misunderstandings, two fundamentally important issues
related to singularities are pursued. Namely, 1) how large are the sets
of singularity points of various generalized functions, and 2) how are
such generalized functions allowed to behave in the neighbourhood of
their point of singularity. Following such a two fold clarification on
singularities, it is further pointed out that, once one represents
generalized functions - thus as well a large class of usual singular functions
- as elements of suitable differential algebras of generalized functions,
one of the main advantages is the resulting freedom to perform
globally arbitrary algebraic and differential operations on such functions,
simply as if they did not have any singularities at all. With the same
freedom from singularities, one can perform globally operations such
as limits, series, and so on, which involve infinitely many generalized
functions. The property of a space of generalized functions of being
a flabby sheaf proves to be essential in being able to deal with large
classes of singularities. The first and third type of the mentioned
differential algebras of generalized functions are flabby sheaves, while the
second type fails to be so. The fourth type has not yet been studied
in this regard.
Category: Functions and Analysis
[13] viXra:1007.0005 [pdf] replaced on 13 Nov 2011
Authors: Jose Javier Garcia Moreta
Comments: 20 pages.
We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1
Category: Functions and Analysis
[12] viXra:1007.0005 [pdf] replaced on 3 Nov 2011
Authors: Jose Javier Garcia Moreta
Comments: 19 pages.
We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1
Category: Functions and Analysis
[11] viXra:1007.0005 [pdf] replaced on 4 Oct 2011
Authors: Jose Javier Garcia Moreta
Comments: 16 pages.
We give a possible interpretation of the Xi-function
of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one
dimension (see paper) for a real-valued function V(x),
this potential V is related to the half-integral of
the logarithmic derivative for the Riemann Xi-function,
through the paper we will assume that the reduced
Planck constant is defined in units where h-bar = 1
and that the mass is 2m = 1
Category: Functions and Analysis
[10] viXra:1007.0005 [pdf] replaced on 28 Jun 2011
Authors: Jose Javier Garcia Moreta
Comments: 14 pages.
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
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
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
Authors: Jose Javier Garcia Moreta
Comments: 15 pages.
We give a possible interpretation of the Xi-function of Riemann as the Functional determinant det(E-H)
for a certain Hamiltonian quantum operator in one dimension () for a real-valued function V(x) ,
this potential V is related to the half-integral of the logarithmic derivative for the Riemann
Xi-function, through
the paper we will assume that the reduced Planck constant is defined in units where h-bar = 1 and that the mass is 2m = 1
Category: Functions and Analysis
[6] viXra:1007.0005 [pdf] replaced on 18 Nov 2010
Authors: Jose Javier Garcia Moreta
Comments: 13 pages.
We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)
Category: Functions and Analysis
[5] viXra:1007.0005 [pdf] replaced on 3 Aug 2010
Authors: Jose Javier Garcia Moreta
Comments: 10 pages.
We give a possible interpretation of the Xi-function of Riemann as the
Functional determinant det (E - H) for a certain Hamiltonian quantum operator in
one dimension ... (see paper for full abstract)
Category: Functions and Analysis
[4] viXra:1007.0005 [pdf] replaced on 27 Jul 2010
Authors: Jose Javier Garcia Moreta
Comments: 9 pages.
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
Authors: Jose Javier Garcia Moreta
Comments: 13 pages
Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of Dirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[2] viXra:1005.0071 [pdf] replaced on 15 Jan 2011
Authors: Jose Javier Garcia Moreta
Comments: 13 pages
Using the theory of distributions and Zeta regularization we manage to give
a definition of product for Dirac delta distributions, we show how the fact of one can be
define a coherent and finite product of Dirac delta distributions is related to the
regularization of divergent integrals ... (see paper for full abstract)
Category: Functions and Analysis
[1] viXra:1003.0166 [pdf] replaced on 20 Mar 2010
Authors: Florentin Smarandache
Comments: 7 pages
A great number of articles widen known scientific results (theorems, inequalities,
math/physics/chemical etc. propositions, formulas), and this is due to a simple procedure,
of which it is good to say a few words:
Let suppose that we want to generalizes a known mathematical proposition P(a) ,
where a is a constant, to the proposition P(n) , where n is a variable which belongs to
subset of N .
To prove that P is true for n by recurrence means the following: the first step is
trivial, since it is about the known result P(a) (and thus it was already verified before by
other mathematicians!). To pass from P(n) to P(n + 1) , one uses too P(a) : therefore one
widens a proposition by using the proposition itself, in other words the found
generalization will be paradoxically proved with the help of the particular case from
which one started!
We present below the generalizations of Hölder, Minkovski, and respectively
Tchebychev inequalities.
Category: Functions and Analysis