Functions and Analysis
[1]
vixra:2401.0010 [
pdf]
Calculus and Applications
This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:mathbb Rtomathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and of the integral $int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:mathbb Ctomathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:mathbb R^Ntomathbb R^M$ or $f:mathbb R^Ntomathbb C^M$ or $f:mathbb C^Ntomathbb C^M$, with general theory, integration results, maximization questions, and basic applications to physics.
[2]
vixra:2312.0167 [
pdf]
Complete Integers: Extending Integers to Allow Real Powers Have Discontinuities in Zero
We will define a superset of integers (the complete integers), which contains the dual of integers along parity (e.g. the odd zero, the even one, ...). Then we will see how they form a ring and how they can be used as exponents for real numbers powers, in order to write functions which have a discontinuity in zero (the function itself or one of its derivates), as for example |x| and sgn(x).
[3]
vixra:2312.0160 [
pdf]
Wellposedness for the Homogeneous Periodic Navier-Stokes Equation
We use the strichartz estimate and commutator estimates to prove the decay property of the solution. The global results can be obtained by the decay property.
[4]
vixra:2312.0132 [
pdf]
Homogenization of the First Initial-Boundary Value Problem for Periodic Hyperbolic Systems. Principal Term of Approximation
Let $mathcal{O}subset mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $ L_2(mathcal{O};mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $A_{D,varepsilon}$ with the Dirichlet boundary condition. Here $varepsilon >0$ is a small parameter. The coefficients of the operator $A_{D,varepsilon}$ are periodic and depend on $mathbf{x}/varepsilon$. The principal terms of approximations for the operator cosine and sine functions are given in the $(H^2ightarrow L_2)$- and $(H^1ightarrow L_2)$-operator norms, respectively. The error estimates are of the precise order $O(varepsilon)$ for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation $(partial _t^2+A_{D,varepsilon})mathbf{u}_varepsilon =mathbf{F}$.
[5]
vixra:2312.0125 [
pdf]
Quadratic Phase Quaternion Domain Fourier Transform
Based on the quaternion domain Fourier transform (QDFT)of 2016 and the quadratic-phase Fourier transform of 2018, we introduce the quadratic-phase quaternion domain Fourier transform (QPQDFT) and study some of its properties, like its representation in terms of the QDFT, linearity, Riemann-Lebesgue lemma, shift and modulation, scaling, inversion, Parseval type identity, Plancherel theorem, directional uncertainty principle, and the (direction-independent) uncertainty principle. The generalization thus achieved includes the special cases of QDFT, a quaternion domain (QD) fractional Fourier transform, and a QD linear canonical transform.
[6]
vixra:2312.0112 [
pdf]
On Wilker-Type Inequalities
In this paper, we present elementary proofs of Wilker-type inequalities involving trigonometric and hyperbolic functions. In addition, we propose some conjectures which extend and generalize the Wilker-type inequalities.
[7]
vixra:2312.0111 [
pdf]
On Generalized li-Yau Inequalities
We generalize the Li-Yau inequality for second derivatives and we also establish Li-Yau type inequality for fourth derivatives. Our derivation relies on the representation formula for the heat equation.
[8]
vixra:2311.0036 [
pdf]
Roots of Real Polynomial Functions and of Real Functions
The Newton-Raphson method is the most widely used numerical calculation method to determine the roots of Real polynomial functions, but it has the drawback that it does not always converge. The method proposed in this work establishes the convergence condition and the development of its application, and therefore will always converge towards the roots of the function. This will mean a conclusive advance for the determination of roots of Real polynomial functions. According to interpretation of the Abel-Ruffini theorem, the roots of polynomial functions of degree greater than 4 can only be determined by numerical calculation.
[9]
vixra:2310.0144 [
pdf]
Inversions (Mirror Images) With Respect to the Unit Circle and Division by Zero
In this note, we will consider the interesting inversion formula that was discovered by Yoichi Maeda with respect to the unit circle on the complex plane from the viewpoint of our division by zero: $1/0=0/0=0$.
[10]
vixra:2310.0045 [
pdf]
Infinity Tensors, the Strange at Tractor, and the Riemann Hypothesis: An Accurate Rewording of The Riemann Hypothesis Yields Forma L Proof
The Riemann Hypothesis can be reworded to indicate that the real part of one half always balanced at the infinity tensor by stating that the Riemann zeta function has no more than an infinity tensor’s worth of zeros on the critical line. For something to be true in proof, it often requires an outside perspective. In other words, there must be some exterior, alternate perspective or system on or applied to the hypothesis from which the proof can be derived. Two perspectives, essentially must agree. Here, a fractal web with infinitesimal 3D strange attractor is theorized as present at the solutions to the Riemann Zeta function and in combination with the infinity tensor yields an abstract, mathematical object from which the rewording of the Riemann Zeta function can be derived. From the rewording, the law that mathematical sequences can be expressed in more concise and manageable forms is applied and the proof is manifested. The mathematical law that any mathematical sequence can be expressed in simpler and more concise terms: ∀s∃su2032⊆s: ∀φ: s⊆φ ⇒ su2032⊆φ, is the final key to the proof when comparing the real and imaginary parts. Parker Emmerson is affiliated with now defunct, Marlboro College, as he attained his B.A. in Psychology and Philosophy with a focus on mathematics of perception in 2010.
[11]
vixra:2309.0055 [
pdf]
Common Points of Parallel Lines and Division by Zero Calculus
In this note, we will consider some common points of two parallel lines on the plane from the viewpoint of the division by zero calculus. Usually, we will consider that there are no common points or the common point is the point at infinity for two parallel lines. We will, surprisingly, introduce a new common point for two parallel lines from the viewpoint of the division by zero calculus.
[12]
vixra:2308.0124 [
pdf]
Embedding of Octonion Fourier Transform in Geometric Algebra of R^3 and Polar Representations of Octonion Analytic Signals in Detail
We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three-dimensional Euclidean space Cl(3,0). We apply a new form of dimensionally minimal embedding of octonions in geometric algebra, that expresses octonion multiplication non-associativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas.
[13]
vixra:2306.0098 [
pdf]
The Series Limit of Sum_k Cos(a Log K)/[k Log k]
The slowly converging series sum_{k>=2} cos(a log k)/[k log k] is evaluated numerically for a=1/2, 1, 3/2, ..., 4. After some initial terms, the infinite tail of the sum is replaced by the integral of the associated interpolating function, an Exponential Integral, and the "second form" of the Euler-Maclaurin corrections is derived from the analytic equations for higher order derivatives.
[14]
vixra:2304.0169 [
pdf]
Integrability of Continuous Functions in 2 Dimensions
In this paper it is shown that the Banach space of continuous, R^2- or C-valued functions on a simply connected either 2-dimensional real or 1-dimensional complex compact region can be decomposed into the topological direct sum of two subspaces, a subspace of integrable (and conformal) functions, and another one of unintegrable (and anti-conformal) functions. It is shown that complex integrability is equivalent to complex analyticity. This can be extended to real functions. The existence of a conjugation on that Banach space will be proven, which maps unintegrable functions onto integrable functions.The boundary of a 2-dimensional simply connected compact region is defined by a Jordan curve, from which it is known to topologically divide the domain into two disconnected regions. The choice of which of the two regions is to be the inside, defines the orientation.The conjugation above will be seen to be the inversion of orientation.Analyticity, integrability, and orientation on R^2 (or C) therefore are intimately related.
[15]
vixra:2304.0153 [
pdf]
Serious Problems in Standard Complex Analysis Texts From The Viewpoint of Division by Zero Calculus
In this note, we shall refer to some serious problems for the standard complex analysis text books that may be considered as common facts for many years from the viewpoint of the division by zero calculus. We shall state clearly our opinions with the new book: V. Eiderman, An introduction to complex analysis and the Laplace transform (2022).
[16]
vixra:2304.0120 [
pdf]
On the Equation X + (X/X) = X
In this note, we shall refer to the equation X + (X/X) = X from our division by zero and division by zero calculus ideas against the Barukčić's idea.
[17]
vixra:2303.0144 [
pdf]
Solving Triangles Algebraically
Quaterns are a new measure of rotation. Since they are defined in terms of rectangular coordinates, all of the analogue trigonometric functions become algebraic rather than transcendental. Rotations, angle sums and differences, vector sums, cross and dot products, etc., all become algebraic. Triangles can be solved algebraically. Computer algorithms use truncated infinite sums for the transcendental calculations of these quantities. If rotations were expressed in quaterns, these calculations would be simplified by a few orders of magnitude. This would have the potential to greatly reduce computing time. The archaic Greek letter koppa is used to represent rotations in quaterns, rather than the traditional Greek letter theta. Because calculations utilizing quaterns are algebraic, simple rotation in the first two quadrants can be done "by hand" using "pen and paper." Using the approximate methods outlined towards the end of the paper, triangles may be approximately solved with an error of less than 3% using algebra and a few simple formulas.
[18]
vixra:2302.0097 [
pdf]
Complex Analysis and Theory of Reproducing Kernels
The theory of reproducing kernels is very fundamental, beautiful and will have many applications in analysis, numerical analysis and data sciences. In this paper, some essential results in complex analysis derived from the theory of reproducing kernels will be introduced, simply.
[19]
vixra:2302.0019 [
pdf]
Some Deep Properties of the Green Function of Q. Guan on the line of Suita-Saitoh-Yamada's Conjectures
In this paper, we would like to refer to some deep results of the Green function of Q. Guan on the conjugate analytic Hardy $H_2$ norm and on the line of Oikawa-Sario's problems; Suita's conjecture, Saitoh's conjecture and Yamada's conjecture and to propose new related open problems. In particular, Q. Guan examined the deep properties of the inversion of the normal derivative of the Green function, the property of the level curve of the Green function and the magnitude of the logarithm capacity with his colleagues.
[20]
vixra:2302.0017 [
pdf]
A Monte Carlo Packing Algorithm for Poly-Ellipsoids and Its Comparison with Packing Generation Using Discrete Element Model
Granular material is showing very often in geotechnical engineering, petroleum engineering, material science and physics. The packings of the granular material play a very important role in their mechanical behaviors, such as stress-strain response, stability, permeability and so on. Although packing is such an important research topic that its generation has been attracted lots of attentions for a long time in theoretical, experimental, and numerical aspects, packing of granular material is still a difficult and active research topic, especially the generation of random packing of non-spherical particles. To this end, we will generate packings of same particles with same shapes, numbers, and same size distribution using geometry method and dynamic method, separately. Specifically, we will extend one of Monte Carlo models for spheres to ellipsoids and poly-ellipsoids.
[21]
vixra:2210.0030 [
pdf]
Approximation by Power Series of Functions
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented ideas may have applications in numerical mathematics.
[22]
vixra:2210.0026 [
pdf]
Reduction Formulas of the Cosine of Integer Fractions of Pi
The power of some cosines of integer fractions pi/n of the half circle allow a reduction to lower powers of the same angle. These are tabulated in the format sum_{i=0}^[n/2] a_i^n cos^i(pi/n)=0; n=2,3,4,...Related expansions of Chebyshev Polynomials T_n(x) and factorizations of T_n(x)+1 are also given.
[23]
vixra:2209.0134 [
pdf]
New Principles of Dierential Equations VI
This paper uses Z transformations to get the general solutions of many second-order, third-order and fourth-order linear PDEs for the first time, and uses the general solutions to obtain the exact solutions of many typical definite solution problems. We present the Z4 transformation for the first time and use it to solve a specific case. We successfully get the Fourier series solution by the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famed unresolved debate in the history of mathematics.
[24]
vixra:2208.0111 [
pdf]
Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice
The integrals R_{n,n}$ obtained by Atkinson and van Steenwijkfor the resistance between points of an infinite set ofunit resistors on the triangular latticeobey P-finite recurrences. The main causeof these are similarities uncovered by partial integrations of theirintegral representations with algebraic kernels. All R_{n,p} resistancesto points with integer coordinates n and p relative to an originin the lattice can be derived recursively.
[25]
vixra:2208.0050 [
pdf]
An Example of the Division by Zero Calculus Appeared in Conformal Mappings
We introduce an interesting example of conformal mappings (Joukowski transform) from the view point of the division by zero calculus. We give an interpretation of the identity, for a larger than b larger than 0 frac{rho + 1/rho}{rho - 1/rho} = frac{a}{b}, quad rho = sqrt{frac{a+b}{a - b}}, for the case a=b.
[26]
vixra:2208.0019 [
pdf]
Approximating Roots and π Using Pythagorean Triples
Methods approximating the square root of a number use recursive sequences. They do not have a simpleformula for generating the seed value for the approximation, so instead they use various algorithms for choosing the first term of the sequences. Section 1 introduces a new option, based upon the number of digits of the radicand, for selecting the first term. This new option works well at all scales. This first term will then be used in a traditional recursive sequence used to approximate roots. Section 2 will apply the method shown in Section 1 to approximate pi using Archimedes’ method, which then no longer requires different algorithms at different scales for seed values. Section 3 will introduce new recursive sequences for approximating rootsusing Pythagorean triples. Section 4 will then use the same new method to approximate pi.
[27]
vixra:2207.0148 [
pdf]
Erratum to "Tables of Integral Transforms" by A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi (1953), p. 61 (4)
The integral (4) on page 61 in the "Tables of Integral Transforms", the Fourier Cosine Transform of a product of a Gaussian and a symmetric sum of two Parabolic-Cylinder Functions, is erroneous. A more general integral is derived here.
[28]
vixra:2207.0071 [
pdf]
On the Integral Inequality of Some Trigonometric Functions in $mathbb{r}^n$
In this note, we prove the inequality begin{align}bigg| int limits_{|a_n|}^{|b_n|} int limits_{|a_{n-1}|}^{|b_{n-1}|}cdots int limits_{|a_1|}^{|b_1|}cos bigg(frac{sqrt[4s]{sum limits_{j=1}^{n}x^{4s}_j}}{||vec{a}||^{4s+1}+||vec{b}||^{4s+1}}bigg)dx_1dx_2cdots dx_nbigg| leq frac{bigg|prod_{i=1}^{n}|b_i|-|a_i|bigg|}{|Re(langle a,b angle)|}onumberend{align}and begin{align}bigg|int limits_{|a_n|}^{|b_n|} int limits_{|a_{n-1}|}^{|b_{n-1}|}cdots int limits_{|a_1|}^{|b_1|}sin bigg(frac{sqrt[4s]{sum limits_{j=1}^{n}x^{4s}_j}}{||vec{a}||^{4s+1}+||vec{b}||^{4s+1}}bigg)dx_1dx_2cdots dx_nbigg| leq frac{bigg|prod_{i=1}^{n}|b_i|-|a_i|bigg|}{|Im(langle a,b angle)|}onumberend{align}under some special conditions.
[29]
vixra:2207.0052 [
pdf]
Numerical Derivatives
The idea of this work is to present the software that allows us to quickly and numerically calculate values of f 0, f 00 , f 000 and f IV at the points where they are required, especially thinking about the estimation of the error in problems that involve differential equations. ordinary and partial differential equations. The calculation of these values by means of numerical methods is of great. It helps in solving these problems, as it saves a lot of time. The routines presented have been written in Google Inc.'s Go language, following our policy of making the most of "21st century C", which is a very fast, comfortable tool with sufficient accuracy for the proposed applications. We hope that this study will be useful for professional mathematicians as well as scientists from other areas and engineers who need to calculate the error in their equations or the rates of change associated with a whole potential of physical applications. <p> La idea de este trabajo es presentar el software que nos permite calcular rápida y numéricamente valores de f 0, f 00 , f 000 y f IV en los puntos donde se les requiera, sobre todo pensando en la estimación del error en problemas que involucran ecuaciones diferenciales ordinarias y ecuaciones en derivadas parciales. El cálculo de estos valores mediante métodos numéricos es de gran ayuda en la resolución de estos problemas, pues ahorra mucho tiempo. Las rutinas presentadas han sido escritas en lenguaje Go de Google Inc., siguiendo nuestra polı́tica de usufructuar al máximo el “C del siglo XXI”, que es una herramienta muy rápida, cómoda y con la exactitud suficiente para las aplicaciones propuestas. Esperamos que este estudio sea de utilidad tanto para matemáticos profe-sionales como cientı́ficos de otrás áreas e ingenieros que requieran calcular el error en sus ecuaciones o las ratas de cambio asociadas con todo un potencial de aplicaciones fı́sicas.
[30]
vixra:2206.0076 [
pdf]
A Lower Bound for Multiple Integral of Normalized Log Distance Function in $\mathbb{R}^n$
In this note we introduce the notion of the local product on a sheet and associated space. As an application, we prove that for $\langle a,b \rangle>e^e$ then \begin{align} \int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \ints_{|a_1|}^{|b_1|}\bigg|\log \bigg(i\frac{\sqrt[4s]{\sum \limits_{j=1}^{n}x^{4s}_j}}{||\vec{a}||^{4s+1}+||\vec{b}||^{4s+1}}\bigg)\bigg| dx_1dx_2\cdots dx_n\nonumber \\ \geq \frac{\bigg| prod_{j=1}^{n}|b_j|-|a_j|\bigg|}{\log \log (\langle a,b\rangle)}\nonumber \end{align}for all $s\in \mathbb{N}$, where $\langle,\rangle$ denotes the inner product and $i^2=-1$.
[31]
vixra:2205.0117 [
pdf]
A Proof of the Line Like Kakeya Maximal Function Conjecture
In this paper we will prove the Kakeya maximal function conjecture in a special case when tube intersections behave like points. We achieve this by showing there exist large essentially disjoint tube-subsets.
[32]
vixra:2205.0090 [
pdf]
The Generating Function Technique and Algebraic Ordinary Differential Equations
In the past, theorems have shown that individuals can implement a (formal) power series method to derive solutions to algebraic ordinary differential equations, or AODEs. First, this paper will give a quick synopsis of these “bottom-up” approaches while further elaborating on a recent theorem that established the (modified) generating function technique, or [m]GFT, as a powerful method for solving differentials equations. Instead of building a (formal) power series, the latter method uses a predefined set of (truncated) Laurent series comprised of polynomial linear, exponential, hypergeometric, or hybrid rings to produce an analytic solution. Next, this study will utilize the [m]GFT to create several analytic solutions to a few example AODEs. Ultimately, one will find [m]GFT may serve as a powerful "top-down" method for solving linear and nonlinear AODEs.
[33]
vixra:2205.0006 [
pdf]
General Solutions of Ordinary Differential Equations and Division by Zero Calculus - New Type Examples
We examined many examples of the relation between general solutions with singular points in ordinary differential equations and division by zero calculus, however, here we will introduce a new type example that was appeared from some general solution of an ordinary differential equation.
[34]
vixra:2204.0014 [
pdf]
Asymptotics of Solutions of Differential Equations with a Spectral Parameter
The main goal of this paper is to construct the so-called Birkhoff-type solutions for linear ordinary differential equations with a spectral parameter. Such solutions play an important role in direct and inverse problems of spectral theory. In Section 1, we construct the Birkhoff-type solutions for n-th order differential equations. Section 2 is devoted to first-order systems of differential equations.
[35]
vixra:2203.0070 [
pdf]
What is the Value of the Function X/x at X=0? What is 0/0?
It will be a very pity that we have still confusions on the very famous problem on 0/0 and the value of the elementary function of x/x at x=0. In this note, we would like to discuss the problems in some elementary and self contained way in order to obtain some good understanding for some general people.
[36]
vixra:2203.0001 [
pdf]
One Century since Bergman, Szego and Bochner on Reproducing Kernels
In this note, we wrote the preface for the first volume of the International Journal of Reproducing Kernels (The Roman Science Publications and Distributions (RSPD): https://romanpub.com/ijrk.php). Incidentally, this year is one century since the origin of reproducing kernels at Berlin. For some detailed information of the origin and some global situation of the theory of reproducing kernels with the content of the first volume are introduced.
[37]
vixra:2202.0094 [
pdf]
Folium of Descartes and Division by Zero Calculus - An Open Question
In this note, in the folium of Descartes, with the division by zero calculus we will see some interesting results at the point at infinity with some interesting geometrical property. We will propose an interesting open question.
[38]
vixra:2201.0204 [
pdf]
The Local Product and Local Product Space
In this note we introduce the notion of the local product on a sheet and associated space. As an application we prove under some special conditions the following inequalities \begin{align} 2\pi \frac{|\log(\langle \vec{a},\vec{b}\rangle)|}{(||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4})|\langle \vec{a},\vec{b}\rangle|}\bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s+3]{\sum \limits_{i=1}^{n}x^{4s+3}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq \bigg|\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\mathbf{e}\bigg(-i\frac{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}\bigg)dx_1dx_2\cdots dx_n\bigg|\nonumber \end{align} and \begin{align} \bigg|\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\mathbf{e}\bigg(i\frac{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}\bigg)dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq 2\pi \frac{|\langle \vec{a},\vec{b}\rangle|\times |\log(\langle \vec{a},\vec{b}\rangle)|}{(||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4})}\bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s+3]{\sum \limits_{i=1}^{n}x^{4s+3}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \end{align} and \begin{align} \bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s]{\sum \limits_{i=1}^{n}x^{4s}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq \frac{|\langle \vec{a},\vec{b}\rangle|}{2\pi |\log(\langle \vec{a},\vec{b}\rangle)|}\times (||\vec{a}||^{4s+1}+||\vec{b}||^{4s+1}) \times \bigg|\prod_{i=1}^{n}|b_i|-|a_i|\bigg|\nonumber \end{align}for all $s\in \mathbb{N}$, where $\langle,\rangle$ denotes the inner product and where $\mathbf{e}(q)=e^{2\pi iq}$.
[39]
vixra:2111.0167 [
pdf]
Seeking the Analytic Quaternion
By combining the complex analytic Cauchy-Riemann derivative with the Cayley-Dickson construction of a quaternion, possible formulations of a quaternion derivative are explored with the goal of finding an analytic quaternion derivative having conjugate symmetry. Two such analytic derivatives can be found. Although no example is presented, it is suggested that this finding may have significance in areas of quantum mechanics where quaternions are fundamental, especially regarding the enigmatic phenomenon of complementarity, where a quantum process seems to present two essential aspects.
[40]
vixra:2111.0124 [
pdf]
PhD's Thesis of Youcef Naas
Le troisième chapitre concerne le cas L p (0, 1; X). Plus précisément, on s’intéresse à l’équation différentielle abstraite du second ordre de type elliptique (1) avec les conditions aux limites de type mêlé (4) où A est un opérateur linéaire fermé sur un espace de Banach complexe X et u0, u 0 1 sont des éléments donnés dans X. Ici f ∈ L p (0, 1; X), 1 < p < ∞, 5 INTRODUCTION INTRODUCTION et X a la proporiété géométrique dite UMD. On suppose que A est un opérateur Bip et on montre que (1)-(4) admet une unique solution stricte, sous certaines hypothèses naturelles d’ellipticité de l’opérateur et de régularité sur les données, on donne alors, une représentation explicite de la solution stricte. La formule de représentation de la solution est donnée par deux méthodes, la première se base sur le calcul fonctionnel de Dunford et la deuxième sur la méthode de Krein[27], l’unicité de la représentation est démontrée. Dans ce chapitre, on fait une nouvelle approche du problème (1)-(4) en utilisant le théorème de Mikhlin. Dans cette partie on utilise les techniques des multiplicateurs de Fourier et la théorie de Mikhlin pour majorer les puissances imaginaires pures d’opérateurs. Le quatrième chapitre illustre notre théorie abstraite par quelques exemples concrets d’applications en EDP dans le cas des espaces L p et C α .
[41]
vixra:2111.0072 [
pdf]
Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
Recent analysis has uncovered a broad swath of rarely considered real numbers called real numbers in the neighborhood of infinity. Here, we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior. The main results are (1) to prove with modest axioms that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity. We define numbers in the neighborhood of infinity with a Cartesian product of Cauchy equivalence classes of rationals. We axiomatize the arithmetic of such numbers, prove the operations are well-defined, and then make comparisons to the similar axioms of a complete ordered field. After developing the many underlying foundations, we present a basis for a topology.
[42]
vixra:2110.0004 [
pdf]
On the Method of Dynamical Balls
In this paper we introduce and develop the notion of dynamical systems induced by a fixed $a\in \mathbb{N}$ and their associated induced dynamical balls. We develop tools to study problems requiring to determine the convergence of certain sequences generated by iterating on a fixed integer.
[43]
vixra:2108.0165 [
pdf]
Differential Coefficients at Corners and Division by Zero Calculus
For a $C_1$ function $y=f(x)$ except for an isolated point $x=a$ having $f^\prime(a-0)$ and $f^\prime(a+0)$, we shall introduce its natural differential coefficient at the singular point $x=a$. Surprisingly enough, the differential coefficient is given by the division by zero calculus and it will give the gradient of the natural tangential line of the function $y=f(x)$ at the point $x=a$.
[44]
vixra:2108.0148 [
pdf]
Two-Dimensional Fourier Transformations and Double Mordell Integrals II
Several Fourier transforms of functions of two variables are calculated. They enable one to calculate integrals that contain trigonometric and hyperbolic functions and also evaluate certain double Mordell integrals in closed form.
[45]
vixra:2107.0134 [
pdf]
A Geometric Variational Problem on a Periodic Domain
A diblock copolymer melt is a soft material, characterized by fluid-like disorder on the molecular scale and a high degree of order at a longer length scale. A molecule in a diblock copolymer is a linear sub-chain of A-monomers grafted covalently to another subchain of B-monomers. The Ohta-Kawasaki density functional theory of diblock copolymer gives rise to a non local free boundary problem. We will work on a periodic lattice in C generated by two complex numbers and we will assume periodic boundary conditions. In this thesis we will find two stationary sets of the energy functional of the problem. The first set is a perturbation of a round disk in C. More specifically, we will perturb a round disk under the polar coordinates. The radius of this perturbed disk will be sufficiently small. Also, we will minimize the energy of this stationary set with respect to the shape and size of the lattice. Additionally, we will show that for every K >= 2, K 2 N there exists a stationary set of the free energy functional that is the union of K disjoint perturbed disks in C. Later, we will assume that K = 2 and we will deal with the problem of finding the centers of these two perturbed disks. We will show that the centers of these disks are close to a global minimum of the Green’s function of the problem. We will minimize the Green’s function of the problem looking at some special cases of lattice structures. These lattices are the hexagonal lattice and a family of rectangular lattices.
[46]
vixra:2107.0053 [
pdf]
On the Elementary Function y=|x| and Division by Zero Calculus
In this paper, we will consider the elementary function $y=|x|$ from the viewpoint of the basic relations of the normal solutions (Uchida's hyper exponential functions) of ordinary differential equations and the division by zero calculus. In particular, $y^\prime(0) =0$ in our sense and this function will show the fundamental identity with the natural sense $$ \frac{0}{0} =0 $$ with the sense $$ \frac{1}{0} =0 $$ that may be considered as $0$ as the inversion of $0$ through the Uchida's hyper exponential function.
[47]
vixra:2106.0108 [
pdf]
Division by Zero Calculus in Figures - Our New Space Since Euclid -
We will show in this paper in a self contained way that our basic idea for our space is wrong since Euclid, simply and clearly by using many simple and interesting figures.
[48]
vixra:2106.0085 [
pdf]
On the State of Convergence of the Flint Hill Series
In this paper we study the convergence of the flint hill series of the form \begin{align} \sum \limits_{n=1}^{\infty}\frac{1}{(\sin^2n) n^3}\nonumber \end{align}via a certain method. The method works essentially by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular we show that the convergence of the flint hill series relies very heavily on the condition that for any small $\epsilon>0$ \begin{align} \bigg|\sum \limits_{i=0}^{\frac{n+1}{2}}\sum \limits_{j=0}^{i}(-1)^{i-j}\binom{n}{2i+1} \binom{i}{j}\bigg|^{2s} \leq |(\sin^2n)|n^{2s+2-\epsilon}\nonumber \end{align}for some $s\in \mathbb{N}$.
[49]
vixra:2103.0092 [
pdf]
An Idea of Fermat for the Stop and Division by Zero Calculus
In this note we will consider an idea of Fermat for the stop in connection with the division by zero calculus. Here, in particular, we will see some mysterious logic on the stop in connection with the concepts of differential and differential coefficient.
[50]
vixra:2103.0039 [
pdf]
History of the Division by Zero Calculus
Today is the 7th birthday of the division by zero calculus as stated in details in the Announcement 456(2018.10.15) of the Institute of Reproducing Kernels and the book was published recently. We recall simply a history of the division by zero calculus. Division by zero has a long and mysterious history since the origins of mathematics by Euclid and Brahmagupta. We will see that they are very important in mathematics, however they had the serious problems; that is, on the point at infinity and the division by zero, respectively.
[51]
vixra:2102.0136 [
pdf]
Division by Zero Calculus and Hyper Exponential Functions by K. Uchida
In this paper, we will consider the basic relations of the normal solutions (hyper exponential functions by K. Uchida) of ordinary differential equations and the division by zero calculus. In particular, by the concept of division by zero calculus, we extend the concept of Uchida's hyper exponential functions by considering the equations and solutions admitting singularities. Surprisingly enough, by this extension, any analytic functions with any singularities may be considered as Uchida's hyper exponential functions. Here, we will consider very concrete examples as prototype examples.
[52]
vixra:2102.0114 [
pdf]
Rotation Without Imaginary Numbers, Transcendental Functions, or Infinite Sums
Abstract. Quaterns are introduced as a new measure of rotation. Rotation in quaterns has an advantage in that only simple algebra is required to convert back and forth between rectangular and polar coordinates that use quaterns as the angle measure. All analogue trigonometric functions also become algebraic when angles are expressed in quaterns. This paper will show how quatern measure can be easily used to approximate trigonometric functions in the first quadrant without recourse to technology, innite sums, imaginary numbers, or transcendental functions. Using technology, these approximations can be applied to all four quadrants to any degree of accuracy. This will also be shown by approximating u to any degree of accuracy desired without reference to any traditional angle measure at all.
[53]
vixra:2101.0049 [
pdf]
Filter Exhaustiveness and Lter Limit Theorems for K-Triangular Lattice Group-Valued Set Functions
We give some limit theorems for sequences of lattice group-valuedk-triangular set functions,in the setting of filter convergence, and some results about their equivalence. We use the toolof filter exhaustiveness to get uniform (s)-boundedness, uniform continuity and uniform regular-ity of a suitable subsequence of the given sequence, whose indexes belong to the involved filter. Furthermore we pose some open problems.
[54]
vixra:2011.0202 [
pdf]
On Linear Ordinary Differential Equations of Second Order and Their General Solutions
We have worked out a new geometric approach to linear ordinary differential equations of second order which makes it possible to obtain general solutions to infinite number of equations of this sort. No need new families of special functions and their theories arose, solutions are composed straightforwardly. In this work we present a number of particular cases of equations with their general solutions. The solutions are divided into four groups the same way one encounters in any book on special functions.
[55]
vixra:2011.0182 [
pdf]
A Note on Lp-Convergence and Almost Everywhere Convergence
It is a classical but relatively less well-known result that, for every given measure space and every given $1 \leq p \leq +\infty$, every sequence in $L^{p}$ that converges in $L^{p}$ has a subsequence converging almost everywhere. The typical proof is a byproduct of proving the completeness of $L^{p}$ spaces, and hence is not necessarily ``application-friendly''. We give a simple, perhaps more ``accessible'' proof of this result for all finite measure spaces.
[56]
vixra:2011.0181 [
pdf]
New Principles of Differential Equations Ⅳ
In previous papers, we proposed several new methods to obtain general solutions or analytical solutions of some nonlinear partial differential equations. In this paper, we will continue to propose a new effective method to obtain general solutions of certain nonlinear partial differential equations for the first time, such as nonlinear wave equation, nonlinear heat equation, nonlinear Schrödinger equation, etc.
[57]
vixra:2011.0163 [
pdf]
A General Definition of Means and Corresponding Inequalities
This paper proves inequalities among generalised f-means and provides formal conditions which a function of several inputs must satisfy in order to be a `meaningful' mean. The inequalities we prove are generalisations of classical inequalities including the Jensen inequality and the inequality among the Quadratic and Pythagorean means. We also show that it is possible to have meaningful means which do not fall into the general category of f-means.
[58]
vixra:2011.0131 [
pdf]
Topological Stationarity and Precompactness of Probability Measures
We prove the precompactness of a collection of Borel probability measures over an arbitrary metric space precisely under a new legitimate notion, which we term \textit{topological stationarity}, regulating the sequential behavior of Borel probability measures directly in terms of the open sets. Thus the important direct part of Prokhorov's theorem, which permeates the weak convergence theory, admits a new version with the original and sole assumption --- tightness --- replaced by topological stationarity. Since, as will be justified, our new condition is not vacuous and is logically independent of tightness, our result deepens the understanding of the connection between precompactness of Borel probability measures and metric topologies.
[59]
vixra:2011.0101 [
pdf]
New Principles of Differential Equations Ⅲ
Using the new method proposed in this paper, in theory, it is possible to obtain general or exact solutions of an infinite number of ordinary differential equations and partial differential equations. These equations can be linear or nonlinear. We enumerate some typical cases and use the new method to prove that some equations do not have certain forms of solutions.
[60]
vixra:2011.0090 [
pdf]
An Untold Story of Brownian Motion
Although the concept of Brownian motion or Wiener process is quite popular, proving its existence via construction is a relatively deep work and would not be stressed outside mathematics. Taking the existence of Brownian motion in $C([0,1], \R)$ ``for granted'' and following an existing implicit thread, we intend to present an explicit, simple treatment of the existence of Brownian motion in the space $C([0, +\infty[, \R)$ of all continuous real-valued functions on the ray $[0, +\infty[$ with moderate technical intensity. In between the developments, some informative little results are proved.
[61]
vixra:2011.0052 [
pdf]
Another Topological Proof for Equivalent Characterizations of Continuity
To prove the equivalence between the $\eps$-$\delta$ characterization and the topological characterization of the continuity of maps acting between metric spaces, there are two typical approaches in, respectively, analysis and topology. We provide another proof that would be pedagogically informative, resembling the typical proof method --- principle of appropriate sets --- associated with sigma-algebras.
[62]
vixra:2011.0044 [
pdf]
How Likely Is It for Countably Many Almost Sure Events to Occur Simultaneously?
Given a countable collection of almost sure events, the event that at least one of the events occurs is ``evidently'' almost sure. It is, however, not so trivial to assert that the event for every event of the collection to occur is almost sure. Measure theory helps to furnish a simple, definite, and affirmative answer to the question stated in the title. This useful proposition seems to rarely, if not never, occur in a teaching material regarding measure-theoretic probability; our proof in particular would help the beginning students in probability theory to get a feeling of almost sure events.
[63]
vixra:2011.0030 [
pdf]
Distribution of Integrals of Wiener Paths
With a new proof approach, we show that the normal distribution with mean zero and variance $1/3$ is the distribution of the integrals $\int_{[0,1]}W_{t}\df t$ of the sample paths of Wiener process $W$ in $C([0,1], \R)$.
[64]
vixra:2011.0029 [
pdf]
Growth Order of Standardized Distribution Functions
Denote by $\CDF^{0,1}(\R)$ the class of all (cumulative) distribution functions on $\R$ with zero mean and unit variance; if $F \in \CDF^{0,1}(\R)$, we are interested in the asymptotic behavior of the function sequence $(x \mapsto nF(x/\sqrt{n}))_{n \in \N}$. We show that $\inf_{F \in \CDF^{0,1}(\R)}\liminf_{n \to \infty}nF(x/\sqrt{n}) \geq \Phi(x)$ for all $x \in \R$, which in particular would be a result obtained for the first time regarding the growth order of an arbitrary standardized distribution function on $\R$ near the origin.
[65]
vixra:2011.0005 [
pdf]
Some New Type Laurent Expansions and Division by Zero Calculus; Spectral Theory
In this paper we introduce a very interesting property of the Laurent expansion in connection with the division by zero calculus and Euclid geometry by H. Okumura. The content may be related to analytic motion of figures. We will refer to some similar problems in the spectral theory of closed operators.
[66]
vixra:2010.0050 [
pdf]
Division by Zero Calculus and Laplace Transform
In this paper, we will discuss the Laplace transform from the viewpoint of the division by zero calculus with typical examples. The images of the Laplace transform are analytic functions on some half complex plane and meanwhile, the division by zero calculus gives some values for isolated singular points of analytic functions. Then, how will be the Laplace transform at the isolated singular points? For this basic question, we will be able to obtain a new concept for the Laplace integral.
[67]
vixra:2009.0194 [
pdf]
RLC Circuits and Division by Zero Calculus
In this paper, we will discuss an RLC circuit for missing the capacitor from the viewpoint of the division by zero calculus as a typical example.
[68]
vixra:2009.0149 [
pdf]
Representations of the Division by Zero Calculus by Means of Mean Values
In this paper, we will give simple and pleasant introductions of the division by zero calculus by means of mean values that give an essence of the division by zero. In particular, we will introduce a new mean value for real valued functions in connection with the Sato hyperfunction theory.
[69]
vixra:2009.0051 [
pdf]
Mirror Images and Division by Zero Calculus
Very classical results on the mirror images of the centers of circles and bolls should be the centers as the typical results of the division by zero calculus. For their importance, we would like to discuss them in a self-contained manner.
[70]
vixra:2009.0045 [
pdf]
Quantum Permutations and Quantum Reflections
The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We review the known facts regarding $S_N^+$, and its versions $S_F^+$, with $F$ being a finite quantum space. We discuss then the structure of the closed subgroups $G\subset S_N^+$ and $G\subset S_F^+$, with particular attention to the quantum reflection groups.
[71]
vixra:2009.0039 [
pdf]
On the Number of Intersections of Tubes
In this article we will always assume that the number of $\delta$-tubes is $N = \delta^{1-n}.$ Moreover, we will assume that if any two $\delta$-tubes intersect, then they intersect in the unit ball. We will show that the number of their intersections of order $\mu$ is bounded by $\frac{|B(0,1+\delta)|}{|B(0,1)|}(1+ 2\delta)2^{n-1}|B_{n-1}(0,1)|\frac{N^{n/(n-1)}}{\mu}$. After making a dyadic decomposition and summing the orders together we will find that the number of tube intersections of $N$ tubes is bounded by $\frac{|B(0,1+N^{-1/(n-1)})|}{|B(0,1)|}2^{n-1}(1+ 2N^{-1/(n-1)})|B_{n-1}(0,1)|N^{n/(n-1)}.$ Moreover, we will prove a generalized lemma of C{\'o}rdoba and we will prove the Kakeya sets have greater dimension than $n-1.$
[72]
vixra:2009.0013 [
pdf]
New Principles of Differential Equations Ⅱ
This is the second part of the total paper. Three kinds of Z Transformations are used to get many laws for general solutions of mth-order linear partial differential equations with n variables in the present thesis. Some general solutions of first-order linear partial differential equations, which cannot be obtained by using the characteristic equation method, can be solved by the Z Transformations. By comparing, we find that the general solutions of some first-order partial differential equations got by the characteristic equation method are not complete.
[73]
vixra:2009.0005 [
pdf]
Liouville-Type Theorems Outside Small Sets
We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.
[74]
vixra:2008.0177 [
pdf]
A Conjecture On Some ds Periods On The Complex Plane
Here we will propose a simple and very difficult open question like the Fermat's problem on some $ds$ periods on the complex pane. This very elementary problem will create a new field on the complex plane.
[75]
vixra:2008.0002 [
pdf]
On Evaluation of an Improper Real Integral Involving a Logarithmic Function
In this paper we use the methods in complex analysis to evaluate an improper real integral involving the natural logarithmic function. Our presentation is somewhat unique because we use traditional notation in performing the calculations.
[76]
vixra:2007.0190 [
pdf]
Global Stability for a System of Parabolic Conservation Laws Arising from a Keller-Segel Type Chemotaxis Model
In this paper, we investigate the time-asymptotically nonlinear stability to the initial-boundary value problem for a coupled system in (p, q) of parabolic conservation laws derived from a Keller-Segel type repulsive model for chemotaxis with singular sensitivity and nonlinear production rate of g(p) = p γ , where γ > 1. The proofs are based on basic energy method without any smallness assumption. We also show the zero chemical diffusion limit (ε → 0) of solutions in the case ¯p = 0.
[77]
vixra:2007.0180 [
pdf]
Time-periodic Solution to the Compressible Viscoelastic Flows in Periodic Domain
In this paper, we are concerned with the time-periodic solutions to the threedimensional compressible viscoelastic flows with a time-periodic external force in a periodic domain. By using an approach of parabolic regularization and combining with the topology degree theory, we show the existence and uniqueness of the time-periodic solution to the model under some smallness and symmetry assumptions on the external force.
[78]
vixra:2007.0178 [
pdf]
Initial-boundary Value Problems for a System of Parabolic Conservation Laws Arising From a Keller-segel Type Chemotaxis Model
In this paper, we investigate the time-asymptotically nonlinear stability to the initial-boundary value problem for a coupled system in (p, q) of parabolic conservation laws derived from a Keller-Segel type repulsive model for chemotaxis with singular sensitivity and nonlinear production rate of g(p) = p γ , where γ > 1. The proofs are based on basic energy method without any smallness assumption.
[79]
vixra:2007.0121 [
pdf]
Quasinilpotent Operators on Separable Hilbert Spaces Have Nontrivial Invariant Subspaces
The invariant subspace problem is a well known unsolved problem in funtional analysis. While many partial results are known, the general case for complex, infinite dimensional separable Hilbert spaces is still open. It has been shown that the problem can be reduced to the case of operators which are norm limits of nilpotents. One of the most important subcases is the one of quasinilpotent operators, for which the problem has been extensively studied for many years. In this paper, we will prove that every quasinilpotent operator has a nontrivial invariant subspace. This will imply that all the operators for which the ISP has not been established yet are norm-limits of operators having nontrivial invariant subspaces.
[80]
vixra:2007.0088 [
pdf]
Limited Polynomials
In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the zeros are real and are of the same sign.
[81]
vixra:2007.0036 [
pdf]
Differential Quotients and Division by Zero
In this very short note, a pleasant relation of the basic idea of differential quotients $dy/dx$ of Leipniz and division by zero $1/0=0$. This will give a natural interpretation of the important result $\tan (\pi/2)=0$.
[82]
vixra:2006.0263 [
pdf]
Approximation of Harmonic Series
Background : Harmonic Series is the sum of Harmonic Progression. There have been multiple formulas to approximate the harmonic series, from Euler's formula to even a few in the 21st Century. Mathematicians have concluded that the sum cannot be calculated, however any approximation better than the previous others is always needed. In this paper we will discuss the flaws in Euler's formula for approximation of harmonic series and provide a better formula. We will also use the infinite harmonic series to determine the approximations of finite harmonic series using the Euler-Mascheroni constant. We will also look at the Leibniz series for Pi and determine the correction factor that Leibniz discussed in his paper which he found using Euler numbers. Each subsequent approximation we find in this paper is better than all previous ones. Different approximations for different types of harmonic series are calculated, best fit for the given type of harmonic series. The correction factor for Leibniz series might not provide any applied results but it is a great way to ponder some other infinite harmonic series.
[83]
vixra:2006.0206 [
pdf]
Majorization in the Framework of 2-Convex Systems
We define a 2-convex system by the restrictions $x_{1} + x_{2} + \ldots + x_{n} = ns$, $e(x_{1}) + e(x_{2}) + \ldots + e(x_{n}) = nk$, $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ where $e:I \to \RR$ is a strictly convex function. We study the variation intervals for $x_k$ and give a more general version of the Boyd-Hawkins inequalities. Next we define a majorization relation on $A_S$ by $x\preccurlyeq_p y$ $\Leftrightarrow$ $T_k(x) \leq T_k(y) \ \ \forall 1 \leq k \leq p-1$ and $B_k(x) \leq B_k(y) \ \ \forall p+2 \leq k \leq n$ (for fixed $1 \leq p \leq n-1$) where $T_k(x) = x_1 + \ldots + x_k$, $B_k(x) = x_k + \ldots + x_n$. The following Karamata type theorem is given: if $x, y \in A_S$ and $x\preccurlyeq_p y$ then $f(x_1) + f(x_2) + \ldots + f(x_n) \leq f(y_1) + f(y_2) + \ldots + f(y_n)$ $\forall$$f:I \to \RR$ 3-convex with respect to $e$. As a consequence, we get an extended version of the equal variable method of V. Cîrtoaje
[84]
vixra:2006.0105 [
pdf]
First Steps of Vector Differential Calculus
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.
[85]
vixra:2005.0256 [
pdf]
Newton's Limit Operator Has no Sense
The Limits and infinitesimal numbers were invented by the fathers of Science like Newton and Leibniz. However, a hypothetical being from another star system could have developed more realistic mathematics [in my opinion the mathematics should be defined via numbers of our fingers and the actions (like adding) with them]. In this note, I am showing the paradox of the current version of ``highest mathematics''.
[86]
vixra:2005.0159 [
pdf]
Quantum Isometries and Noncommutative Geometry
The space $\mathbb C^N$ has no free analogue, but we can talk instead about the free sphere $S^{N-1}_{\mathbb C,+}$, as the manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. We discuss here the structure and hierarchy of the submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, with particular attention to the manifolds having an integration functional $tr:C(X)\to\mathbb C$.
[87]
vixra:2004.0325 [
pdf]
On Attractivity for $\psi$-Hilfer Fractional Differential Equations Systems
In this paper, we investigate the existence of a class of globally attractive solutions of the Cauchy fractional problem with the $\psi$-Hilfer fractional derivative using the measure of noncompactness. An example is given to illustrate our theory.
[88]
vixra:2004.0323 [
pdf]
Attractivity for Differential Equations Systems of Fractional Order
This paper investigates the overall solution attractivity of the fractional differential equation introduced by the $\psi$-Hilfer fractional derivative and the Krasnoselskii's fixed point theorem. We highlight some particular cases of the result investigated here, especially involving the Riemann- Liouville and Katugampola fractional derivative, elucidating the fundamental property of the $\psi$-Hilfer fractional derivative, that is, the broad class of particular cases of fractional derivatives that consequently apply to the results investigated herein.
[89]
vixra:2004.0287 [
pdf]
Why Quasi-Interpolation onto Manifold has Order 4
We consider approximations of functions from samples where the functions take values on a submanifold of $\mathbb{R}^n$. We generalize a common quasi-interpolation scheme based on cardinal B-splines by combining it with a projection $P$ onto the manifold. We show that for $m\geq 3$ we will have approximation order $4$. We also show why higher approximation order can not be expected when the control points are constructed as projections of the filtered samples using a fixed mask.
[90]
vixra:2004.0234 [
pdf]
Relative Uniform Convergence of a Sequence of Functions at a Point and Korovkin-Type Approximation Theorems
We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence.
[91]
vixra:2004.0232 [
pdf]
On Matrix Methods of Convergence of Order Alpha in L-Groups
We introduce a concept of convergence of order alpha, which is positive and strictly less than one, with respect to a summability matrix method A for sequences, taking values in lattice groups. Some main properties and dierences with the classical A-convergence are investigated. A Cauchy-type criterion and a closedness result for the space of convergent sequences according our notion is proved.
[92]
vixra:2003.0318 [
pdf]
Division by Zero Calculus in Ford Circles
We will refer to an application of the division by zero calculus in Ford circles that have the relations to some criteria of irrational numbers as covering problems and to the Farey sequence $F_n$ for any positive integer $n$. Division by zero, division by zero calculus, $1/0=0/0=z/0=\tan(\pi/2) =0, [(z^n)/n]_{n=0} = \log z$, $[e^{(1/z)}]_{z=0} = 1$, Ford circle, Farey series, Farey intermediate number, packing by circle, criteria of irrational number.
[93]
vixra:2003.0071 [
pdf]
Ankur Tiwari's Great Discovery of the Division by Zero $1/0 = \tan (\pi/2) = 0$ on $ 2011$
We got an important information on the Ankur Tiwari's great discovery of the division by zero $1/0 = \tan (\pi/2) = 0$ on $ 2011$. Since the information was not known for us and among many colleagues, we would like to state our opinions on his great discovery.
[94]
vixra:2002.0508 [
pdf]
On Certain Finite Sums of Inverse Tangents
An identity is proved connecting two finite sums of inverse tangents. This identity is discretized version of Jacobi's imaginary transformation for the modular angle from the theory of elliptic functions. Some other related identities are discussed.
[95]
vixra:2002.0366 [
pdf]
Division by Zero Calculus For Differentiable Functions in Multiply Dimensions
Based on the preprint survey paper, we will give a fundamental relation among the basic concepts of division by zero calculus and derivatives as a direct extension of the preprints which gave the generalization of the division by zero calculus to differentiable functions. Here, we will consider the case of multiply dimensions. In particular, we will find a new viewpoint and applications to the gradient and nabla.
[96]
vixra:2002.0060 [
pdf]
A Short Remark on the Result of Jozsef Sandor
We point out that Corollary 2.2 in the recently published article, ’On the Iyengar-Madhava Rao-Nanjundiah inequality and it’s hyperbolic version’ [3] by Jozsef Sandor is slightly incorrect since its proof contains a gap. Fortunately, the proof can be corrected and this is the main aim of this note.
[97]
vixra:2001.0590 [
pdf]
On Entire Functions-Minorants for Subharmonic Functions Outside of a Small Exceptional Set
Let u be an arbitrary subharmonic function of finite order on the complex plane. We construct a nonzero entire function f such that ln|f| does not exceed the function u everywhere outside some very small exceptional set E.
[98]
vixra:2001.0586 [
pdf]
Division by Zero Calculus, Derivatives and Laurent's Expansion
Based on a preprint survey pape, we will give a fundamental relation among the basic concepts of division by zero calculus, derivatives and Laurent's expansion as a direct extension of the preprint which gave the generalization of the division by zero calculus to differentiable functions. In particular, we will find a new viewpoint and applications to the Laurent expansion, in particular, to residures in the Laurent expansion. $1/0=0/0=z/0=\tan(\pi/2) =\log 0 =0, (z^n)/n = \log z$ for $n=0$, $e^{(1/z)} = 1$ for $z=0$.
[99]
vixra:2001.0376 [
pdf]
On a Certain Identity Involving the Gamma Function
The goal of this paper is to prove the identity \begin{align}\sum \limits_{j=0}^{\lfloor s\rfloor}\frac{(-1)^j}{s^j}\eta_s(j)+\frac{1}{e^{s-1}s^s}\sum \limits_{j=0}^{\lfloor s\rfloor}(-1)^{j+1}\alpha_s(j)+\bigg(\frac{1-((-1)^{s-\lfloor s\rfloor +2})^{1/(s-\lfloor s\rfloor +2)}}{2}\bigg)\nonumber \\ \bigg(\sum \limits_{j=\lfloor s\rfloor +1}^{\infty}\frac{(-1)^j}{s^j}\eta_s(j)+\frac{1}{e^{s-1}s^s}\sum \limits_{j=\lfloor s\rfloor +1}^{\infty}(-1)^{j+1}\alpha_s(j)\bigg)=\frac{1}{\Gamma(s+1)},\nonumber \end{align}where \begin{align}\eta_s(j):=\bigg(e^{\gamma (s-j)}\prod \limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)\nonumber \\e^{-(s-j)/m}\bigg)\bigg(2+\log s-\frac{j}{s}+\sum \limits_{m=1}^{\infty}\frac{s}{m(s+m)}-\sum \limits_{m=1}^{\infty}\frac{s-j}{m(s-j+m)}\bigg), \nonumber \end{align}and \begin{align}\alpha_s(j):=\bigg(e^{\gamma (s-j)}\prod \limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)e^{-(s-j)/m}\bigg)\bigg(\sum \limits_{m=1}^{\infty}\frac{s}{m(s+m)}-\sum \limits_{m=1}^{\infty}\frac{s-j}{m(s-j+m)}\bigg),\nonumber \end{align}where $\Gamma(s+1)$ is the Gamma function defined by $\Gamma(s):=\int \limits_{0}^{\infty}e^{-t}t^{s-1}dt$ and $\gamma =\lim \limits_{n\longrightarrow \infty}\bigg(\sum \limits_{k=1}^{n}\frac{1}{k}-\log n\bigg)=0.577215664\cdots $ is the Euler-Mascheroni constant.
[100]
vixra:2001.0103 [
pdf]
Integrals of Entire, Meromorphic and Subharmonic Functions on Small Sets of the Positive Semiaxis
In this note, we announce the results on estimates of integrals of entire, meromorphic, and subharmonic functions on small subsets of the positive semiaxis. These results develop one classical theorem of R. Nevanlinna and the well-known lemmas on small arcs or intervals of A. Edrei, W.H.J. Fuchs, A.F. Grishin, M.L. Sodin and T.I. Malyutina.
[101]
vixra:2001.0091 [
pdf]
Division by Zero Calculus for Differentiable Functions L'Hôpital's Theorem Versions
We will give a generalization of the division by zero calculus to differentiable functions and its basic properties. Typically, we can obtain l'Hôpital's theorem versions and some deep properties on the division by zero. Division by zero, division by zero calculus, differentiable, analysis, Laurent expansion, l'Hôpital's theorem, $1/0=0/0=z/0=\tan(\pi/2) =\log 0 =0, (z^n)/n = \log z$ for $n=0$, $e^{(1/z)} = 1$ for $z=0$.
[102]
vixra:1912.0537 [
pdf]
Surgical Analysis of Functions
In this paper we introduce the concept of surgery. This concept ensures that almost all discontinuous functions can be made to be continuous without redefining their support. Inspite of this, it preserves the properties of the original function. Consequently we are able to get a handle on the number of points of discontinuities on a finite interval by having an information on the norm of the repaired function and vice-versa.
[103]
vixra:1912.0347 [
pdf]
Q-Analogs of Sinc Sums and Integrals
$q$-analogs of sum equals integral relations $\sum_{n\in\mathbb{Z}}f(n)=\int_{-\infty}^\infty f(x)dx$ for sinc functions and binomial coefficients are studied. Such analogs are already known in the context of $q$-hypergeometric series. This paper deals with multibasic `fractional' generalizations that are not $q$-hypergeometric functions.
[104]
vixra:1912.0300 [
pdf]
Essential Problems on the Origins of Mathematics; Division by Zero Calculus and New World
Based on the preprint survey paper (What Was Division by Zero?; Division by Zero Calculus and New World, viXra:1904.0408 submitted on 2019-04-22 00:32:30) we will give a viewpoint of the division by zero calculus from the origins of mathematics that are the essences of mathematics. The contents in this paper seem to be serious for our mathematics and for our world history with the materials in the preprint. So, the author hopes that the related mathematicians, mathematical scientists and others check and consider the topics from various viewpoints.
[105]
vixra:1912.0030 [
pdf]
Zeros of the Riemann Zeta Function Within the Critical Strip and Off the Critical Line
In a recent paper, the author demonstrated the existence of real numbers in the neighborhood of infinity. It was shown that the Riemann zeta function has non-trivial zeros in the neighborhood of infinity but none of those zeros lie within the critical strip. While the Riemann hypothesis only asks about non-trivial zeros off the critical line, it is also an open question of interest whether or not there are any zeros off the critical line yet still within the critical strip. In this paper, we show that the Riemann zeta function does have non-trivial zeros of this variety. The method used to prove the main theorem is only the ordinary analysis of holomorphic functions. After giving a brief review of numbers in the neighborhood of infinity, we use Robinson's non-standard analysis and Eulerian infinitesimal analysis to examine the behavior of zeta on an infinitesimal neighborhood of the north pole of the Riemann sphere. After developing the most relevant features via infinitesimal analysis, we will proceed to prove the main result via standard analysis on the Cartesian complex plane without reference to infinitesimals.
[106]
vixra:1911.0453 [
pdf]
Existence and Continuous Dependence for Fractional Neutral Functional Differential Equations
In this paper, we investigate the existence, uniqueness and continuous dependence of solutions of fractional neutral functional differential equations with infinite delay and the Caputo fractional derivative order, by means of the Banach's contraction principle and the Schauder's fixed point theorem.
[107]
vixra:1911.0115 [
pdf]
General Order Differentials and Division by Zero Calculus
In this paper, we will give several examples that in the general order $n$ differentials of functions we find the division by zero and by applying the division by zero calculus, we can find the good formulas for $n=0$. This viewpoint is new and curious at this moment for some general situation. Therefore, as prototype examples, we would like to discuss this property. Why division by zero for zero order representations for some general differential order representations of functions does happen?
[108]
vixra:1910.0414 [
pdf]
Divergence Series and Integrals From the Viewpoint of the Division by Zero Calculus
In this short note, we would like to refer to the fundamental new interpretations that for the fundamental expansion $1/(1-z) = \sum_{j=0}^{\infty} z^j$ it is valid in the sense $0=0$ for $z=1$, for the integral $\int_1^{\infty} 1/x dx $ it is zero and in the formula $\int_0^{\infty} J_0(\lambda t) dt = 1/\lambda$, it is valid with $0=0$ for $\lambda =0$ in the sense of the division by zero.
[109]
vixra:1910.0245 [
pdf]
Complex Hadamard Matrices and Applications
A complex Hadamard matrix is a square matrix $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects.
[110]
vixra:1909.0658 [
pdf]
On the Value of the Function $\exp {(ax)}/f(a)$ at $a=0$ for $f(a)=0$
In this short note, we will consider the value of the function $\exp {(ax)}/f(a)$ at $a=0$ for $f(a)=0$. This case appears for the construction of the special solution of some differential operator $f(D)$ for the polynomial case of $D$ with constant coefficients. We would like to show the power of the new method of the division by zero calculus, simply and typically.
[111]
vixra:1909.0366 [
pdf]
Free Quantum Groups and Related Topics
The unitary group $U_N$ has a free analogue $U_N^+$, and the closed subgroups $G\subset U_N^+$ can be thought of as being the ``compact quantum Lie groups''. We review here the general theory of such quantum groups. We discuss as well a number of more advanced topics, selected for their beauty, and potential importance.
[112]
vixra:1908.0542 [
pdf]
Envelopes in Function Spaces with Respect to Convex Sets
We discuss the existence of an envelope of a function from a certain subclass of function space. Here we restrict ourselves to considering the model space of functions locally integrable with respect to the Lebesgue measure in a domain from the finite dimensional Euclidean space
[113]
vixra:1908.0511 [
pdf]
Affine Balayage of Measures in Domains of the Complex Plane with Applications to Holomorphic Functions
Let u and M are two non-trivial subharmonic functions in a domain D in the complex plane. We investigate two related but different problems. The first is to find the conditions on the Riesz measures of functions u and M respectively under which there exists a non-trivial subharmonic function h on D such that u+h< M. The second is the same question, but for a harmonic function h on D. The answers to these questions are given in terms of the special affine balayage of measures introduced in our recent previous works. Applications of this technique concern the description of distribution of zeros for holomorphic functions f on the domain D satisfying the restriction |f|< exp M.
[114]
vixra:1908.0436 [
pdf]
Balayage of Measures and Their Potentials: Duality Theorems and Extended Poisson-Jensen Formula
We investigate some properties of balayage of measures and their potentials on domains or open sets in finite-dimensional Euclidean space. Main results are Duality Theorems for potentials of balayage of measures, for Arens-Singer and Jensen measures and potentials, and also a new extended and generalized variant of Poisson-Jensen formula for balayage of measure and their potentials.
[115]
vixra:1907.0491 [
pdf]
Fractional Calculus
This paper generalises the limit definitions of calculus to define differintegrals of complex order, calculates some differintegrals of elementary functions, and introduces the notion of a fractional differential equation. An application to quantum theory is explored, and we conclude with some operator algebra. Functions in this paper will only have one variable.
[116]
vixra:1907.0454 [
pdf]
A Weak Extension of Complex Structure on Hilbert Spaces
The purpose of this paper is to try to replicate what happens in C on spaces where there are more then one of immaginary units. All these spaces, in our definition, will have the same Hilbert structure. At first we will introduce the sum and product operations on C(H):=RxH (where H is an Hilbert space), then we'll investigate on its algebraic properties. In our construction we lose only the associative of multiplication regardless of H, exept when dim H=1 (in this case RxH = C), and this is why we say "weak extension". One of the most important result of this study is the Weak Integrity Theorem according to which in particular conditions there exist zero divisors. The next result is the Foundamental Theorem according to which for all z in C(H) there exists w in C(H) such that z=w^2. Afterwards we will study tranformations between these spaces which keep operation (that's why we will call them C-morphisms). At the end we will look at the "commutative" functions, i.e. maps C(H) to C(H') which can be rapresented by complex transformations C to C
[117]
vixra:1906.0329 [
pdf]
Some Conjectures On Inequalities In Operator Axioms
The Operator axioms have deduced number systems. In this paper, we conjecture some inequalities in Operator axioms. The general inequalities show the value of Operator axioms.
[118]
vixra:1906.0185 [
pdf]
Division by Zero Calculus in Multiply Dimensions and Open Problems (An Extension)
In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from one dimensional case.
[119]
vixra:1906.0163 [
pdf]
Maximal Generalization of Lanczos' Derivative Using One-Dimensional Integrals
Derivative of a function can be expressed in terms of integration over a small neighborhood of the point of differentiation, so-called differentiation by integration method. In this text a maximal generalization of existing results which use one-dimensional integrals is presented together with some interesting non-analytic weight functions.
[120]
vixra:1906.0148 [
pdf]
On Some Isoperimetric Inequalities for Dirichlet Integrals; Green's Function and Dirichlet Integrals
In this paper, as a direct application of Q. Guan's result on the conjugate analytic Hardy $H_2$ norm we will derive a new type isoperimetric inequality for Dirichlet integrals of analytic functions.
[121]
vixra:1905.0485 [
pdf]
Approximation of Sum of Harmonic Progression
Background: The harmonic sequence and the infinite harmonic series have been a topic of great interest to mathematicians for many years. The sum of the infinite harmonic series has been linked to the Euler-Mascheroni constant. It has been demonstrated by Euler that, although the sum diverges, it can be expressed as the Euler-Mascheroni constant added to the natural log of infinity. Utilizing the Euler-Maclaurin method, we can extend the expression to approximate the sum of finite harmonic series with a fixed first term and a variable last term. However, a natural extension is not possible for a variable value of the first term or the common difference of the reciprocals.Aim: The aim of this paper is to create a formula that generates an approximation of the sum of a harmonic progression for a variable first term and common difference. The objective remains that the resultant formula is fundamentally similar to Euler's equation of the constant and the result using the Maclaurin method. Method: The principle result of the paper is derived using approximation theory. The assertion that the graph of harmonic progression closely resembles the graph of y=1/x is key. The subsequent results come through a comparative view of Euler's expression and by using numerical manipulations on the Euler-Mascheroni Constant.Results: We created a general formula that approximates the sum of harmonic progression with variable components. Its fundamental nature is apparent because we can derive the results of the Maclaurin method from our results.
[122]
vixra:1904.0414 [
pdf]
Unitary Quantum Groups vs Quantum Reflection Groups
We study the intermediate liberation problem for the real and complex unitary and reflection groups, namely $O_N,U_N,H_N,K_N$. For any of these groups $G_N$, the problem is that of understanding the structure of the intermediate quantum groups $G_N\subset G_N^\times\subset G_N^+$, in terms of the recently introduced notions of ``soft'' and ``hard'' liberation. We solve here some of these questions, our key ingredient being the generation formula $H_N^{[\infty]}=<H_N,T_N^+>$, coming via crossed product methods. Also, we conjecture the existence of a ``contravariant duality'' between the liberations of $H_N$ and of $U_N$, as a solution to the lack of a covariant duality between these liberations.
[123]
vixra:1904.0408 [
pdf]
What Was Division by Zero?; Division by Zero Calculus and New World
In this survey paper, we will introduce the importance of the division by zero and its great impact to elementary mathematics and mathematical sciences for some general people. For this purpose, we will give its global viewpoint in a self-contained manner by using the related references.
[124]
vixra:1904.0259 [
pdf]
Some Hereditary Properties of the E-J Generalized Cesàro Matrices
A countable subcollection of the Endl-Jakimovski generalized Ces\`{a}ro matrices of positive order is seen to inherit posinormality, coposinormality, and hyponormality from the Ces\`{a}ro matrix of the same order.
[125]
vixra:1904.0052 [
pdf]
D\"aumler's Horn Torus Model and\\ Division by Zero \\ - Absolute Function Theory -\\ New World
In this paper, we will introduce a beautiful horn torus model by Puha and D\"aumler for the Riemann sphere in complex analysis attaching the zero point and the point at infinity. Surprisingly enough, we can introduce analytical structure of conformal to the model. Here, some basic opinions on the D\"aumler's horn torus model will be stated as the basic ones in mathematics.
[126]
vixra:1903.0488 [
pdf]
Division by Zero Calculus in Complex Analysis
In this paper, we will introduce the division by zero calculus in complex analysis for one variable at the first stage in order to see the elementary properties.
[127]
vixra:1903.0432 [
pdf]
Division by Zero Calculus and Singular Integrals
What are the singular integrals? Singular integral equations are presently encountered in a wide range of mathematical models, for instance in acoustics, fluid dynamics, elasticity and fracture mechanics. Together with these models, a variety of methods and applications for these integral equations has been developed. In this paper, we will give the interpretation for the Hadamard finite part of singular integrals by means of the division by zero calculus.
[128]
vixra:1903.0409 [
pdf]
Soft and Hard Liberation of Compact Lie Groups
We investigate the liberation question for the compact Lie groups, by using various ``soft'' and ``hard'' methods, based respectively on joint generation with a free quantum group, and joint generation with a free torus. The soft methods extend the ``easy'' methods, notably by covering groups like $SO_N,SU_N$, and the hard methods partly extend the soft methods, notably by covering the real and complex tori themselves.
[129]
vixra:1903.0371 [
pdf]
Division by Zero Calculus in Multiply Dimensions and Open Problems
In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from the one dimensional case.
[130]
vixra:1902.0223 [
pdf]
Horn Torus Models for the Riemann Sphere and Division by Zero
In this paper, we will introduce a beautiful horn torus model by Puha and D\"aumler for the Riemann sphere in complex analysis attaching the zero point and the point at infinity. Surprisingly enough, we can introduce analytical structure of conformality to the model.
[131]
vixra:1901.0341 [
pdf]
Necessary and Sufficient Conditions for a Factorable Matrix to be Hyponormal
Necessary and sufficient conditions are given for a special subclass of the factorable matrices to be hyponormal operators on $\ell^2$. Three examples are then given of polynomials that generate hyponormal weighted mean operators, and one example that does not. Paired with the main result presented here, various computer software programs may then be used as an aid for classifying operators in that subclass as hyponormal or not.
[132]
vixra:1812.0345 [
pdf]
New Equations of the Resolution of The Navier-Stokes Equations
This paper represents an attempt to give a solution of the Navier-Stokes equations under the assumptions (A) of the problem as described by the Clay Mathematics Institute. After elimination of the pressure, we obtain the fundamental equations function of the velocity vector u and vorticity vector \Omega=curl(u), then we deduce the new equations for the description of the motion of viscous incompressible fluids, derived from the Navier-Stokes equations, given by: \nu \Delta \Omega -\frac{\partial \Omega}{\partial t}=0 \Delta p=-\sum^{i=3}_{i=1}\sum^{j=3}_{j=1}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i} Then, we give a proof of the solution of the Navier-Stokes equations u and p that are smooth functions and u verifies the condition of bounded energy.
[133]
vixra:1812.0321 [
pdf]
Positivity of the Fourier Transform of the Shortest Maximal Order Convolution Mask for Cardinal B-splines
Positivity of the Fourier transform of a convolution mask can be used to define an inverse convolution and show that the spatial dependency decays exponentially. In this document, we consider, for an arbitrary order, the shortest possible convolution mask which transforms samples of a function to Cardinal B-spline coefficients and show that it is unique and has indeed a positive Fourier transform. We also describe how the convolution mask can be computed including some code.
[134]
vixra:1812.0178 [
pdf]
The Zeta Induction Theorem: The Simplest Equivalent to the Riemann Hypothesis?
This paper presents an uncommon variation of proof by induction. We call it deferred induction by recursion. To set up our proof, we state (but do not prove) the Zeta Induction Theorem. We then assume that theorem is true and provide an elementary proof of the Riemann Hypothesis (showing their equivalence).
[135]
vixra:1811.0496 [
pdf]
Dieudonné-Type Theorems for Lattice Group-Valued K-Triangular Set Functions
Some versions of Dieudonne-type convergence and uniform boundedness theorems are proved, for k-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case.
[136]
vixra:1811.0244 [
pdf]
Remark on the paper of Zheng Jie Sun and Ling Zhu
In this short review note we show that the new proof of theorem 1.1 given by Zheng Jie Sun and Ling Zhu in the paper Simple proofs of the Cusa-Huygens-type and Becker-Stark-type inequalities is logically incorrect and present another simple proof of the same.
[137]
vixra:1811.0222 [
pdf]
Real Numbers in the Neighborhood of Infinity
We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals, we prove that numbers in the neighborhood of infinity are ordinary real numbers of the type detailed in Euclid's Elements. We show that real numbers in the neighborhood of infinity obey the Archimedes property of real numbers. The main result is an application in complex analysis. We show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line in the neighborhood of infinity.
[138]
vixra:1810.0312 [
pdf]
Riemann Integration On R^n
Throughout these discussions the numbers epsilon > 0 and delta > 0 should be thought of as very small numbers. The aim of this part is to provide a working definition for the integral of a bounded function f(x) on the interval [a, b]. We will see that the real number "f(x)dx" is really the limit of sums of areas of rectangles.
[139]
vixra:1810.0308 [
pdf]
Existence of Solutions for a Nonlinear Fractional Langevin Equations with Multi-Point Boundary Conditions on an Unbounded Domain
In this work, we apply the fixed point theorems, we study the existence and uniqueness of solutions for Langevin differential equations involving two ractional orders with multi-point boundary conditions on the half-line.
[140]
vixra:1810.0170 [
pdf]
Existence of Solutions for Langevin Differential Equations Involving Two Fractional Orders on the Half-Line
In this paper, we study the existence and uniqueness of solutions for Langevin differential equations of Riemman-Liouville fractional derivative with boundary value conditions on the half-line. By a classical fixed point theorems, several new existence results of solutions are obtained.
[141]
vixra:1810.0169 [
pdf]
Existence of Solutions for Fractional Langevin Equations with Boundary Conditions on an Infinite Interval
In this paper, we investigate the existence and uniqueness of solutions for the following fractional Langevin equations with boundary conditions $$\left\{\begin{array}{l}D^{\alpha}( D^{\beta}+\lambda)u(t)=f(t,u(t)),\text{ \ \ \ }t\in(0,+\infty),\\ \\u(0)=D^{\beta}u(0)=0,\\ \\ \underset{t\rightarrow+\infty}{\lim}D^{\alpha-1}u(t)=\underset{t\rightarrow+\infty}{\lim}D^{\alpha +\beta-1}u(t)=au(\xi),\end{array}\right.$$ where $1<\alpha \leq2$ and$\ 0<\beta \leq1,$ such that $1<\alpha +\beta \leq2,$ with $\ a,b\in\mathbb{R},$ $\xi \in\mathbb{R}^{+},$\ and $D^{\alpha}$, $D^{\beta }$ are the Riemman-Liouville fractional derivative. Some new results are obtained by applying standard fixed point theorems.
[142]
vixra:1810.0168 [
pdf]
Existence of Solutions for a Class of Nonlinear Fractional Langevin Equations with Boundary Conditions on the Half-Line
In this work, we use the fixed point theorems, we investigate the existence and uniqueness of solutions for a class of fractional Langevin equations with boundary value conditions on an infinite interval.
[143]
vixra:1809.0481 [
pdf]
The Riemann Hypothesis
The Riemann Hypothesis is a famous unsolved problem dating from 1859. This paper will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).
[144]
vixra:1809.0234 [
pdf]
Proof of the Limits of Sine and Cosine at Infinity
We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we will show that the new representation has special properties which allow for a modification to the transformation law for the variation which preserves, in certain cases, the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. We use the modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.
[145]
vixra:1808.0641 [
pdf]
On Some Ser's Infinite Product
I derive some Ser's infinite product for exponential function and exponential of the digamma function; as well as an integral representation for the digamma function.
[146]
vixra:1808.0576 [
pdf]
A Joint Multifractal Analysis of Finitely Many Non Gibbs-Ahlfors Type Measures
In the present paper, new multifractal analysis of vector valued Ahlfors type measures is developed. Mutual multifractal generalizations f fractal measures such as Hausdorff and packing have been introduced with associated dimensions. Essential properties of these measures have been shown using convexity arguments.
[147]
vixra:1808.0116 [
pdf]
Sine Function at Rational Argument, Finite Product of Gamma Functions and Infinite Product Representation
I corrected the Theorem 21 of previous paper, obtaining an identity for sine function at rational argument involving finite sum of the gamma functions; hence, the representation of infinite product arose.
[148]
vixra:1807.0324 [
pdf]
Lyapunov-Type Inequality for the Hadamard Fractional Boundary Value Problem on a General Interval [a;b], (1≤a<b)
In this paper, we studied an open problem, where using two different methods, we obtained several results for a Lyapunov-type and Hartman-Wintner-type inequalities for a Hadamard fractional differential equation on a general interval [a;b],(1≤a<b) with the boundary value conditions.
[149]
vixra:1806.0082 [
pdf]
Derivation of the Limits of Sine and Cosine at Infinity
This paper examines some familiar results from complex analysis in the framework of hypercomplex analysis. It is usually taught that the oscillatory behavior of sine waves means that they have no limit at infinity but here we derive definite limits. Where a central element in the foundations of complex analysis is that the complex conjugate of a C-number is not analytic at the origin, we introduce the tools of hypercomplex analysis to show that the complex conjugate of a *C-number is analytic at the origin.
[150]
vixra:1804.0405 [
pdf]
Mixed Generalized Multifractal Densities for Vector Valued Quasi-Ahlfors Measures
In the present work we are concerned with some density estimations of vector valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures satisfying a weak quasi-Alfors regularity. Mixed multifractal generalizations of densities are then introduced and studied in a framework of relative mixed multifractal analysis.
[151]
vixra:1803.0001 [
pdf]
Expansion Into Bernoulli Polynomials Based on Matching Definite Integrals of Derivatives
A method of function expansion is presented. It is based on matching the definite integrals of the derivatives of the function to be approximated by a series of (scaled) Bernoulli polynomials. The method is fully integral-based, easy to construct and presumably slightly outperforms Taylor series in the convergence rate. Text presents already known results.
[152]
vixra:1802.0126 [
pdf]
A Note on the Possibility of Icomplete Theory
In the paper it is demonstrated that Bells theorem is an unprovable theorem. This inconsistency is similar to concrete mathematical incompleteness. The inconsistency is purely mathematical. Nevertheless the basic physics requirements of a local model are fulfilled.
[153]
vixra:1802.0120 [
pdf]
Analyticity and Function Satisfying :$\displaystyle \ F'=e^{{f}^{-1}}$
In this note we present some new results about the analyticity of the functional-differential equation $ f'=e^{{f}^{-1}}$ at $ 0$ with $f^{-1}$ is a compositional inverse of $f$ , and the growth rate of $f_-(x)$ and $f_+(x)$ as $x\to \infty$ , and we will check the analyticity of some functional equations which they were studied before and had a relashionship with the titled functional-differential and we will conclude our work with a conjecture related to Borel- summability and some interesting applications of some divergents generating function with radius of convergent equal $0$ in number theory
[154]
vixra:1801.0096 [
pdf]
Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters.
[155]
vixra:1712.0539 [
pdf]
Integrals Containing the Infinite Product $\prod_{n=0}^\infty\left[1+\left(\frac{x}{b+n}\right)^3\right]$
We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation $\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$ and to some other formulas.
[156]
vixra:1712.0486 [
pdf]
Finite and Infinite Product Transformations
Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite products are motivated by solution of Dirichlet boundary problem on a rectangular grid. These finite product formulas give an elementary proof of several modular transformations.
[157]
vixra:1712.0478 [
pdf]
Two-Dimensional Fourier Transformations and Double Mordell Integrals
Several Fourier transformations of functions of one and two variables are evaluated andthen used to derive some integral and series identities. It is shown that certain double Mordellintegrals can be reduced to a sum of products of one-dimensional Mordell integrals. As aconsequence of this reduction, a quadratic polynomial identity is found connecting productsof certain one-dimensional Mordell integrals. An integral that depends on one real valuedparameter is calculated reminiscent of an integral previously calculated by Ramanujan andGlasser. Some connections to elliptic functions and lattice sums are discussed.
[158]
vixra:1710.0083 [
pdf]
Non-Standard General Numerical Methods for the Direct Solution of Differential Equations not Cleared in Canonical Forms
In this work I develop numerical algorithms that can be applied directly to differential equations of the general form f (t, x, x ) = 0, without the need to cleared x . My methods are hybrid algorithms between standard methods of solving differential equations and methods of solving algebraic equations, with which the variable x is numerically cleared. The application of these methods ranges from the ordinary differential equations of order one, to the more general case of systems of m equations of order n. These algorithms are applied to the solution of different physical-mathematical equations. Finally, the corresponding numerical analysis of existence, uniqueness, stability, consistency and convergence is made, mainly for the simplest case of a single ordinary differential equation of the first order.
[159]
vixra:1710.0036 [
pdf]
Laws of General Solutions of Partial Differential Equations
In this paper, four kinds of Z Transformations are proposed to get many laws of general solutions of mth-order linear and nonlinear partial differential equations with n variables. Some general solutions of first-order linear partial differential equations, which cannot be obtained by using the characteristic equation method, can be solved by the Z Transformations. By comparing, we find that the general solutions of some first-order partial differential equations got by the characteristic equation method are not complete.
[160]
vixra:1709.0393 [
pdf]
Neumann Series Seen as Expansion Into Bessel Functions J_{n} Based on Derivative Matching.
Multiplicative coefficients of a series of Bessel functions of the first kind can be adjusted so as to match desired values corresponding to a derivatives of a function to be expanded. In this way Neumann series of Bessel functions is constructed. Text presents known results.
[161]
vixra:1709.0304 [
pdf]
Boys' Function Computed by Gauss-Jacobi Quadrature
Boys' Function F_m(z) that appears in the quantum mechanics of Gaussian Type Orbitals is a special case of Kummer's confluent hypergeometric function. We evaluate its integral representation of a product of a power and an exponential function over the unit interval with the numerical Gauss-Jacobi quadrature. We provide an implementation in C for real values of the argument z which basically employs a table of the weights and abscissae of the quadrature rule for integer quantum numbers m <= 129.
[162]
vixra:1705.0410 [
pdf]
New Principles of Differential Equations Ⅰ
This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct.
[163]
vixra:1705.0028 [
pdf]
An Efficient Computational Method for Handling Singular Second-Order, Three Points Volterra Integrodifferenital Equations
In this paper, a powerful computational algorithm is developed for the solution of classes of singular second-order, three-point Volterra integrodifferential equations in favorable reproducing kernel Hilbert spaces. The solutions is represented in the form of series in the Hilbert space W₂³[0,1] with easily computable components. In finding the computational solutions, we use generating the orthogonal basis from the obtained kernel functions such that the orthonormal basis is constructing in order to formulate and utilize the solutions. Numerical experiments are carried where two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. Error estimates are proven that it converge to zero in the sense of the space norm. Several computational simulation experiments are given to show the good performance of the proposed procedure. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to multipoint singular boundary value problems restricted by Volterra operator.
[164]
vixra:1704.0282 [
pdf]
The Asymptotic Behavior of Defocusing Nonlinear Schrödinger Equations
This article is concerned with the scattering problem for the defocusing nonlinear Schrödinger equations (NLS) with a power nonlinear |u|^p u where 2/n < p < 4/n. We show that for any initial data in H^{0,1} x the solution will eventually scatter, i.e. U(-t)u(t) tends to some function u+ as t tends to innity.
[165]
vixra:1703.0295 [
pdf]
Higher Order Derivatives of the Inverse Function
A general recursive and limit formula for higher order derivatives of the inverse function is presented. The formula is next used in couple of mathematical applications: expansion of the inverse function into Taylor series, solving equations, constructing random numbers with a given distribution from uniformly distributed randomnumbers and expanding a function in the neighborhood of a given point in an alternative way to the Taylor expansion.
[166]
vixra:1612.0413 [
pdf]
Area of Torricelli's Trumpet or Gabriel's Horn, Sum of the Reciprocals of the Primes, Factorials of Negative Integers
In our previous work [1], we defined the method for computing general limits of functions at their singular points and showed that it is useful for calculating divergent integrals, the sum of divergent series and values of functions in their singular points. In this paper, we have described that method and we will use it to calculate the area of Torricelli's trumpet or Gabriel's horn, the sum of the reciprocals of the primes and factorials of negative integers.
[167]
vixra:1611.0368 [
pdf]
Infinite Product Representations for Gamma Function and Binomial Coefficient
In this paper, I demonstrate one new infinite product for binomial coefficient and news Euler's and Weierstrass's infinite product for Gamma function among other things.
[168]
vixra:1611.0073 [
pdf]
Coefficient-of-determination Fourier Transform CFT
This algorithm is designed to perform Discrete Fourier Transforms (DFT) to convert temporal data into spectral data. What is unique about this DFT algorithm is that it can produce spectral data at any user-defined resolution; existing DFT methods such as FFT are limited in resolution proportional to the temporal resolution. This algorithm obtains the Fourier Transforms by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate.
[169]
vixra:1611.0049 [
pdf]
In nite Product Representations for Binomial Coefcient, Pochhammer's Symbol, Newton's Binomial and Exponential Function
In this paper, I demonstrate one infinite product for binomial coefficient, Euler's and Weierstrass's infinite product for Pochhammer's symbol, limit formula for Pochhammer's symbol, limit formula for exponential function, Euler's and Weierstrass's infinite product for Newton's binomial and exponential function, among other things.
[170]
vixra:1610.0244 [
pdf]
Operator Exponentials for the Clifford Fourier Transform on Multivector Fields in Detail
In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit i∈C is replaced by a multivector square root of −1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator.
[171]
vixra:1608.0155 [
pdf]
Exact Solutions for Sine-Gordon Equations and F-Expansion Method
A large number of methods have been proposed for solving nonlinear differential equations. The Jacobi elliptic function method and the f-expansion methods are generalizations from a few of them. These methods produce not only single-solitons but also multi-soliton solutions. In this work we applied the f -expansion method and found novel solutions besides those known for three main equations of the kind sine-Gordon: Triple Sine-Gordon (TSG), Double Sine-Gordon (DSG) and Simple Sine-Gordon (SSG).
[172]
vixra:1606.0324 [
pdf]
The Theory of N-Scales
We provide a theory of $n$-scales previously called as $n$ dimensional time scales. In previous approaches to the theory of time scales, multi-dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. $n$-scales make the mathematical structure more flexible and appropriate to real world applications in physics and related fields. Here we define an $n$-scale as an arbitrary closed subset of $\mathbb R^n$. Modified forward and backward jump operators, $\Delta$-derivatives and multiple integrals on $n$-scales are defined.
[173]
vixra:1606.0141 [
pdf]
Some Definite Integrals Over a Power Multiplied by Four Modified Bessel Functions
The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x)K_1^v(x) dx are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are Modified Bessel Functions. There are essentially 15 types of the 4-fold product. Partial integration of each of these types leads correlations between these integrals. The main result are (forward) recurrences of the integrals with respect to the exponent j of the power.
[174]
vixra:1604.0204 [
pdf]
On Zeros of Some Entire Functions
Let \begin{equation*} A_{q}^{(\alpha)}(a;z)=\sum_{k=0}^{\infty}\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}}, \end{equation*} where $\alpha >0,~0<q<1.$ In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire function $A_{q}^{(\alpha)}(a;z)$ are all real and established some results on the zeros of $A_{q}^{(\alpha)}(a;z)$ which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that $A_{q}^{(\alpha)}(q^l;z),~l\geq 2$ has only infinitely many negative zeros that gives a partial answer to Zhang's question. In addition, we establish some results on zeros of certain entire functions involving the Rogers-Szeg\H{o} polynomials and the Stieltjes-Wigert polynomials.
[175]
vixra:1604.0001 [
pdf]
General One-Sided Clifford Fourier Transform, and Convolution Products in the Spatial and Frequency Domains
In this paper we use the general steerable one-sided Clifford Fourier transform (CFT), and relate the classical convolution of Clifford algebra-valued signals over $\R^{p,q}$ with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the CFTs of the factor functions. In full generality do we express the classical convolution of Clifford algebra signals in terms of a linear combination of Mustard convolutions, and vice versa the Mustard convolution of Clifford algebra signals in terms of a linear combination of classical convolutions.
[176]
vixra:1602.0235 [
pdf]
Performances Piecewise Defined Functions in Analytic Form, Prime-Counting Function
The article discusses the representation of discrete functions defined in an analytic form without the use of approximations, namely the Heaviside function, identity function, the Dirac delta function and the prime-counting function.
[177]
vixra:1602.0044 [
pdf]
General Two-Sided Clifford Fourier Transform, Convolution and Mustard Convolution
In this paper we use the general steerable two-sided Clifford Fourier transform (CFT), and relate the classical convolution of Clifford algebra-valued signals over $\R^{p,q}$ with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the CFTs of the factor functions. In full generality do we express the classical convolution of Clifford algebra signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of Clifford algebra signals in terms of finite linear combinations of classical convolutions.
[178]
vixra:1601.0289 [
pdf]
Korovkin-Type Theorems for Abstract Modular Convergence
We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are lter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.
[179]
vixra:1601.0165 [
pdf]
General Two-Sided Quaternion Fourier Transform, Convolution and Mustard Convolution
In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over $\R^2$ with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.
[180]
vixra:1511.0302 [
pdf]
The Quaternion Domain Fourier Transform and its Properties
So far quaternion Fourier transforms have been mainly defined over $\mathbb{R}^2$ as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalarvector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.
[181]
vixra:1511.0247 [
pdf]
General Method for Summing Divergent Series Using Mathematica and a Comparison to Other Summation Methods
We are interested in finding sums of some divergent series using the general method for summing divergent series discovered in our previous work and symbolic mathematical computation program Mathematica. We make a comparison to other five summation methods implemented in Mathematica and show that our method is the stronger method than methods of Abel, Borel, Cesaro, Dirichlet and Euler.
[182]
vixra:1511.0246 [
pdf]
ACourse of Numerical Analysis and Applied Mathematics
It is a course of numerical analysis and applied mathematics for the students of the geomatic and topography option of the ESAT School, Tunis, Tunisia.
[183]
vixra:1511.0077 [
pdf]
Abscissa and Weights for Gaussian Quadratures of Modified Bessel Functions Integrated from Zero to Infinity
We tabulate the abscissae and associated weights for numerical integration of integrals with kernels which contain a power of x and Modified Bessel Functions K_nu(x). The first family of integrals contains the factor x^m K_nu(x) with integer indices nu=0 or 1 and integer powers nu <= m <= 3. The second family of integrals contains the factor x^m K_nu(x)K_nu'(x) with integer indices 0 <= nu, nu' <= 1 and integer powers nu+nu' <= m <= 3.
[184]
vixra:1508.0204 [
pdf]
New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both finite and infinite sums. The high points of this paper are perhaps the discovery of several previously unknown infinite summation results involving {\em non-linear} generalized harmonic number terms and the derivation of interesting alternating summation formulas involving these numbers.
[185]
vixra:1506.0019 [
pdf]
Analysis of Generals Algorithms of Numeric Solutions of Ordinary Differential Equations of Higher Order One with Initial Conditions.
In this paper several existing numerical methods of solution of ordinary differential equations of first order with initial conditions are modified, so that they can be generalized as methods of solution of ordinary differential equations of order n, according to the theory.
[186]
vixra:1503.0032 [
pdf]
New Integral Representation for Inverse Sine Function, the Rate of Catalan's Constant by Archimedes Constant and Other Functions
In present article, we developed infinite series representations for inverse sine function and other functions. Our main goal is to get the hypergeometric representation for Catalan constant and hyperbolic sine function; and new integral representation for inverse sine function.
[187]
vixra:1502.0074 [
pdf]
General Method for Summing Divergent Series. Determination of Limits of Divergent Sequences and Functions in Singular Points
In this work I am going to mention historical development of divergent series theory, and to give a number of different examples, as some of the methods for their summing. After that, I am going to introduce the general method, which I discovered, for summing divergent series, which we can also consider as a method for computing limits of divergent sequences and functions in divergent points, In this case, limits of sequences of their partials sums. Through the exercises, I am going to apply this method on given examples and prove its validity. Then I'm going to apply the method to compute the value of some divergent integrals.
[188]
vixra:1412.0001 [
pdf]
Improvement On Operator Axioms And Fundamental Operator Functions
The Operator axioms have been contructed to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions and pose some conjectures. Real operators can give new equations and inequalities so as to precisely describe the relation of mathematical objects or scientific objects.
[189]
vixra:1409.0125 [
pdf]
Extreme Values of the Sequence of Independent and Identically Distributed Random Variables with Mixed Asymmetric Distributions
In this paper, we derive the extreme value distributions of independent identically distributed random variables with mixed distributions of two and finite components, which include generalized logistic, asymmetric Laplace and asymmetric normal distributions.
[190]
vixra:1408.0084 [
pdf]
Analytic Functions for Clifford Algebras
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.)
[191]
vixra:1407.0169 [
pdf]
New Developments in Clifford Fourier Transforms
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.
[192]
vixra:1404.0072 [
pdf]
On The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions
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.
[193]
vixra:1403.0977 [
pdf]
Solid Angle of the Off-Axis Circle Sector
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.
[194]
vixra:1403.0310 [
pdf]
Operator Exponentials for the Clifford Fourier Transform on Multivector Fields
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.
[195]
vixra:1310.0255 [
pdf]
Demystification of the Geometric Fourier Transforms
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.<BR> <b>Keywords:</b> geometric algebra, Clifford algebra, Fourier transform, trigonometric transform, convolution theorem.
[196]
vixra:1310.0249 [
pdf]
Extending Fourier Transformations to Hamilton’s Quaternions and Clifford’s Geometric Algebras
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.<br> <b>Keywords:</b> Clifford geometric algebra, quaternion Fourier transform, Clifford Fourier transform, Clifford Fourier-Mellin transform, Mulitvector wavepackets, Spacetime Fourier transform.<br> AMS Subj. Class. 15A66, 42A38
[197]
vixra:1310.0248 [
pdf]
The Quest for Conformal Geometric Algebra Fourier Transformations
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.<BR> <b>Keywords:</b> Clifford geometric algebra, Clifford Fourier transform, conformal geometric algebra, horosphere.<BR> AMS Subj. Class. 15A66, 42A38
[198]
vixra:1306.0133 [
pdf]
Tutorial on Fourier Transformations and Wavelet Transformations in Cliord Geometric Algebra
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.
[199]
vixra:1306.0130 [
pdf]
The Clifford Fourier Transform in Real Clifford Algebras
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.
[200]
vixra:1306.0127 [
pdf]
Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions N = 2 (Mod 4) and N = 3 (Mod 4)
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.
[201]
vixra:1306.0126 [
pdf]
Uncertainty Principle for Clifford Geometric Algebras Cl(n,0), N = 3 (Mod 4) Based on Clifford Fourier Transform
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.
[202]
vixra:1306.0096 [
pdf]
Windowed Fourier Transform of Two-Dimensional Quaternionic Signals
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
[203]
vixra:1306.0095 [
pdf]
Clifford Algebra Cl(3,0)-valued Wavelets and Uncertainty Inequality for Clifford Gabor Wavelet Transformation
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.
[204]
vixra:1306.0094 [
pdf]
Clifford Algebra Cl(3,0)-valued Wavelet Transformation, Clifford Wavelet Uncertainty Inequality and Clifford Gabor Wavelets
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.
[205]
vixra:1306.0092 [
pdf]
Two-Dimensional Clifford Windowed Fourier Transform
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.
[206]
vixra:1306.0091 [
pdf]
An Uncertainty Principle for Quaternion Fourier Transform
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
[207]
vixra:1306.0089 [
pdf]
Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl(3,0)
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.
[208]
vixra:1305.0174 [
pdf]
Rational Structure, General Solution and Naked Barred Galaxies
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.
[209]
vixra:1305.0147 [
pdf]
Creator's Standard Equation, General Solution and Naked Barred 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.
[210]
vixra:1305.0094 [
pdf]
The Creator's Equation System Without Composite Functions
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.
[211]
vixra:1305.0082 [
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Numerical Solution of Nonlinear Sine-Gordon Equation with Local RBF-Based Finite Difference Collocation Method
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.
[212]
vixra:1305.0052 [
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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 are very old and are not experts in mathematics. Please help us humans to resolve the question.
[213]
vixra:1304.0158 [
pdf]
Products of Generalised Functions
An elementary algebra of products of generalised functions is constructed. A way of multiplying the defined generalised functions with polynomials is also given. The theory is given for single-variable functions but it can be easily generalised to the multi-variable case.
[214]
vixra:1303.0038 [
pdf]
Gaussian Quadrature of the Integrals Int_(-Infty)^infty F(x) dx / Cosh(x)
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.
[215]
vixra:1303.0013 [
pdf]
Gauss-Laguerre and Gauss-Hermite Quadrature on 64, 96 and 128 Nodes
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.
[216]
vixra:1302.0138 [
pdf]
Integral Mean Estimates for the Polar Derivative of a Polynomial
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.
[217]
vixra:1206.0005 [
pdf]
Fractional Geometric Calculus: Toward A Unified Mathematical Language for Physics and Engineering
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.
[218]
vixra:1203.0078 [
pdf]
Apparent Measure and Relative Dimension
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.
[219]
vixra:1203.0065 [
pdf]
On the Growth of Meromorphic Solutions of a type of Systems of Complex Algebraic Differential Equations
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.
[220]
vixra:1203.0029 [
pdf]
Local Fractional Improper Integral in Fractal Space
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.
[221]
vixra:1202.0015 [
pdf]
Volume of the Off-center Spherical Pyramidal Trunk
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.
[222]
vixra:1008.0025 [
pdf]
Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test
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.
[223]
vixra:0703.0011 [
pdf]
The Total Differential Integral of Calculus
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.