Functions and Analysis

The Fractional Invariance Analysis and Applications

Authors: Theophilus Agama
Comments: 5 Pages.

In this note, we introduce and develop the analysis of the fractional invariance. This analysis is used for estimating the partial sums of arithmetic functions $f:mathbb{N}longrightarrow mathbb{R}$ of the form $sum limits_{substack{nleq xin mathbb{A}}}f(n)$ for $mathbb{A}subseteq mathbb{N}$. This analysis can be applied to a broad class of arithmetic functions.
Category: Functions and Analysis

Representation of an Integral Involving Trigonometric Functions by Triple Integral

Authors: Edigles Guedes
Comments: 3 Pages.

In this paper, we present an integral representation involving trigonometric functions and variable transformation techniques to turn it into a triple integral. The proposed integral is initially simplified using trigonometric identities, so we rewrite the original integral in terms of a three-variable integral representation. The main theorem demonstrates the equivalence between the initial integral and the resulting triple integral, illustrating the applicability of trigonometric identities in calculations of complicated integrals.
Category: Functions and Analysis

Real Constants Index of Derivatives and Integrals

Authors: Juan Elias Millas Vera
Comments: 1 Page.

I taught myself Calculus seriously the last months and I want to share some thoughts about Real constants Calculus with a single variable. It will may be mostly know this results by professionals but, in any case it could be interesting to share my conclusions in a short paper.
Category: Functions and Analysis

Uniform Continuity

Authors: Subrat Kumar Verma
Comments: 7 Pages. CC-BY

This article discusses the notion of uniform continuity, its relation with the derivative of differentiable functions and Lipschitzcontinuity or even more weakly Hölder continuity related in some way to how wildly the function oscillates. It also discusses its connectionwith compactness for the very large general class of functions - continuous functions. Further, a few properties of uniform continuous function especially with regards to unique continuous extension of functions are discussed.
Category: Functions and Analysis

Optimization of Response Curve by Chebyshev Polynomials

Authors: Kang Kum Phyong, Ji Ryong Hwan, Kang Hyok Chol
Comments: 9 Pages.

In control system synthesis, it is interesting to use orthogonal basis functions such as various polynomials and series.However, there are still no examples of applying spectral methods to closed-loop control systems and poor applications to nonlinear plants.In this paper, we proposed a control method of allowing the state of the plant to pass through the desired points that the user suggests subjectively in case a model of the control plant is given and the boundary values at a given time point are given.In other words, to generate the optimal response curve of the Volza problem, we used the standard Chebyshev pseudo-spectral (PS) method, which deals with the state and control of the plant with Chebyshev polynomial approximation, based on which the optimization problem is considered to be a nonlinear programming problem. At that time we obtained the desired control quantity u with the polynomial coefficients and thus implemented the control.In this paper, we demonstrated the practical applicability of this method by showing not only examples of linear plants but also applications of nonlinear plants.
Category: Functions and Analysis

Research on the Control of Time-Varying Systems Using Lattice Matrix Operators and Integral Equations

Authors: Ri Kum Ju, Kang Yu Song, Kang Hyok Chol
Comments: 18 Pages.

In this paper, we propose a response optimization method for time-varying or nonlinear systems using lattice matrix operators.In order to compare the effectiveness and accuracy of this method against previous nonlinear optimal control methods, simulation results for nonlinear plants with time-varying and gap nonlinearity are presented in this paper.This method overcomes the drawbacks of previous controller design methods that have been complicated by the characteristics of the plant and allows easy and general development of controllers by matrix algebraic equations for objects with time-varying or nonlinear or uncertain parameters, which have strong robustness to variations in disturbances, environmental noise and parameters.
Category: Functions and Analysis

A New Method to Estimate the State of Lithium Ion Battery Capacity Using Chaos Optimization-Least Squares Support Vector Machine

Authors: Kang Hun Won, Chol Sik Ryang, Un Jong Hong, Kang Song Ham, Yu Gang Song
Comments: 11 Pages.

In this paper, new algorithm is proposed to estimate the state of lithium ion battery capacity using Chaos optimization-least squares support vector machine. Here ,at first we had set parameters of the least squares support vector machine using Chaos optimization algorithms. Next we had made the regression model of least squares support vector machine using Gaussian kernel function. Also it had estimated the state of lithium ion battery capacity using Chaos optimization-least squares support vector machine.And the exactness on the estimation model of Chaos optimization-least squares support vector machine had verified through the simulation analysis.
Category: Functions and Analysis

A Novel Offline and Online Parameter Identification Technique of Nonlinear Fractional Order Systems Using Approximated Fractional Order Derivative and the Intelligent Optimization Methods

Authors: Chol-Sik Ryang, Kyong-Min Yun, Tae-Song Kwan, Il Song Rim, Hyon Ho Han
Comments: 19 Pages.

This paper makes an accurate fractional model of the existing non-linear systems using fractional order theory and various intelligent optimization methods and proposes a novel method to identify time-varying parameters of the fractional non-linear system offline and online. More accurate mathematical model of the proposed system was made by applying approximated fractional derivative into the state space model of the classical non-linear system. The initial parameter values of the proposed non-linear fractional system were identified offline by using hybrid particle swarm optimization-genetic algorithm method that is a combination of particle swarm optimization(PSO) and genetic algorithm(GA) that are typical intelligent optimization methods. The time-varying parameters of the non-linear fractional order systems were identified online in real-time by using the output error technique and the recursive least square method. In order to verify the efficiency of the proposed identification technique, we made a simulation experiment for offline and online identification of the time-varying parameters in the existing nonlinear fractional Lorentz system and nonlinear fractional lithium-ion battery system. Simulation results show that the proposed novel identification method can be effectively used for offline and online parameter identification of many complicated non-linear fractional order systems in practice.
Category: Functions and Analysis

Image Denoising and Edge Detection Method Using Least Squares Support Vector Machine

Authors: Jo Sok Chol, Ryang Chol Sik, Choe Yu Song, Kim Hyok Jin, U Hyok Chol
Comments: 5 Pages.

This paper proposes a novel denoising and edge detection algorithms for image using least squares support vector machine (LS-SVM) with Gaussian radial basis functions (RBF) kernel. The new filter, called least squares support vector machine filter (LS-SVMF) for image denoising, is based on the general concept of binary filters and machine learning theory. Using the LS-SVM, a set of the new gradient operators and the corresponding second derivative operators are obtained. Computer experiments are carried out for denoising and extracting edge information from real images. The results obtained for the applications show that the proposed algorithms outperform many other existing methods in the image denoising task and the traditional edge detectors. The proposed algorithms can be successfully applied for the processing of images corrupted with impulsive noise while maintaining the visual quality and a low reconstruction error.
Category: Functions and Analysis

Improvement of the Harmonic Disturbance Rejection Performance in the Linear Active Disturbance Rejection Control System Using the m-th Order Extended State Observer with Frequency Estimation

Authors: Tae Hyok Kim, Yun Hak Ri, Yong Ho Kim
Comments: 17 Pages.

This paper proposes a method to improve the disturbance rejection performance and guarantee the high-precision tracking performance in the active disturbance rejection control(ADRC) system using the m-th order extended state observer(ESO) with frequency e.
Category: Functions and Analysis

Integration of Tensor Fields with Angular Components: An Analytical and Computational Study

Authors: Parker Emmerson
Comments: 8 Pages.

This paper presents a mathematical framework for integrating tensor fields with angular components, combining linear and angular integrands to form a comprehensive expression. We focus on the integration over spatial variable ( x_i ) and angular variable ( theta ), deriving a combined integrand that reflects the interplay between these dimensions. The methods are implemented computationally, and the resulting combined integrand is visualized to provide insights into its behavior.
Category: Functions and Analysis

Some Properties of Iterated Brownian Motion and Weak Approximation

Authors: Shumpei Sakuragi
Comments: 25 Pages.

Algorithms by stochastic methods to partial differential equations of the fourth order involving biharmonic operators are stated. The author considered a construction of the solution of a partial differential equation using a certain probability space and stochastic process. There are two algorithms for the fourth-order partial differential equations by stochastic methods. The first one is the method using signed measures. This is a methodwhich constructs a signed measure by a solution using the Fourier transform and obtains a coordinate mapping process. The second method uses iterated Brownian motion. The latter is treated in this paper. The definition ofiterated Brownian motion was modified to investigate the properties of its distribution. The author also defined an iterated random walk corresponding to discretization of that, and showed that it converges to an iteratedBrownian motion in law to the iterated Brownian motion, and obtained its order. In the conventional method, the partial differential equation of the fourth order corresponding to iterated Brownian motion, the Laplacian ofthe boundary condition arises in the remainder term. In other words, if the boundary condition is harmonic, the representation of the partial differential equation involving the biharmonic operator is possible.By focusing on the distribution of the iterated Brownian motion, the representation of the partial differential equation including the biharmonic operator is possible when the boundary condition is biharmonic.
Category: Functions and Analysis

A New Continued Fraction Approximation and Bounds for the Psi Function

Authors: CholBok Ri, Kwang Ri, CholJun Choe
Comments: 10 Pages.

In this paper, we provide some useful lemmas for construction continued fraction based on a given power series. Then we establish a new continued fraction approximation and bounds for the psi function. Especially, we analytically determine all parameters of the continued fraction by Bernoulli numbers.
Category: Functions and Analysis

An Interpretation of Ramanujan Mock θ-Functions

Authors: Juan Elias Millas Vera
Comments: 5 Pages.

In this paper I show a several generalizations of the Ramanujan’s Mock θ-functions. Using sigma and pi operators and defining the series.
Category: Functions and Analysis

An IMC-like Design of LQR-PID Controller Using Optimal Weight of Sensitivity Function

Authors: Hyon Sung-Yun, Sung Chol U, Sung Chol Kim, Myong hyok Sin
Comments: 15 Pages.

Proportional-Integral-Derivative controller is remained as the most widely used controller for many industrial applications even though it was developed decades ago. This is because of its simplicity, satisfactory control performance and robustness. The classical and empirical PID tuning rules are applicable only to the FOPDT and cannot guarantee its optimality. The Ziegler-Nichols frequency response method is regarded as a basis of relay feedback auto-tuning because it uses the only information at the phase crossover frequency. The Ziegler-Nichols tunings are quite aggressive for lag- dominant processes but sluggish for delay-dominant processes. The IMC-PID settings yield good servo performance and robustness, they result in poor LD (load disturbance) rejection for lag-dominant plants. We have proposed a new method of determining LQ index in consideration of dominant pole placement for desired performance and derive a simple LQR-PID tuning formula via IMC-like H∞ approach for first order plus dead-time systems. We have determined the weight for sensitivity function so that LQ optimization is equivalent to the H∞ optimization of the weighted sensitivity function. The proposed PID controller has the same performance as LQR controller and the tuning method is simple, since it does not need to solve the Riccati algebraic equation. We also present other two tuning methods for PID controller: LQR-like and pole-placement-like ones. The new contributions in this paper are: determination of LQ index for dominant poles placement and optimal weight for sensitivity, and derivation of LQR-PID tuning methods via IMC-like approach, LQR and pole-placement approaches. The effectiveness of the proposed methodology and the identity of the PID parameters tuned by those three methods have been demonstrated via simulation.
Category: Functions and Analysis

Global Wellposedness for the Homogeneous Periodic Navier-Stokes Equation Small Initial Data

Authors: En-Lin Liu
Comments: 14 Pages. If there is any error in the proof, please let me know.

We consider the homogeneous incompressible Navier-Stokes equations on periodic domain mathbb{T}^{d} with sufficiently small initial datum. For dgeq3 and sgeqfrac{d}{2}-1, the equations are globally wellposed in the energy space L_{t}^{infty}dot{H}_{x}^{s}left(mathbb{R}^{+};mathbb{T}^{d}ight)cap L_{t}^{2}dot{H}_{x}^{s+1}left(mathbb{R}^{+};mathbb{T}^{d}ight) in the critical sense if the initial data u_{0} is divergence free, mean zero and leftVert u_{0}ightVert _{dot{H}_{x}^{s}left(mathbb{T}^{d}ight)} is sufficiently small. We use Strichartz estimates for the heat kernel, bilinear Strichartz estimates to obtain an iteration scheme critically depending on the value of e^{ttriangle}u_{0} in L_{t}^{2}dot{H}_{x}^{s}left(left[0,Tight];mathbb{T}^{d}ight) norm. Use such iteration scheme, we can prove uleft(tight) is decreasing in dot{H}_{x}^{s}left(mathbb{T}^{d}ight) with time t. The decay property guarantees the global existence and wellposedness.
Category: Functions and Analysis

Identification of Conformable Fractional Order System with Input Delay

Authors: KumSosng Jang, YongKwon Pak, MyongHyok Sin1, NamHo Kim
Comments: 10 Pages.

Classical fractional derivative does not reflect hysteresis characteristic, the unique characteristic of fractional derivative. In order to overcome it, identification approach for the system with input delay expressed by conformable fractional order derivative is proposed. Simulation results shows that the proposed approach infers systematic parameters with high accuracy.
Category: Functions and Analysis

Motivic Operators and M-Posit Transforms on Spinors

Authors: Parker Emerson
Comments: 34 Pages.

Spinor theory and its applications are indispensable in many areas of theoretical physics, especiallyin quantum mechanics, general relativity, and string theory. Spinors are complex objects thattransform under specific representations of the Lorentz or rotation groups, capturing the intrinsicspin properties of particles. Recent developments in mathematical abstraction have provided newinsights and tools for exploring spinor dynamics, particularly through the lens of motivic operatorsand M-Posit transforms.This paper delves into the intricate dynamics of spinors subjected to motivic operators and MPosit transforms. Motivic operators encapsulate intrinsic algebraic properties and perturbations,leading to highly evolved spinor states without reliance on external coordinate systems. The M-Posittransform, a novel operator designed for spinors, leverages fractal morphic properties, topologicalcongruence, and quantum-inspired perturbations to manipulate spinor structures within an infinitedimensional oneness geometry calculus.Drawing on the foundations laid by twistor theory, we aim to redefine the evolution of spinorsusing intrinsic properties derived from phenomenological velocity equations. By interpreting spinorsas self-propelled twistors, we offer new perspectives on spinor transformations and dynamics. Thisintrinsic approach not only simplifies the mathematical treatment but also enhances the physicaland geometric interpretation of spinor behaviors.The structure of this paper is organized as follows: We begin with the formal definition andcomputation of spinor components using motivic operators, highlighting the steps involved in theirtransformations. Following this, we introduce the M-Posit transform and explore its applicationto spinors, providing detailed mathematical formulations and examples. We also examine theimplications of these transformations in higher-dimensional twistor spaces and non-commutativestructures. Finally, we extend our analysis to practical applications in quantum computing, fractalimage processing, and quantum field theory.The potential of spinning theory redefined through motivic operators and M-posit transformsoffers promising avenues for further research in various domains of theoretical physics and mathematics. This paper sets a foundation for these explorations, emphasizing the importance of intrinsicproperties and algebraic dynamics in understanding complex spinor evolutions.
Category: Functions and Analysis

An Extension of the Mollifier

Authors: Bin Wang
Comments: 10 Pages.

We extend the convergence for mollifiers to that for differential forms of arbitrary degrees.
Category: Functions and Analysis

New Continued Fraction Approximations for the Gamma Function Based on the Tri-Gamma Function

Authors: SonUng Hong, Kwang Ri, CholRyong Kim
Comments: 10 Pages.

In this paper, we provide some useful lemmas to construct continued fraction based on a given power series. Then we establish new continued fraction approximations for the gamma function via the Tri-gamma function. Especially, we analytically determine all parameters of the continued fraction by Bernoulli numbers.
Category: Functions and Analysis

Numerical Evaluation of Three Integrals of the Kind Int_0^oo X^m Dx/(1+x^n*sin^2 x)

Authors: Richard J. Mathar
Comments: 11 Pages.

Definite integrals along the real axis from zero to infinity with functions with denominator 1+x^n*sin^2 x suffer from dominant peaks at all x-values that are close to Pi, which impedes sampling the function with generic discrete numerical methods. We demonstrate the method of integrating along a closed contour around a circular sector in the complex x-plane and collecting the sum of all (infinitely many) residues inside the sector with an adapted series acceleration.
Category: Functions and Analysis

Vector Product Approach for Optimization Resolution

Authors: Mamadou Ndao
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references)

This document presents a new method for solving constrained optimization problems, an alternative to the Lagrange multipliers. We introduce the Vector Product Approach for Optimization Resolution (VPAOR), which uses properties of vector products to simplify optimization problems. Our results demonstrate that this approach is an alternative to traditional methods, offering an effective solution for various problems.
Category: Functions and Analysis

A Fourier Derivative Collocation Method for the Solution of the Navier—Stokes Problem

Authors: Daniel Thomas Hayes
Comments: 6 Pages.

A proposed solution to the millennium problem on the existence and smoothness of the Navier—Stokes equations.
Category: Functions and Analysis

Convergence of a Rather Particular Series

Authors: Mame Goumba Amar, Aba Lo Amar
Comments: 2 Pages.

This article aims to examine the convergence of a particular series. This series is defined by a sequence governed by a recurrence relation which we will analyze in detail. Its specificity lies in the fact that it is made up of blocks whose terms are selected according to a jump pattern, which accentuates its elegance. The establishment of the described form gives this series remarkable and distinctive properties. We will thus explore the convergence of this series and the interesting relationships which link the blocks together.
Category: Functions and Analysis

Connected Prime Digit Factorial Occurrences to The Riemann Zeta Function

Authors: Parker Emmerson
Comments: 8 Pages.

This paper explores the theoretical relationship between the frequency of prime digits in factorial representations and the non-trivial zeros of the Riemann Zeta function. By defining the prime digit frequency within fac- torials and aggregating these frequencies, we propose a hypothesis where such aggregated prime digit frequencies exhibit periodic patterns that mirror the distribution of the non-trivial zeros of the Riemann Zeta func- tion. Utilizing Fourier transform analysis, we identify periodic compo- nents in the digit frequencies that may correspond to these zeros. Sta- tistical tests, including Chi-Squared and Kolmogorov-Smirnov tests, are employed to validate this connection. This study suggests that the nature of prime digit frequencies in number sequences, such as factorials, may reflect deeper mathematical structures influenced by the Riemann Zeta functions zeros.
Category: Functions and Analysis

Complete and Rigorous Proof of the Collatz Conjecture

Authors: Hans Rieder
Comments: 101 Pages. In German

In this article, we present a complete and rigorous proof of the Collatz Conjecture. The conjecture states that for any natural number, the sequence generated by repeated application of the Collatz function will eventually reach the number 1. We use mathematical induction to confirm the validity of the conjecture for all natural numbers.
Category: Functions and Analysis

The Nature of Semi-Exponentials

Authors: Warren D. Smith
Comments: 9 Pages.

A "semi-exponential" is a function F(z) such that F(F(z))=exp(z).We show that(a) no entire-analytic semi-exponential F(z) exists;(b) no semi-exponential F(z) exists that is analytic within any interior-connected domainthat includes both the real axis, and all complex Q obeying Q=exp(Q), in its interior, and whichmaps reals→reals;(c) Analytic semiexponentials do exist that map most reals to complex numbers and which have non-analytic points;(d) We also construct a useful piecewise-analytic real→real semi-exponentialsuch that F, F', and F'' all are continuous,and F(x) is strictly increasing and strictly concave-∪, for all real x;and indeed the domain of definition of this F(z) may beslightly expanded to a long and thin complex set that includes the real axis in its interior,albeit then F becomes discontinuous at an infinite set of nonreal points.(e) But we show that no piecewise-analytic, with piece boundaries being nonemptyrectifiable differentiable curves, semi-exponentialthat maps reals→reals can be defined within any domain that includes thestrip 0≤im(z)<π.Many of our arguments may be repurposed for many other "semi" functions besides the exponential.Finally (f) we show that a real-valuedC^∞-smooth,strictly increasing, strictly concave-∪semi-exponential exists, which under certain asymptotic analyticity demands is unique.
Category: Functions and Analysis

The Proof of the Riemann Hypothesis

Authors: Zuher El Ahmed
Comments: 5 Pages.

The problem of finding the zeros of functions is one of the important issues in mathematics, and I would not be exaggerating if I said that all of mathematics is based on such problems. Here, curiosity struck me to understand the mechanism or the secret behind these functions. I never expected that this curiosity would lead me to encounter an important function like the Riemann zeta function, starting from the Taylor and Maclaurin series, which at least enabled me to find a function that links the point belonging to a certain domain and the values of this domain with an exponential function, as demonstrated in the proof. In conclusion, I believe that if this function cannot find the zeros of the Riemann zeta function, it will at least allow us to look at the zeta function from another perspective that is easier to deal with.
Category: Functions and Analysis

Sliding Discrete Fourier Transform of Windowed Samples

Authors: Jon Un Song, Kwon Ryong Il
Comments: 8 Pages.

In general the sliding discrete Fourier transform (SDFT) for analyzing the frequency characteristics of shift data is not applied to the windowed shift data and so in this case a lot of calculations are needed. But in practical applications calculating the spectrum of the windowed shift data efficiently is needed.This paper proposes a SDFT of the windowed shift data using the window functions. Window functions have the symmetric property. Using this symmetric property, discrete Fourier transform (DFT) of the windowed shift data can be calculated by recursive algorithm.And this paper verifies its correctness and analyzes the number of calculations through the simulation using MATLAB. Several window functions, i.e. Bartlett, hanning, hamming, Blackman window functions are used to prove the proposed algorithm. These algorithms can be efficiently used in the analysis of frequency characteristics of the shift data and implemented using dsPIC, ARM and FPGA.
Category: Functions and Analysis

Generalisation of Lim(x→0)u2061(x/sinu2061x)=1

Authors: Yudai Sakuma
Comments: 4 Pages.

It is known that most of the formulae that hold for ordinary trigonometric functions hold for generalised trigonometric functions. In this study, we succeeded in generalizing lim(x→0)u2061(x/sinu2061x)=1 . This makes it possible to discuss the generalised case in unsolved problems involving trigonometric functions, such as the generalisation of the Flint Hills series.
Category: Functions and Analysis

Effectiveness of Navier-Stokes Equations in Modeling Turbulence or Chaos

Authors: Bertrand Wong
Comments: 2 Pages.

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. Although they are relatively simple-looking, the three-dimensional Navier-Stokes equations misbehave very badly. Even with nice, smooth, reasonably harmless initial conditions, the solutions could wind up being extremely unstable. The field of fluid mechanics would be dramatically altered through a mathematical understanding of the outrageous behavior of these equations. The three-dimensional Navier-Stokes equations are apparently not solvable, i.e., the equations cannot be used to model turbulence or chaos (which is a three-dimensional phenomenon).
Category: Functions and Analysis

Particle Swarm Optimization Algorithm Using Exponential Function Way-Decreased Inertia Weight

Authors: Rim Ung Jang, Yong Chon Jang, Se Yong Chon, Hak Mun Kim. Song Hak Hong
Comments: 12 Pages.

In this article we assumed that during the particle swarm optimization (PSO)process, the inertia weight value of the velocity vector calculating equation would be changed by non-liner way. And also this way reflects PSO’s real nature very well. The inertia weight factor’s non-liner-changed equation that is proposed is the flowing []. This equation is an exponential function.
Category: Functions and Analysis

An Integral Collocation Method

Authors: Daniel Thomas Hayes
Comments: 4 Pages.

A new method is developed of which is applied to a problem involving a 1D wave equation in disguise.
Category: Functions and Analysis

Cobweb Model with Conformable Fractional Derivative in Liouville-Caputo Sense

Authors: Hyon Sung-Yun, Kwang Min-Sok, Myong Hyok-Sin1, Nam Ho-Kim
Comments: 12 Pages.

In this paper, we formulate a continuous-time cobweb model expressed as a conformable fractional derivative in Liouville-Caputo sense, and a continuous-time cobweb model expressed as a beta-type conformable fractional derivative in Liouville-Caputo sense, and obtain an analytical solution of this model and analyze the properties of the solution.We also compare the results of the previous cobweb model solutions with several examples.
Category: Functions and Analysis

A New Continued Fraction Approximation for the Lugo and Euler—Mascheroni Constants

Authors: Sin Ryu Song, Ri Kwang, Yun Chol
Comments: 9 Pages.

In this paper, we provide a remarkable method for construction of continued fraction based on a given power series. Then we establish a new continued fraction approximation for the Lugo and Euler—Mascheroni constants. Especially, we analytically determine the coefficients of the Lugo’s asymptotic formula and all parameters of the continued fraction by Bernoulli numbers.
Category: Functions and Analysis

A Software Reliability Growth Model Considering Imperfect Debugging and Disagreement Between Operation Environments

Authors: Ji Won Pak, Kwang Chol Kim, Kwang Song Han
Comments: 15 Pages.

Many software reliability growth models are proposed to be used in practice. However, most software reliability growth models suffer in the realistic software testing environment due to the unrealistic assumptions, such as perfect debugging, constant fault detection rate and regular changes. In fact, considering more reasonable assumptions in the reliability modeling may further improve the fitting and predictive power of software reliability growth models. It is affected by many factors, such as tester’s skill, test plans, testing tools and runtime environment. Thus, software debugging is an imperfect process. And software testing for getting fault data set is done under the assumption that user’s operation environment is the same as the testing one. However, in practice, it is exactly the same. This paper deals with a software reliability growth model which considers imperfect debugging and disagreement between operation environments. The better performance of proposed model is illustrated with fault data sets from software development project.
Category: Functions and Analysis

About an Equation with Radicals

Authors: Edgar Valdebenito
Comments: 3 Pages.

In this note we solve an equation with radicals and give two series for Pi.
Category: Functions and Analysis

Existence and Smoothness of Solutions to the Navier-Stokes Equations Using Fourier Series Representation

Authors: Biruk Alemayehu Petros
Comments: 11 Pages. This is continuation of published result.

This paper presents an analytic solution to the Navier-Stokes equations for incompressible fluid flow with a periodic initial velocity vectorfield. Leveraging Fourier series representations, the velocity fields are expressed as expansions, accounting for their temporal evolution. Thesolution’s existence and smoothness are verified by demonstrating its consistency with the Navier-Stokes equations, including the incompressibilitycondition and pressure compatibility. The proposed solution contributes to understanding fluid dynamics and offers insights into the millennium prize problem related to the Navier-Stokes equations. This work lays the groundwork for further investigations into fluid flow behavior under various conditions and geometries, combininganalytical and numerical approaches to advance our understanding of fluid dynamics.
Category: Functions and Analysis

Riemann Sums of Sin X and Cos X

Authors: Pawel Piskorz
Comments: 4 Pages.

Riemann integrals of the trigonometric functions sin x and cos x have been computed directly from the appropriate Riemann sums.
Category: Functions and Analysis

Proof of Convergence of the Fourier Series

Authors: Pawel Piskorz
Comments: 6 Pages.

Absolute and uniform convergence of any Fourier series has been proven using integration with substitution of variables and limits and the Dirichlet integral value. Our proof of the convergence of the Fourier series requiresdirect computations.
Category: Functions and Analysis

Investor Satisfaction Through the Service Quality Dimensions of Brokerage Firms of Pokhara

Authors: Bidhan Adhikari, Bhumika Gurung, Subash Khatri
Comments: 4 Pages.

The brokerage industry plays a crucial role in the capital market, acting as an intermediary for the buying and selling of securities. This study examines the service quality dimensions provided by brokerage firms in Pokhara, Nepal, and their impact on investor satisfaction. The study focuses on three independent variables: reliability, responsiveness, and assurance, and their relationship with the dependent variable, investor satisfaction.The study utilized primary data collected through a survey of 105 respondents from various age groups, academic qualifications, occupations, and levels of experience. The responses were measured on a five-point Likert scale, and statistical analyses, including mean, standard deviation, correlation, and regression, were conducted using SPSS software.The descriptive analysis revealed that all the independent variables, except for the statement "The service is not delayed," had a mean value greater than 3, indicating that they are strong predictors of investor satisfaction. The correlation analysis showed a positive relationship between reliability, responsiveness, and assurance of service delivery with investor satisfaction in Pokhara valley.The findings suggest that service quality and investor satisfaction are interrelated, with higher service quality leading to higher investor satisfaction. The study also identified assurance of service delivery as the most important predictor of investor satisfaction in this context.
Category: Functions and Analysis

Miscellaneous Summation, Integration, and Transformation Formulas

Authors: Martin Nicholson
Comments: 16 Pages.

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $sum_{ninmathbb{Z}} c_ne^{pi i gamma n^2}$; Fusion of integrals, and in particular fusion of $q$-beta integrals related to Gauss-Fourier transform, and a related family of eigenfunctions of the cosine Fourier transform; Summation formulas of the type $sum_{nge 1}frac{chi(n)}{n},varphi(n)$ with Dirichlet characters; Trigonometric Fourier series expansion of hypergeometric functions of the argument $sin^2x$; Modifications of the inverse tangent integral and identities for corresponding infinite products.
Category: Functions and Analysis

Formulae of Vector Analysis

Authors: A. V. Serghienko
Comments: 3 Pages.

We derive the formulae for the sine and the cosine of the sum, not using the notions of scalar and vector products, and using only the definitions of the sine and the cosine. We derive the formulae for the gradient operator, the divergence and the Laplace operator in different orthogonal coordinate systems, not using any additional constructions like Lame coefficients, and using only the definitions of the sine and the cosine.
Category: Functions and Analysis

Advanced Continued Fraction Approximations and Bounds for the Gamma Function and the Generalized Wallis

Authors: YunJong Kang, HyonChol Kim HyonChol Kang, Kwang Ri
Comments: 26 Pages.

In this paper, we provide a main method for construction of continued fraction based on a given power series using Euler connection. Then we establish very innovative results in continued fraction approximation for the Gamma function as applications of our method. Also new continued fraction bounds for the Gamma function are obtained. Finally new continued fraction approximations and bounds for Wallis ratio are established.
Category: Functions and Analysis

Continued Fraction Approximation and Bounds for the Psi Function

Authors: Pak SongBom, Han CholSok, Kim HyonChol
Comments: 11 Pages.

In this paper, we provide a new continued fraction approximation for the psi function. Then we establish continued fraction bounds for the psi function.
Category: Functions and Analysis

Unification of the Hiperoperators Theory and the Serial Operators Theory

Authors: Juan Elias Millas Vera
Comments: 4 Pages.

In this paper I combined all the hiperoperations theory with the theory of series of functions developing new notation when it was necessary.
Category: Functions and Analysis

Zernike Expansion of Chebyshev Polynomials of the First Kind

Authors: Richard J. Mathar
Comments: 8 Pages.

The even Chebyshev Polynomials T_i(x) can be expanded into sums of even Zernike Polynomials R_n^0(x), and the odd Chebyshev Polynomials can be expanded into sums of odd Zernike Polynomials R_n^1(x). This manuscripts provides closed forms for the rational expansion coefficients as products of Gamma-functions of integer and half-integer arguments.
Category: Functions and Analysis

A proof of the Kakeya Maximal Function Conjecture Via Big Bush Argument

Authors: Johan Aspegren
Comments: 4 Pages.

In this paper we reduce the Kakeya maximal function conjecture to the tube sets of unit measure. We show that the Kakeya maximal function is essentially monotonic. So by adding tubes we can reduce the conjecture to the case of unit measure tube set if we allow the technicality that there are possibly two tubes on the same direction. Then we proof by Kakeya maximal function conjecture by a density argument.
Category: Functions and Analysis

A New Numerical Interpretation of the Concept of Exponentory (θ Notation)

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper I show a possible change in the theory of series beyond product. Instead of a resolution Bottom-to-Top we will see a necessary application of the method for exponents that is a process Top-to-Bottom. That implies a change in the numerical results in a same proposition of a series.
Category: Functions and Analysis

Simple, Mysterious, New Type and Best Possible Integral Inequalities

Authors: Saburou Saitoh
Comments: 7 Pages.

In this note, we give simple, mysterious, new type and best possible integral inequalities.
Category: Functions and Analysis

A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-Hydrodynamic Boundary Value Problems

Authors: SongBom Kim, KwangJin Ro, MyongHyok Sin
Comments: 14 Pages.

In this paper, we have studied the weak Galerkin finite element method for the incompressible viscous Magneto-hydrodynamic(MHD) equations.A weak Galerkin finite element methods are based on new concept called discrete weak gradient, discrete weak divergence and discrete weak rotation, which are expected to play an important role in numerical methods for magneto-hydrodynamic equation.This article intends to provide a general framework for managing differential, divergence, rotation operators on generalized functions. With the proposed method, solving the magneto-hydrodynamic (MHD) equation is that the classical gradient, divergence, rotation operators are replaced by the discrete weak gradient, divergence, rotation and apply the Galerkin finite element method. It can be seen that the solution of the weak Galerkin finite element method is not only continuous function but also totally discontinuous function. For the proposed method, optimal order error estimates are established in various norms.
Category: Functions and Analysis

Is the Field of Real Numbers Really Complete?

Authors: Yinghao Luo
Comments: 3 Pages.

The definition of a limit can only be applied to real numbers rather than infinity, and infinity is independent, necessary and important, so the definition of the limit is incomplete. Based on the incomplete definition of the limit, we can rigorously conclude that the field of real numbers is complete. However, from the point of view that the set of real numbers and the open interval (0, 1) are topologically equivalent but the (0, 1) interval is not a complete space, the completeness of the field of real numbers is inconsistent. We can give a complete definition of a limit through revision. According to the revised definitions, we can rigorously deduce that the field of real numbers is incomplete.
Category: Functions and Analysis

On a Solution of the Inverse Spectral Problem for Differential Operators on a Finite Interval with Complex Weights

Authors: V. A. Yurko
Comments: 7 Pages.

Non-self-adjoint second-order ordinarydifferential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established andthe inverse problem of recovering operators from their spectral characteristics are investigated. For this class of nonlinear inverse problems an algorithm for constructing the global solution is obtained. To study this class of inverse problems, we develop ideas of the method of spectral mappings.
Category: Functions and Analysis

Proof of the Reimann Hypothesis

Authors: Ayoub Zaroual
Comments: 38 Pages. (Correction made by viXra Admin to conform with the requirements of viXra.org)

This article shows that zeta function is a spiral on the complexe plane, then based on the general equation of the spiral we define an analytic continuation for zeta function and finally we prove that Reimann hypothesis is true.
Category: Functions and Analysis

Spiral of the Riemann Zeta Function

Authors: Ayoub Zaroual
Comments: 36 Pages. In French

This paper proves that the Reimann zeta function on the complexe plane is a spiral, it shows the equation of the spiral and provide an analytic continuation of the Reimann zeta function. It also proves that the Reimann hypothesis is true based on some conditions.
Category: Functions and Analysis

A Convergent Subsequence of Theta_n(x+iy) in a Half Strip

Authors: Young Deuk Kim
Comments: 7 Pages.

For $frac{1}{2}0$ and $ninmathbb{N}$, let $displaystyletheta_n(x+iy)=sum_{i=1}^nfrac{{mbox{sgn}}, q_i}{q_i^{x+iy}}$,where $Q={q_1,q_2,q_3,cdots}$ is the set of finite product of distinct odd primes and${mbox{sgn}}, q=(-1)^k$ if $q$ is the product of $k$ distinct primes. In this paper we prove that there exists an ordering on $Q$ such that $theta_n(x+iy)$ has a convergent subsequence.
Category: Functions and Analysis

Calculus and Applications

Authors: Teo Banica
Comments: 400 Pages.

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.
Category: Functions and Analysis

Complete Integers: Extending Integers to Allow Real Powers Have Discontinuities in Zero

Authors: Davide Peressoni
Comments: 8 Pages.

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).
Category: Functions and Analysis

Homogenization of the First Initial-Boundary Value Problem for Periodic Hyperbolic Systems. Principal Term of Approximation

Authors: Yulia Meshkova
Comments: 17 Pages.

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 $(partial _t^2+A_{D,varepsilon})mathbf{u}_varepsilon =mathbf{F}$.
Category: Functions and Analysis

Quadratic Phase Quaternion Domain Fourier Transform

Authors: Eckhard Hitzer
Comments: 12 Pages. Published in: Bin Sheng, Lei Bi, J inman Kim, Nadia Magnenat Thalmann, Daniel Thalmann (eds), Advances in Computer Graphics. CGI 2023. LNCS, vol 14498. Springer, Cham, First Online: 24 Dec. 2023. https://doi.org/10.1007/978-3-031-50078-7_21

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.
Category: Functions and Analysis

On Wilker-Type Inequalities

Authors: Yi-Chieh Huang, Li-Chang Hung
Comments: 6 Pages.

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.
Category: Functions and Analysis

On Generalized li-Yau Inequalities

Authors: Li-Chang Hung
Comments: 11 Pages.

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.
Category: Functions and Analysis

The Infinite Series on the Lviv Scottish Book is Bounded

Authors: Amine Oufaska
Comments: 2 Pages.

In this article we prove that the infinite series on the Lviv Scottish book is bounded , consequently it is convergent.
Category: Functions and Analysis

Extension Formulas and Norm Inequalities in Sobolev Hilbert Spaces

Authors: Saburou Saitoh
Comments: 11 Pages.

In this note, we shall consider extension formulas and norm inequalities for some typical Sobolev Hilbert spaces. We see many related openproblems.
Category: Functions and Analysis

Roots of Real Polynomial Functions and of Real Functions

Authors: Juan Jorge Isaac Lopez
Comments: 4 Pages.

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.
Category: Functions and Analysis

Inversions (Mirror Images) With Respect to the Unit Circle and Division by Zero

Authors: Saburou Saitoh
Comments: 6 Pages.

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: $1/0=0/0=0$.
Category: Functions and Analysis

Infinity Tensors, the Strange at Tractor, and the Riemann Hypothesis: An Accurate Rewording of The Riemann Hypothesis Yields Forma L Proof

Authors: Parker Emmerson
Comments: 13 Pages.

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.
Category: Functions and Analysis

Change of Variable in Integration

Authors: Jouni Puuronen
Comments: 19 Pages.

We find one proof for one form of the change of variable in integration result with Lebesgue integrals.
Category: Functions and Analysis

Common Points of Parallel Lines and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 4 Pages.

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.
Category: Functions and Analysis

Calculation of the Mathematical Expectancy of Cauchy's Law and Extension to Other Improper Integrals

Authors: André Pérennec, Gilles Burel
Comments: 2 Pages.

It is well known that the calculation of the mathematical expectancy of Cauchy'slaw in probability generates an indeterminate form. We show here that this indeterminacy canbe lifted and the calculation leads to a fixed value. Moreover, we show that other improperintegrals with an indeterminate result can be computed.
Category: Functions and Analysis

Embedding of Octonion Fourier Transform in Geometric Algebra of R^3 and Polar Representations of Octonion Analytic Signals in Detail

Authors: Eckhard Hitzer
Comments: 17 Pages. 3 tables, 1 figure. Published in Mathematical Methods in the Applied Sciences, 2023. DOI: 10.1002/mma.9639.

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.
Category: Functions and Analysis

Basic Non-Archimedean Functional Analysis Over Non-Archimedean Field ^{∗}ℝ_{c}^{}

Authors: Jaykov Foukzon
Comments: 35 Pages.

Definitions and theorems related to non-Archimedean functional analysis on non-Archemedean field ┊^{∗}ℝ_{c}^{}┊ and on complex field ┊^{∗}ℂ_{c}^{}┊=┊^{∗}ℝ_{c}^{}┊+i┊^{∗}ℝ_{c}^{}┊areconsidered.Definitions and theorems appropriate to analysis on non-Archemedean field ┊^{∗}ℝ_{c}^{}┊ and on complex field ┊^{∗}ℂ_{c}^{}┊=┊^{∗}ℝ_{c}^{}┊+i┊^{∗}ℝ_{c}^{}┊are given in [1]-[2]
Category: Functions and Analysis

Correcting Rudin's Perfect Sets Are Uncountable Proof

Authors: Timothy W. Jones
Comments: 4 Pages.

It is fascinating fact that the reals are uncountably infinite. Usually Cantor's diagonal method is used to show this. Rudin gives a second proof that promises to be more rigorous than this method. But his proof is a little confusing, if not incorrect. His proof does not stipulate that the perfect set be bounded, but its proof hinges on a local, bounded phenomenon. We duplicate Rudin's proof and argue using two examples that assuming any indexing scheme for the presumed countable set can't work. We then give two proofs: one re-indexes points and the other indexes in the course of the proof.
Category: Functions and Analysis

New Series and Concise Algorithm for "Lambert W Function"

Authors: Warren D. Smith
Comments: 2 Pages.

We give a new series expression, and code up a concise algorithm,for the "Lambert W function" W(X) such that We^W=X with W≥-1.
Category: Functions and Analysis

The OPi Transform and Applications

Authors: Oussama Basta
Comments: 3 Pages. First draft

Abstract- The OPi Transform is a mathematical concept that utalizes the equation f(x) = ln|sec(1/(6x^2) + 1/(4x))| and its counterpart f(x) = ln|sin(1/(6x^2) + 1/(4x))|. In both cases, the equation f(x) equals zero (f(x) = 0) for certain values of n that can be represented as m + ik, where m + ik are known as OPi prime numbers. These prime numbers are complex numbers and exhibit unique divisibility properties, being divisible only by themselves, 1, and i. The OPi Transform serves as a generalization of the Laplace transform and is specifically designed to handle nonlinear functions. By exploring the properties and characteristics of OPi prime numbers and employing the OPi Transform, these mathematical concepts offer a deeper understanding of the equations and provide tools for analyzing and manipulating nonlinear functions with complex numbers.
Category: Functions and Analysis

New Improved Classes Converging Towards the Generalized Euler-Mascheroni Constant

Authors: YunJong Kang, HyonChol Kim, JinSong Kim, JinSong Yu
Comments: 5 Pages.

In this paper, we provide new quicker sequences convergent to the generalized Euler-Mascheroni constant, which is a generalization of the Euler-Mascheroni constant.
Category: Functions and Analysis

On New Type of Sequences Convergent to the Euler-Mascheroni Constant

Authors: ChungIl Kim, HyonChol Kim, JinSong Yu
Comments: 5 Pages.

In this paper, we present a new sequence that converges to the Euler constant. We use the Cramer’s rule to determine the best possible constants of this sequence.
Category: Functions and Analysis

Euler-Gamma Function

Authors: Edgar Valdebenito
Comments: 2 Pages.

The gamma function was first introduced by the Swiss mathematician Leonhard Euler (1707-1783) in his goal to generalize the factorial to non integer values.
Category: Functions and Analysis

The Series Limit of Sum_k Cos(a Log K)/[k Log k]

Authors: Richard J. Mathar
Comments: 7 Pages.

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.
Category: Functions and Analysis

Lineal Systems Identification

Authors: Enrique Domínguez Pinos
Comments: 6 Pages. In Spanish

Review of current methods and new method proposal.
Category: Functions and Analysis

On the Convergence of Polynomial Interpolation

Authors: Abdelmajid Ben Hadj Salem
Comments: 6 Pages. In French

In this note, we present Runge's phenomenon concerning the approximation of a function by polynomials.
Category: Functions and Analysis

Bandlimited Functions and Timelimited Functions on Adeles

Authors: Gorou Kaku
Comments: 17 Pages.

D. Slepian and H.O. Pollak argue "Bandlimited functions" and "Timelimited functions" on the whole real line. In this paper, we will think of them on the ring of adeles
Category: Functions and Analysis

Integrability of Continuous Functions in 2 Dimensions

Authors: Hans Detlef Hüttenbach
Comments: 16 Pages.

In this paper it is shown that the Banach space of continuous, $mathbb{R}^2$- or $mathbb{C}$-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 integrability is equivalent to analyticity.The existence of a conjugation on that Banach space will be proven, which maps unintegrable functionsonto integrable functions.The boundary of a 2-dimensional simply connected compact region is commonly called 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 $mathbb{R}^2$ (or $mathbb{C}$) thereforeare equivalently related.
Category: Functions and Analysis

Use Euler's Identity to Prove the Existence of Natural Logarithms of Numbers Approaching Zero on the Complex Plane

Authors: Shikhar Sehgal
Comments: 1 Page. All Rights Reserved, Shikhar Sehgal 2023 (Abstract added to the article by viXra Admin - Please conform)

This paper provides an overview of using Euler's identity to prove that natural logarithms of numbers approaching zero exist on the complex plane.
Category: Functions and Analysis

Navier-Stokes Equations Analytic 3D Solution for Incompressible Viscous Fluids in the Absence of External Forces for a Given Periodic Initial Velocity Vector

Authors: Biruk Alemayehu Petros
Comments: 6 Pages. Mathematically meaningful solutions for Navier Stokes equation with complete proof is provided

This study proves the existence of smooth periodic solutions for Navier- Stokes three-dimensional equations under the assumption of a given pe- riodic initial velocity vector field with positive viscosity. The solution proposed solves the equation by utilizing a Fourier series representation of periodic initial velocity vector fields and predicting the velocity vec- tor field at all times. The significance of this finding is that it con- tributes positively towards understanding the behavior of solutions of Navier-Stokes equations and suggests that smooth periodic solutions for the given problem can indeed exist under certain conditions. Addition- ally, the authors suggest that their solution can be used to settle the Clay Mathematics Millennium Prize Problem, which seeks to find a solution for Navier-Stokes equations meeting specific criteria. It is important to note, however, that this study does not provide a complete solution to the problem, but it provides a significant contribution to the understanding of the behavior of solutions of Navier-Stokes equations. Overall, this re- search demonstrates that the smooth periodic solutions for Navier-Stokes equations can exist for a given initial velocity vector field with positive viscosity, and it presents a new approach for the Navier-Stokes equation.
Category: Functions and Analysis

Serious Problems in Standard Complex Analysis Texts From The Viewpoint of Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 7 Pages.

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).
Category: Functions and Analysis

On the Equation X + (X/X) = X

Authors: Saburou Saitoh
Comments: 4 Pages.

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.
Category: Functions and Analysis

The Fundamental Reformulation of the Concept of a Weak Solution to the Navier-stokes Problem (the Preliminary Version)

Authors: Jiří Souček
Comments: 19 Pages. [Constructive criticism is welcomed]

At first we identify the main error in the formulation of the concept of the weak solution to Navier-Stokes (NS) equations which is the completely insufficient treatment of the incompressibility condition on the fluid (expressed in the standard way by div u = 0). The repair requires the complete reformulation of the NS problem. The basic concept must be the generalized motion (i.e. the generalized flow) which replaces the standard velocity field. Here we define the generalized flow on the bases of Geometric measure theory extended to the theory of Cartesian currents and weak diffeomorphisms (see [1], [2]). Then the key concept of the complete weak solution to the NS problem is defined and the two conjectures (the existence and the regularity ones) concerning the complete weak solutions are formulated. In two appendices many technical details are described (concerning e.g. Cartesian currents, homology conditions, weak diffeomorphisms, etc.). Our approach is based on the unification of the standard analysis of NS equations with the methods of Geometric measure theory and of the theory of Cartesian currents.
Category: Functions and Analysis

Nontrivial Zeros of the Riemann Zeta Function

Authors: James C. Austin
Comments: 6 Pages.

The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by solving an integral form of the zeta function for the real parts and showing that a ratio of divergent terms can only be finite and nonzero, as required, when the real parts are exactly 1/2.
Category: Functions and Analysis

[A Note Mathematician] Brook Taylor

Authors: Edgar Valdebenito
Comments: 11 Pages.

Brook Taylor was an English mathematician, invented integration by parts, and discovered the celebrated formula known as Taylor's expansion.
Category: Functions and Analysis

Solving Triangles Algebraically

Authors: Joseph Bakhos
Comments: 10 Pages. Published April 19, 2023. Applied Mathematical Sciences, Vol. 17, 2023, no. 8, 379-390 doi: 10.12988/ams.2023.917399 pdf available at: http://www.m-hikari.com/ams/ams-2023/ams-5-8-2023/p/bakhosAMS5-8-2023.pdf

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.
Category: Functions and Analysis

Graphical Representation of Quaternions and Their Concomitant Functions

Authors: Stephen C. Pearson
Comments: 33 Pages. (Note by viXra Admin: Future hand-written submission will not be accepted))

In this particular paper we will demonstrate that, by invoking the concept of a 'quaternionic quasi-complex component', it is possible to graphically represent all quaternions and their concomitant functions with the aid of specific quaternionic analogues of the Argand diagram from complex variable analysis, bearing in mind that the algebraic and analytic properties of the aforesaid numbers and functions have been comprehensively elucidated in the author's antecedent papersIn this particular paper we will demonstrate that, by invoking the concept of a ‘quaternionic quasi-complex component’, it is possible to graphically represent all quaternions and their concomitant functions with the aid of specific quationic analogues of the Argand diagram from complex variable analysis, bearing in mind that the algebraic and analytic properties of the aforesaid numbers and functions have been comprehensively elucidated in the author’s antecedent papers [2]; [3]; [4] & [5].
Category: Functions and Analysis

Counting Differents Types Of Relations

Authors: Israel Julins Andersen Yañez
Comments: 19 Pages. (Correction made by viXra Admin)

This paper is about sequences. The author uses the new theorems to approximate better the values on the sequence, the bigger the number, the better the approximation.
Category: Functions and Analysis

Complex Analysis and Theory of Reproducing Kernels

Authors: Saburou Saitoh
Comments: 18 Pages.

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.
Category: Functions and Analysis

Gradient Descent using Fixed Point Theorem

Authors: Sing Kuang Tan
Comments: 8 Pages.

In this paper, I am going to propose a gradient descent algorithm using fixed point theorem. Fixed point theorem comes from topology mathematics. This gradient descent is able to converge at exponential rate, faster than Netwon method which converges at quadratic rate. Besides that, it does not need second order derivatives. The algorithm is simple and can be implemented using a few lines of equations. It can be used for training Relu deep learning network.
Category: Functions and Analysis

Some Deep Properties of the Green Function of Q. Guan on the line of Suita-Saitoh-Yamada's Conjectures

Authors: Saburou Saitoh
Comments: 17 Pages.

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.
Category: Functions and Analysis

A Monte Carlo Packing Algorithm for Poly-Ellipsoids and Its Comparison with Packing Generation Using Discrete Element Model

Authors: Boning Zhang, Eric B. Herbold, Richard A. Regueiro
Comments: 14 Pages.

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.
Category: Functions and Analysis

Matrix Forms of Normal Linear Differential Equation Systems with Constant Coefficients

Authors: Baran Tuna
Comments: 20 Pages.

This paper will try to explain the fundamentals of matrix solutions for linear differential equation systems. Differential equation systems are commonlyused, simplified, constructed, and notated for every field of science, engineering, and applications of technology.
Category: Functions and Analysis

An Empirical Convergence Phenomenon Related to Riemann Hypothesis

Authors: Jouni Puuronen
Comments: 4 Pages.

We stumble upon an empirical convergence phenomenon that is maybe related to Berry-Keating conjecture and the proof of Riemann hypothesis.
Category: Functions and Analysis

A Visual Proof E^s Larger Than S^e for Certain Conditions

Authors: Sourav Mandal, Amit Basak, Sagar Mandal
Comments: 2 Pages.

The beauty of mathematics fascinates humans and when we are dealing with some special constants that surely encourage us to understand some classy relations through numerous visual proofs of inequalities and when the constants are �� and �� then nothing more to say.There are visual proofs of the beautiful inequality ��^��<��^��. We have provided an alternative visual proof for this inequality using an area argument with the help of the theorem, for all real numbers ��≥0 ������ ��≠��,��^��>��^��.
Category: Functions and Analysis

Theoretical Expansion of Two Operators in Series

Authors: Juan Elias Millas Vera
Comments: 6 Pages.

This paper is a clean and short approximation to the concept of the double serial operators in a theoretical way. It introduces with the help of combinatorics the full extension of theoretical formalism which is necessary to do double serial operators with the 6 basic mathematical tools.
Category: Functions and Analysis

Approximation by Power Series of Functions

Authors: Andrej Liptaj
Comments: 7 Pages.

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, we refer to the literature where such formulas can be found.
Category: Functions and Analysis

Reduction Formulas of the Cosine of Integer Fractions of Pi

Authors: Richard J. Mathar
Comments: 8 Pages.

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.
Category: Functions and Analysis

New Principles of Dierential Equations VI

Authors: Hong Lai Zhu
Comments: 59 Pages.

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.
Category: Functions and Analysis

The Second Order Shannon Total Generalized Variation for Image Restoration

Authors: Alireza Hosseini, Sohrab Bazm
Comments: 26 Pages.

The Second Order Shannon Total Generalized Variation for Image Restoration.
Category: Functions and Analysis

Norm Inequalities for One Dimensional Sobolev Hilbert Spaces (An Extension)

Authors: Saburou Saitoh
Comments: 12 Pages.

In this paper, we shall consider norm inequalities for one dimensional Sobolev Hilbert spaces by using the theory of reproducing kernels as fundamental inequalities.
Category: Functions and Analysis

Norm Inequalities for One Dimensional Sobolev Hilbert Spaces

Authors: Saburou Saitoh
Comments: 8 Pages. In this paper, we shall consider norm inequalities for one dimensional Sobolev Hilbert spaces by using the theory of reproducing kernels as fundamental inequalities.

In this paper, we shall consider norm inequalities for one dimensional Sobolev Hilbert spaces by using the theory of reproducing kernels as fundamental inequalities.
Category: Functions and Analysis

A Multivariate Analogue of Jensen's Inequality Via the Local Product Space

Authors: Theophilus Agama
Comments: 5 Pages. This paper is a multivariate analogue of Jensen's inequality.

In this note we prove a multivariate analogue of Jensen's inequality via the notion of the local product and associated space.
Category: Functions and Analysis

Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice

Authors: Richard J. Mathar
Comments: 8 Pages.

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.
Category: Functions and Analysis

The Riemann Hypothesis Proved

Authors: Marcello Colozzo
Comments: 18 Pages.

The Riemann hypothesis is proved through a theorem on the nature of points critics of the real part and the imaginary part u(x, y), v(x, y) of a holomorphic function having the same zeros of the Riemann zeta function. Precisely, the zeros of thesefunctions are saddle points, and furthermore in these points the partial derivatives of odd order. From this derives a system of infinite identities that are check if and only if the real part of the zeros of the zeta function is equal to 1/2.
Category: Functions and Analysis

An Example of the Division by Zero Calculus Appeared in Conformal Mappings

Authors: Saburou Saitoh
Comments: 4 Pages.

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.
Category: Functions and Analysis

Existence Conditions, Asymptotic Behavior and Properties of a Class Of "Rational-Equivalence" Nonlinear Systems

Authors: Michael C. I. Nwogugu
Comments: 10 Pages. The copyright license-type for this article is CC-BY-NC-ND.

Liptai, Németh, et. al. (2020) supposedly proved that in the diophantine equation (3^a−1)(3^b−1)=(5^c−1)(5^d−1) in positive integers and where a≤b and c≤d, the only solution to the title equation is (a,b,c,d)=(1,2,1,1). This article analyzes the Complexity of, and introduces properties of the equations (3^a−1)(3^b−1)=(5^c−1)(5^d−1) and g^u=f^v, new "Existence Conditions", new theories of "Rational Equivalence", and a new theorem pertaining to the equation g^u=f^v. The class of equations of the type [(X^a−1)(X^b−1)=(Y^c−1)(Y^d−1)] (the "Rational-Equivalence Equation") includes the equation (3^a−1)(3^b−1)=(5^c−1)(5^d−1). This article also introduces simple Java codes for finding solutions to this class of equations for positive-integers up to (10)^2457600000 (and even greater positive-integers depending on available computing power).
Category: Functions and Analysis

On the Collatz Conjecture

Authors: Michael C. I. Nwogugu
Comments: 9 Pages.

This article proves that the Collatz Conjecture is valid for all positive integers. The main formula (and rules) for the Collatz Conjecture is as follows: f(n) = (n/2) or (3n+1).
Category: Functions and Analysis

Equity-Based Incentives and Production/Service Functions in Cyber-Physical Systems: Game Theory and Additional Considerations

Authors: Michael C. I. Nwogugu
Comments: 15 Pages.

Equity Based Incentives include Employee Stock Options (ESOs), and substantially change the traditional production/service function, because ESOs/EBIs have different psychological impacts (motivation, or de-motivation), can create intangible capital (ie. Social Capital, Reputational Capital and Human Capital), and create different economic payoffs. Although Game Theory is a flawed concept, it can be helpful in describing interactions in ESO/EBIs transactions. ESOs/EBIs involve a two-stage game; and there are no perfect Nash Equilibria for the two sub-games. The large number of actual and potential participants in these games significantly complicates resolution of equilibria and increases the dynamism of the game(s), given that players are more sensitive to each other’s moves in such games. This article: i) builds on but differs from Nwogugu (2004; 2006); ii) analyzes how ESOs/EBIs dynamics affect traditional assumptions of production functions (in both the manufacturing and service sectors), iii) develops new models of multi-dimensional/combined games (two-stage games, dynamic games and differential games) inherent in ESO/EBIs transactions, iv) illustrates some of the limitations of game theory.
Category: Functions and Analysis

Systemic Risk, Financial Stability and the Choice Between a Merger/Acquisition and a Strategic- Alliance/Joint-Venture

Authors: Michael C. I. Nwogugu
Comments: 13 Pages. The copyright license-type for this article is CC-BY-NC-ND

The annual volumes of M&A transactions and cross-border M&A transactions around the world are significant and often have Multiplier Effects and Spillover Effects on national economies and households, and a wide range of financial/economic indicators. Similarly, the volumes of Strategic Alliances and Joint Ventures around the world are significant (worth more than US$15 trillion annually) and have Multiplier Effects and Cross-Border Spillover Effects. As noted in Nwogugu (2015), "Synthetic M&As" can be executed using Strategic Alliances or joint Ventures. Conversely Strategic Alliances or Joint Ventures can be structured to provide all the benefits of M&A transactions. This article analyzes critical Dynamical Systems, Nonlinearity, Networks and behavioral issues pertaining to the choice between a Strategic Alliance or joint venture on one hand, and an M&A transaction; and also introduces new decision models.
Category: Functions and Analysis

Additive-Contingent Nonlinearity, Asymptotic Behaviors and Quantum-Causality in a Group of Covariant Systems

Authors: Michael C. I. Nwogugu
Comments: 31 Pages. The copyright license-type for this article is CC-BY-NC-ND

Some properties of the equations x2+y2+z2+v2= rXYZ, x2+y2+z2= rXYZ, x2+y2+z2+v2+u2=rXYZ, X2+Y2+Z2+V2= rXYZ, X2+Y2+Z2 = rXYZ, X2+Y2+Z2+V2 +U2 = rXYZ, Xi+Yi+Zi+Vi= rXYZ, x3+y3+z3=rXYZ, x3+y3+z3+x6+y6+z6=rXYZ, x6+y6+z6=rXYZ, [(x12+y12+z12)-(x6+y6+z6)]=rXYZ, and xi+yi+zi=rXYZ, (i is a positive integer), where x│X (ie. X is a multiple of x), y│Y, and z│Z are real numbers. This article also summarizes the relationships to Homotopy Theory, PDEs, Mathematical Cryptography and Analysis. The proofs are within the context of Sub-Rings. The additional common factor is that each of the variables x,y,z, v and dXYZ are multiples of (n-f), where n and f are real numbers. The solutions derived herein can be extended to other problems wherein (n-f) can take the form of polynomials/functions such as (6d-3), (14-5c), (ai-b2i), etc.. Some of the results are applicable where all variables are Integers.
Category: Functions and Analysis

Erratum to "Tables of Integral Transforms" by A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi (1953), p. 61 (4)

Authors: Richard J. Mathar
Comments: 3 Pages.

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.
Category: Functions and Analysis

On the Equation F(x)=x^2+exp(-2x)-1=0

Authors: Edgar Valdebenito
Comments: 5 Pages.

In this note we give solution of the equation f(x)=x^2+exp(-2x)-1=0
Category: Functions and Analysis

On the Integral Inequality of Some Trigonometric Functions in $mathbb{r}^n$

Authors: Theophilus Agama
Comments: 6 Pages.

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.
Category: Functions and Analysis

Numerical Derivatives

Authors: Horacio Useche
Comments: 50 Pages. In Spanish. Cálculo de derivadas mediante métodos numéricos (Calculation of derivatives using numerical methods)

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.

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.
Category: Functions and Analysis

What Are the Operator Error Estimates?

Authors: Yulia Meshkova
Comments: 2 Pages. (Corrections made by viXra Admin to conform with scholarly norm)

A very brief explanation what are the operator error estimates in periodic homogenization.
Category: Functions and Analysis

A Lower Bound for the Multiple Integral of Harmonic Generalized Distance Function in $\mathbb{r}^n$

Authors: Theophilus Agama
Comments: 5 Pages.

In this note we, we prove the inequality \r\n\\begin{align}\r\n\\int \\limits_{|a_n|}^{|b_n|} \\int \\limits_{|a_{n-1}|}^{|b_{n-1}|}\\cdots \\int \\limits_{|a_1|}^{|b_1|}\\frac{1}{\\sqrt[4s+3]{\\sum \\limits_{j=1}^{n}x^{4s+3}_j}}dx_1dx_2\\cdots dx_n \\geq \\frac{2\\pi \\times |\\log (\\langle a,b \\rangle)|\\bigg|\\prod_{j=1}^{n}|b_j|-|a_j|\\bigg|}{||\\vec{a}||^{4s+4}+||\\vec{b}||^{4s+4}}\\nonumber\r\n\\end{align}under some special conditions.
Category: Functions and Analysis

A Lower Bound for Multiple Integral of Normalized Log Distance Function in $\mathbb{R}^n$

Authors: Theophilus Agama
Comments: 4 Pages.

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 $i^2=-1$.
Category: Functions and Analysis

Hyperoperator Analysis

Authors: Dmitrii V. Guryanov
Comments: 17 Pages.

The purpose of this article as a continuation of development of the Multiplical concept is to give an answer to the earlier raised question of why the place of the operator in the function y = e↗x was taken by the operator - a power tower with left associativity, and not with the generally accepted right associativity (the Tetration). Answering on this question required to conduct an hyperoperator analyze. The hyperoperator nature is considered, definition is made and an alternative way of its development is proposed in the present analysis.
Category: Functions and Analysis

Singular Properties

Authors: Dmitrii V. Guryanov
Comments: 28 Pages.

The purpose of this article as a continuation of development of the multiplical topic is to find a solution for operation of differentiation and factorization of a function with points of interruption, points where function turns to zero. The solution which allows restoring the original function as result of reverse operation of integration and factorial-multiplication of previously obtained derivative and factor-derivative respectively and with an appropriate selection of an arbitrary multiplier B or addend C, respectively. As the result of the work made a number of new classes of function properties and definitions are introduced as function point properties.
Category: Functions and Analysis

Multiplical Concept

Authors: Dmitrii V. Guryanov
Comments: 21 Pages.

The purpose of this article is to introduce and to describe a concept of math calculus “Multiplical”. To my total surprise I have found that currently such a concept does not exist among set of math definitions in its direct and explicit form. Nevertheless there are number of areas of its practical use, where this concept would be suitable and potentially would be naturally used in its direct and explicit form, especially, in statistics, finance and economy researches and analysis and many other areas. Moreover from my perspective this concept perfectly fits into the coherent system of standard mathematical concepts and operators and should take its rightful place there. In this article also other topics are considered and some interesting conclusions are made.
Category: Functions and Analysis

A Proof of the Kakeya Maximal Function Conjecture from a Special Case

Authors: Johan Aspegren
Comments: 5 Pages.

First in this paper we will prove the Kakeya maximal function conjecture in a special case when tube intersections behave like line intersections. This paper highlights how different tube intersections can be than line intersections. However, we show that the general case can be deducted from the linelike case.
Category: Functions and Analysis

The Generating Function Technique and Algebraic Ordinary Differential Equations

Authors: Robert Lloyd Jackson
Comments: 8 Pages. contact info: rljacksonmd@gmail.com

In the past, theorems have shown that a power series can derive solutions to algebraic ordinary differential equations, or AODEs. First, this paper gives a quick synopsis on power series methods while elaborating on a recent theorem that established the generating function technique (GFT) as a powerful method for solving differential equations, such as AODEs. Instead of building a formal power series into an analytic solution, GFT uses a composition of predefined formal power series that form another formal power series, hence the analytic function. Next, the study examines an earlier paper that directly claimed a formal power series method solves AODEs; then, it shows a few partially solved problems via a "bottom-up" algorithm. This study utilizes GFT to create several analytic solutions to the same problems using a "top-down" approach. Ultimately, one finds that GFT, not other methods for solving nonlinear partial differential equations may serves as a powerful method for solving many AODEs.
Category: Functions and Analysis

A Mystery in Conformal Mappings and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 5 Pages.

We introduce a mysterious property in conformal mappings and division by zero calculus with some elementary linear mapping.
Category: Functions and Analysis

On the Integral’s Substitution Rule

Authors: Yang Liu
Comments: 19 Pages.

The first part of the article mainly gives some concepts and lemmas, such as the definition of some function spaces, Radon integral and its basic properties, $L^1-$seminorm, the definition of Lebesgue integral and so on. These basic concepts and lemmas are very helpful in proving the main theorem later. The proof of the main theorem is divided into two parts. The main idea is to locally approximate the transformation $\phi:U\to V$ around a point $a\in U$ by an affine map. In order to be able to use this local approximation meaningfully, we decompose the given function $\psi\in C_c\left(V\right)$ into a sum of functions with very small support, so that the approximation of $\phi:U\to V$ by the local affine approximation is very good. This is done with the help of the $\zeta-$function introduced below, which provides a practical partition of the one on $\mathbb{R}^n$. In the second part, the proof will be performed on suitable approximations of any integrable function.
Category: Functions and Analysis

General Solutions of Ordinary Differential Equations and Division by Zero Calculus - New Type Examples

Authors: Saburou Saitoh
Comments: 6 Pages.

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.
Category: Functions and Analysis

Briefly About the Notion of a Quaternionic Holomorphic Function

Authors: Michael Parfenov
Comments: 6 Pages.

The so-called essentially adequate notion of quaternionic holomorphy is briefly considered. This calls into question the known statement of R. Penrose that there is no satisfactory quaternionic analogue of the notion of a holomorphic function.
Category: Functions and Analysis

Asymptotics of Solutions of Differential Equations with a Spectral Parameter

Authors: Vjacheslav A. Yurko
Comments: 17 Pages.

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.
Category: Functions and Analysis

Derivative of a Sequence at Infinity

Authors: Josef Bukac
Comments: 13 Pages.

When a sequence of real numbers is convergent to some finite number, we may approximate the members of the sequence by its limit provided the subscript is large. But we may want a higher accuracy. If we know the speed of convergence, we define a derivative of the sequence at infinity. We also define the second derivative which enables us even better approximations.
Category: Functions and Analysis

Triangular A-Statistical Approximation by Double Sequences of Positive Linear Operators

Authors: Carlo Bardaro, Antonio Boccuto, Kamil Demirci, Ilaria Mantellini, Sevda Orhan
Comments: 21 Pages.

In the present paper we introduce a new type of statistical convergence for double sequences called triangular A-statistical convergence and we show that triangular A-statistical convergence and A-statistical convergence overlap, neither contains the other. Also, we give a Korovkintype approximation theorem using this new type of convergence. Finally we give some further developments.
Category: Functions and Analysis

Amazing Formulas Related to pi

Authors: Edgar Valdebenito
Comments: 10 Pages.

In this note we give some formulas related to Pi.
Category: Functions and Analysis

What is the Value of the Function X/x at X=0? What is 0/0?

Authors: Saburou Saitoh, Yoshinori Saitoh
Comments: 9 Pages.

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.
Category: Functions and Analysis

One Century since Bergman, Szego and Bochner on Reproducing Kernels

Authors: Heinrich Begehr, Saburou Saitoh
Comments: 8 Pages.

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.
Category: Functions and Analysis

Folium of Descartes and Division by Zero Calculus　- 　An Open Question

Authors: Saburou Saitoh, Yoshinori Saitoh
Comments: 5 Pages. 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.

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.
Category: Functions and Analysis

The Solution of the Invariant Subspace Problem. Part I. Complex Hilbert Space.

Authors: Jaykov Foukzon
Comments: 104 Pages.

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory TG.It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence of inaccessible cardinals [1].In this paper we look at a set theory NC_{∞^{#}}^{#}, based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [2]-[5].In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule.We present a new approach to the In this paper we deal with set theory INC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule.We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace.Non-conservative extension based on set theory NC_{∞}^{} of the model theoretical nonstandard analysis[6] is considered.
Category: Functions and Analysis

The Local Product and Local Product Space

Authors: Theophilus Agama
Comments: 5 Pages.

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}for all $s\in \mathbb{N}$, where $\langle,\rangle$ denotes the inner product and where $\mathbf{e}(q)=e^{2\pi iq}$.
Category: Functions and Analysis

High-Accuracy Approximation of the Voigt Function Based on Fourier Expansion of Exponential Multiplier

Authors: Yihong Wang
Comments: 8 pages, 3 figures

A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The computational test reveals that with only the first 33 terms Fourier expansion of the exponential multiplier, this approximation provides accuracy better than 5.5383×10−19 in the domain of practical interest 0 < x < 40,000 and 10−4 < y < 102 that is needed for applications using the HITRAN molecular spectroscopic database. Compared with the typical approximation algorithms, the proposed approximation still available even if y is very small and the accuracy in the narrow band domain 0 < x < 40,000 ∩ 10−10 < y < 10−4 remains high and better than 5.5385×10−13.
Category: Functions and Analysis

Zero-over-Zero Theorem

Authors: Kyumin Nam
Comments: 2 Pages.

In this paper, we provide proof of the Zero-over-Zero Theorem: For some constant k, if 0/0 = k, then k = 1. This result would be some help for the 0^0 problem, and 0/0 problem.
Category: Functions and Analysis

“3n+1 Problem” Solution Approach Through the Series Convergence Study

Authors: Ruslan Enikeev
Comments: 4 Pages.

We propose a solution approach to the so-called ”3n+1” problem. The iterations of algorithm are represented by a series which convergence analysis gives us confirmation of the conjecture.
Category: Functions and Analysis

The Covariant Helmholtzian

Authors: Claude Michael Cassano
Comments: 46 Pages.

The d'alembertian operator on vector doublets may be factored with a pair of four-by-four matrices via simple partial derivatives as elements. As a generalization of the d'alembertian operator,the Helmholtzian operator on vector doublets may be factored with a pair of four-by-four matrices via simple partial derivatives augmented by adding certain constants as elements thereto. The Covariant Helmholtzian operator generalizes these, where the elements of the pair of four-by-four matrices are covariant derivatives applying to the vector doublet operated on. Thus, the d'alembertian operator is a Covariant Helmholtzian operator operated in a flat rectangular Cartesian space; the Helmholtzian operator is a Covariant Helmholtzian operator operated in a space of curvature where all the Christoffel symbols are appropriate constants.
Category: Functions and Analysis

Seeking the Analytic Quaternion

Authors: Colin Walker
Comments: 12 Pages.

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.
Category: Functions and Analysis

Set Theory NC_{∞^{}}^{} Based on Bivalent Infinitary Logic with Restricted Modus Ponens Rule. Basic Analysis on External Non-Archimedean Field R_{c}^{#}.

Authors: Jaykov Foukzon
Comments: 98 Pages.

In this paper we deal with set theory NC_{∞}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule [1]-[3].The main goal of this paper is to present basic analysis on non Archimedean field R_{c}^{#}.The non Archimedean field ℝ_{c}^{#}consist of Cauchy hyperreals.The non Archimedean external field ℝ_{c}^{#}≠┊^{∗}ℝ┊ is obtained as generalized Cauchy completion of non Archimedean field ℚ^{#} or ^{∗}ℚ.In order to obtain such completion we deal with external hyper infinite Cauchy sequences{x_{n}}_{n∈ℕ^{#}},{x_{n}}_{n∈|^{∗}ℕ|}.Basic Analysis on External Non-Archimedean Field ℝ_{c}^{#}is considered.
Category: Functions and Analysis

Maximality of Linear Operators

Authors: Mohammed Meziane
Comments: 80 Pages.

In this thesis, we show some maximality results about non-necessarily bounded linear operators.
Category: Functions and Analysis

Square Roots of Boudned Operators

Authors: Mohammed Hichem Mortad
Comments: 10 Pages.

This is part of some lectures about square roots of bounded operators.
Category: Functions and Analysis

An Operator Theory Problem Book

Authors: Mohammed Hichem Mortad
Comments: 18 Pages.

This constitutes the preface and references of the book "an operator theory problem book".
Category: Functions and Analysis

Positive Operators. Square Root

Authors: Mohammed Hichem Mortad
Comments: 41 Pages.

This is Chapter 5 of the manuscript "an operator theory problem book".
Category: Functions and Analysis

PhD's Thesis of Youcef Naas

Authors: Youcef Nass
Comments: 73 Pages.

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 .
Category: Functions and Analysis

The Absolute Value of Unbounded Linear Operators

Authors: Imene Boucif
Comments: 61 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Dans cette thèse en théorie des opérateurs, on s’intéresse aux opérateurs normaux, aux opérateurs autoadjoints, aux opérateurs positifs, à la valeur absolue d’un opérateur, à l’inégalité triangulaire. Cette thèse s’articule autour des relations du type |AB| = |A||B|, |A||B| = |B||A|, |A ± B| ≤ |A| + |B|. Dans la première partie on fournit des définitions et des notions élémentaires en théorie des opérateurs. Après cette introduction, dans ce chapitre on s’intéresse aux opérateurs positifs bornés, à la racine carrée d’un opérateur positif, à la valeur absolue d’un opérateur borné, et on donne des résultats sur l’inégalité triangulaire et d’autres relations concernant la somme et le produit de la valeur absolue dans le cas borné. Dans le dernier chapitre, on s’intéresse au cas des opérateurs non-bornés. On commence d’abord par des définitions et des propriétés primordiales des opérateurs non-bornés, ensuite on donne des résultats sur la somme et le produit de la valeur absolue dans le cas des opérateurs non-bornés. On fournit cette partie par quelques exemples. On obtient, comme conséquence intéressante, une caractérisation de l’inversibilité pour la classe des opérateurs normaux non-bornés. On obtient également une preuve très simple de l’inclusion dans R du spectre des opérateurs non-bornés autoadjoints.

In this thesis in operator theory, we are interested in operators normal, to self-assistant operators, to positive operators, to the absolute value of an operator, with triangular inequality. This thesis revolves around the relationships of the type | AB | = | A || B |, | A || B | = | B || A |, | A ± B | ≤ | A | + | B |. In the first part we provide definitions and elementary notions in operator theory. After this introduction, in this chapter we are interested in bounded positive operators, to the square root of a positive operator, to the value absolute of a bounded operator, and we give results on the triangular inequality and other relations concerning the sum and the product of the absolute value in the case thick headed. In the last chapter, we are interested in the case of unbounded operators. We start first with definitions and primordial properties of unbounded operators, then we give results on the sum and the product of the value absolute in the case of unbounded operators. We provide this part with a few examples. As an interesting consequence, we obtain a characterization of invertibility for the class of unbounded normal operators. We obtain also a very simple proof of the inclusion in R of the spectrum of unbounded self-adjoining operators.
Category: Functions and Analysis

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 141 Pages. Formerly viXra:2104.0068

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 as Cartesian products 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.
Category: Functions and Analysis

Kepler's Laws as Properties of the Kinematic Equations of Motion of a Point Along Curves of the Second Order

Authors: Viktor Strohm
Comments: 13 Pages.

Differential equations of motion on curves of the second order are inferred. Solutions to equations are made by computer programs. The results of the calculation are compared with Kepler's laws.
Category: Functions and Analysis

Triple Serial Operators Theory

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper you will see the theoretical structure of triple serial operators. I will build on this work in two of my previous papers to do an easy and clean explanation of what happens when you combine three linked variables in five independent operations.
Category: Functions and Analysis

On the Method of Dynamical Balls

Authors: Theophilus Agama
Comments: 11 Pages.

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.
Category: Functions and Analysis

Double Serial Operators Theory

Authors: Juan Elias Millas Vera
Comments: 27 Pages.

This paper is an advance of my work in serial operators. Here you can see 216 theoretical combinations of two serial operators with a 2-variable operation. The theory covers addition, subtraction, product, division, power and root. Moreover I am going to present 48 numerical examples.
Category: Functions and Analysis

Riemann's Functional Equation When Zeta(s) = 0 = Zeta(1-s)

Authors: Michael C. Dickerson
Comments: 1 Page. Contacting author at michaeldickerson89@gmail.com [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Riemann's Functional equation zeta(s) has values where zeta(s) = 0 at negative even integers of s (-2,-4,-6...) when the function sin(pi*s/2) equals 0. This paper demonstrates that the only other case where zeta(s) = 0 in Riemann's functional equation is when zeta(s) = zeta(1-s) which is only true when the real part of s = 1/2.
Category: Functions and Analysis

Review of the Serial Operators Theory

Authors: Juan Elias Millas Vera
Comments: 13 Pages. If you find an error please send me an e-mail.

In this paper, I want to explain with definitions and examples the 36 possible combinations of conjunction of a serial operator and operation inside it. I am going to remember to the reader the basic theory of the 6 basic serial operators (Summation or Sigma notation, Restory or Rho notation, Productory or Pi notation, Divisory or Delta notation, Exponentory or Theta notation and Divisory or Zeta notation), then I will do the 6 x 6 categories.
Category: Functions and Analysis

Differential Coefficients at Corners and Division by Zero Calculus

Authors: Saburou Saitoh, Keitaroh Uchida
Comments: 5 Pages.

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 $x=a$.
Category: Functions and Analysis

Two-Dimensional Fourier Transformations and Double Mordell Integrals II

Authors: Martin Nicholson
Comments: 6 Pages.

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.
Category: Functions and Analysis

Definition and Application of Anti-Factorial

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper I want to show a new concept, the anti-factorial. This is the inverse operator of the factorial. I introduce a full (and necessary) new notation for this concept. The main idea is to develop an operator (notated by n¡) that is able of do the inverse form of an expanded number n to a contracted number k and if you do the factorial of k you will end up back at n, that is k!=n.
Category: Functions and Analysis

A Unique Step Function and Stitching Piecewise Defined Functions

Authors: William Blickos
Comments: 12 Pages. thoughtfarm@live.com

This paper discusses a unique complex logarithmic function that can be used as an alternative to piecewise defined step functions such as the Heaviside function. It describes how to create second tier logical functions such as multi-steps, boxcars, and valleys, and it provides a method to stitch the components of piecewise functions into a single function.
Category: Functions and Analysis

On the Elementary Function y=|x| and Division by Zero Calculus

Authors: Hiroshi Okumura, Saburou Saitoh, Keitarou Uchida
Comments: 7 Pages.

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.
Category: Functions and Analysis

Division by Zero Calculus in Figures - Our New Space Since Euclid -

Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 54 Pages.

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.
Category: Functions and Analysis

On the State of Convergence of the Flint Hill Series

Authors: Theophilus Agama
Comments: 5 Pages.

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 $s\in \mathbb{N}$.
Category: Functions and Analysis

New Notation in Series of Functions II

Authors: Juan Elias Millas Vera
Comments: 4 Pages.

In this paper we will see the next part of my theory of notation in series. Focusing on summation and productory we will do a defined explanation of an iterated serial operators.
Category: Functions and Analysis

A Modified Mid Point Rule

Authors: D Williams
Comments: 2 Pages.

A (better?) version of the Mid Point Rule for approximating integrals is given and tested against the standard version. As usual there are many unanswered questions for you to solve.
Category: Functions and Analysis

A Marriage Between the Inner-Outer Method and Averaging Methods in a Special Case, or at Least an Attempt At.

Authors: Alon Ray
Comments: 8 Pages.

I had combined the methods of Averaging and Inner-Outer methods of solution in Perturbation theory of Ordinary Differential equations. Obviously one cannot solve analytically this ODE (i.e via elementary functions), perhaps through some sort of Special Functions' transformation which I haven't tried as of yet it may be feasible. What I have done is just take a simple specific example for the general ODE, I believe I was trying to find asymptotics for this specific example. This is my failed attempt for writing my thesis paper for M.Sc in Mathematics, my final attempt. Hope it's worth something to someone...
Category: Functions and Analysis

An Elementary Proof of $0.999\dots=1$

Authors: O. Kurwa
Comments: 2 Pages.

One of the properties distinguishing irrational and rational numbers is the uniqueness (or the lack thereof) of their decimal representations. For example, the numbers $\pi$ and $1$ can be used as specimens of this phenomenon, as $\pi$ has precisely one expression as decimals, but $1=1.0=1.00=\dots$. In this paper, we provide an elementary proof for the fact that $0.999\dots$ is also a decimal representation of $1$, using the Lebesgue measure.
Category: Functions and Analysis

The Complex Curvature Is the Inverse of the Complex Radius

Authors: Abel Cavasi
Comments: 2 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

I don't know if anyone else has noticed, but if we consider that the radius and pitch of the helix is a complex number, then the curvature and torsion can be JUST THE INVERSE of this complex number.
Category: Functions and Analysis

A Formula of Integration

Authors: Antoine Balan
Comments: 1 page, written in french

We propose a formula which characterizes the integral.
Category: Functions and Analysis

Two Exercices of Integration

Authors: Antoine Balan
Comments: 2 pages, written in french

We present two exercices of integration. The aim is to characterize the integral by the integration by parts.
Category: Functions and Analysis

A Proof of the Riemann Hypothesis Using the Two-Sided Laplace Transform

Authors: Seong Won Cha
Comments: 16 Pages.

A proof of the Riemann hypothesis using the two-sided Laplace transform
Category: Functions and Analysis

An Idea of Fermat for the Stop and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 5 Pages.

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.
Category: Functions and Analysis

History of the Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 54 Pages. The concept of division by zero will create the concept of division by zero calculus and this concept will give great impacts to elementary mathematics.

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.
Category: Functions and Analysis

Introduction to Mathematical Series

Authors: Juan Elias Millas Vera
Comments: 4 Pages.

Euler, Leibniz or Ramanujan are some names who have developed mathematical series. In this paper I want to introduce some series of these famous mathematicians and contribute some of my own open series.
Category: Functions and Analysis

The Navier-Stokes Equations and Turbulence or Chaos

Authors: Bertrand Wong
Comments: 4 Pages.

The motion of fluids which are incompressible could be described by the Navier-Stokes differential equations. Although they are relatively simple-looking, the three-dimensional Navier-Stokes equations misbehave very badly. Even with nice, smooth, reasonably harmless initial conditions, the solutions could wind up being extremely unstable. The field of fluid mechanics would be dramatically altered through a mathematical understanding of the outrageous behaviour of these equations. An explanation why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence or chaos (which is a three-dimensional phenomenon), would be provided. [Published in an international journal.]
Category: Functions and Analysis

Division by Zero Calculus and Hyper Exponential Functions by K. Uchida

Authors: Saburou Saitoh, Keitaroh Uchida
Comments: 11 Pages.

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.
Category: Functions and Analysis

Algebraic Conversion Between Rectangular and Polar Coordinates

Authors: Joseph Bakhos
Comments: 6 Pages.

Algebraic equations are derived to approximate the relationship between the rectangular coordinates and the polar coordinates of a vector. These equations can then be used without recourse to imaginary numbers, transcendental functions, or innite sums. This submission incorporates changes to previous work so that the functions presented involving quaterns are now valid from -infinity to +infinity, using a new type of step function
Category: Functions and Analysis

Rotation Without Imaginary Numbers, Transcendental Functions, or Infinite Sums

Authors: Joseph Bakhos
Comments: 4 Pages.

Algebraic equations are derived to approximate the relationship between the rectangular components and the polar coordinates of a vector. These equations can then be used without recourse to imaginary numbers, transcendental functions, or innite sums. This is done with an error of approximately one-half percent.
Category: Functions and Analysis

Kouider Function have Basis a

Authors: Kouider Mohammed Ridha
Comments: 4 Pages.

Josephus function is a new numerical function which presented by Kouider (2019,[1]) by studying Joseph's problem . In this paper we interesting in study of its derived function and some of its related properties for Josephus function. From this point of view we saw that we can define a more comprehensive function than the Josephus function. And we called it the Kouider function with basis a. We have also studied some of its related properties with proof as well.
Category: Functions and Analysis

A Solution to the Riemann Hypothesis

Authors: Bittu Marley
Comments: Pages.

This paper discloses a proof for the Riemann Hypothesis. We define function f(s) such that sum of f(s) and f(1 − s) is zero at the non-trivial zeros of zeta function. Further, f(s) can be written as sum of −1/s and a series function −λ(s) which is obtained on the expansion of integration term in the functional equation of zeta function. Either f(s) and f(1 − s) are individually zero or f(s) = −f(1−s) for an s, off the critical line if the Riemann hypothesis is false. From geometric analysis of λ(s) we find the f(s)!=0 any where in the critical strip. We also prove by contradiction that f(s) != −f(1 − s) for any s, off the critical line
Category: Functions and Analysis

Integration

Authors: Antoine Balan
Comments: 2 pages, written in french

The integration by parts can charaterize the integration. We give a formula for a manifold.
Category: Functions and Analysis

New Notation in Series of Functions

Authors: Juan Elias Millas Vera
Comments: 5 Pages.

In this paper we will see how it is possible to establish a precise and complete notation for operators that describe a series of functions.
Category: Functions and Analysis

Inverse Problems and Theory of Reproducing Kernels Theory

Authors: Saburou Saitoh
Comments: 16 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

At least until about 20 years ago, we had very difficult inverse problems that are important in many practical problems (fundamentals). These problems were indeed difficult in both mathematics and numerical realizations of the solutions and so, they are called ill-posed problems and very famous difficult problems. We were able to solve these problems in both senses of mathematics and numerical problems by using the theory of reproducing kernels applying the Tikhonov regularization. However, for the real inversion formula of the Laplace transform, we needed the great power of computers by H. Fujiwara.
Category: Functions and Analysis

Filter Exhaustiveness and Lter Limit Theorems for K-Triangular Lattice Group-Valued Set Functions

Authors: A. Boccuto, X. Dimitriou
Comments: 11 Pages.

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.
Category: Functions and Analysis

Algebraic Structures for Pairwise Comparison Matrices: Consistency, Social Choices and Arrow's Theorem

Authors: Giuseppina Barbieri, Antonio Boccuto, Gaetano Vitale
Comments: 20 Pages. [Corrections made by viXra Admin to conform with scholarly norm]

We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow's theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e. multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories.
Category: Functions and Analysis

Some New Results on Dieudonné-Type Theorems for K-Triangular Lattice Group-Valued Set Functions

Authors: Antonio Boccuto
Comments: 10 Pages.

Using the Maeda-Ogasawara-Vulikh representation theorem and sliding hump-type techniques, we prove some Dieudonne-type theorems for k-triangular set functions, taking values in lattice groups.
Category: Functions and Analysis

Quaternionic Zeta Function

Authors: Yuly Shipilevsky
Comments: 6 Pages.

We introduce and suggest to study famous Zeta Function, extending it to a quaternionic variable and other hypercomplex variables
Category: Functions and Analysis

Smooth Functions Vanishing at Zero Map $H^s$ with $s>d/2$ Into Itself

Authors: Alon Brook-Ray, Steve Schochet
Comments: 8 Pages.

Some theorems and lemmas on Smooth Functions Vanishing at Zero Map $H^s$ with $s>d/2$ into itself.
Category: Functions and Analysis

Solution to Problems 12 and 13 in Michael Taylor's Volume 3 in PDE.

Authors: Alon Brook-Ray
Comments: 11 Pages.

Solutions to problems 12 and 13 in chapter 16 of Volume 3 of PDE textbook by Michael Taylor.
Category: Functions and Analysis

A Relationship Between Exponential Growing and a Basic Asymptotic Function

Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 6 Pages.

In this paper we study a simple exponential growing problem which leads to a basic asymptotic function. It shows a hidden property of asymptotic functions.
Category: Functions and Analysis

On Linear Ordinary Differential Equations of Second Order and Their General Solutions

Authors: Zafar Turakulov
Comments: 10 Pages. A new method for solving linear ordinary differential equations is discovered. This material exposes examples of general solutions obtained this way.

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.
Category: Functions and Analysis

A Note on Lp-Convergence and Almost Everywhere Convergence

Authors: Yu-Lin Chou
Comments: 3 Pages. expository article

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.
Category: Functions and Analysis

New Principles of Differential Equations Ⅳ

Authors: Hong Lai Zhu
Comments: 30 Pages.

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.
Category: Functions and Analysis

A Simple Proof for Almost Everywhere Convergence to Imply Weak Convergence of Induced Measures

Authors: Yu-Lin Chou
Comments: 2 Pages. expository article

Establishing on a finite measure space the implication for almost everywhere convergence to imply weak convergence of the corresponding induced measures (in particular to imply convergence in distribution) is usually indirect, convergence in measure being the transition. We give a simple, pedagogically informative proof for the implication.
Category: Functions and Analysis

A General Definition of Means and Corresponding Inequalities

Authors: Pranjal Jain
Comments: 10 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

The definitions of the Quadratic, Arithmetic, Geometric and Harmonic means all follow a certain generalisable pattern. The aim of this paper is to explicitly state that pattern, hence generalising the definition of a 'mean', and to prove inequalities for comparing different means (of which those between the four means stated previously is a special case).
Category: Functions and Analysis

Topological Stationarity and Precompactness of Probability Measures

Authors: Yu-Lin Chou
Comments: 6 Pages.

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.
Category: Functions and Analysis

An Untold Story of Brownian Motion

Authors: Yu-Lin Chou
Comments: 6 Pages. expository article

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.
Category: Functions and Analysis

Exact Solution of All Real Bessel LHODE Formula

Authors: Claude Michael Cassano
Comments: 9 Pages. [Corrections made by viXra Admin to conform with scholarly norm]

The Bessel linear homogeneous ordinary differential equation (LHODE) with real parameter may be solved exactly using my "Vector Space Transformation Technique" similarly to the usage to solve the "Exact Solution of All Half-Integer Bessel LHODE Formula" - using "Exact solution of ODEs - Vector Space Transformation Technique - Part 2", Theorem I.1.
Category: Functions and Analysis

Another Topological Proof for Equivalent Characterizations of Continuity

Authors: Yu-Lin Chou
Comments: 2 Pages. expository article

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.
Category: Functions and Analysis

How Likely Is It for Countably Many Almost Sure Events to Occur Simultaneously?

Authors: Yu-Lin Chou
Comments: 4 Pages. expository article

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.
Category: Functions and Analysis

Distribution of Integrals of Wiener Paths

Authors: Yu-Lin Chou
Comments: 3 Pages.

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 $C([0,1], \R)$.
Category: Functions and Analysis

Growth Order of Standardized Distribution Functions

Authors: Yu-Lin Chou
Comments: 5 Pages.

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.
Category: Functions and Analysis

Some New Type Laurent Expansions and Division by Zero Calculus; Spectral Theory

Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 10 Pages.

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.
Category: Functions and Analysis

Exact Solution of All Half-Integer Bessel LHODE Formula

Authors: Claude Michael Cassano
Comments: 6 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

A formula for all the Half-Integer Bessel ODE is produced using my article "Exact solution of ODEs Vector Space Transformation Technique, Part 2". (In an earlier publication I produced an algorithm for the Bessel hallf-integer solutions, but advancements on the Vector Space Transformation Technique allowed an actual formula to be produced (as well as further results)
Category: Functions and Analysis

Exact Solution of ODEs - Vector Space Transformation Technique - Part 2

Authors: Claude Michael Cassano
Comments: 7 Pages. [Corrections made to conform with the requirements on the Submission Form]

Since the number of linearly independent solutions of an HLODE equals the order of the HLODE, the dimension of the solution set of an HLODE is it's order; so the dimension of all HLODEs of the same order is this order. Thus, a first order HLODE may be solved exactly using the single HLODE ; but higher order non-elementary HLODEs require that order number of equations to solve such HLODEs - i.e. usage of linear transformations between linearly independent HLODE solution sets of that same order. The first order HLODE being of dimension 1 may be solved exactly by itself. Higher order HLODEs are subject to condition that the additional dimensional HLODEs must also be satisfied.
Category: Functions and Analysis

Processing of the H-Holomorphic Functions

Authors: Michael Parfenov
Comments: 16 Pages.

To automate cumbersome, error-prone and tedious manual procedures of calculations with quaternionic holomorphic (ℍ -holomorphic) functions we have developed and present here a special programmes pack in the Wolfram Mathematica® programming language. By using this pack a lot of examples of ℍ -holomorphic functions is processed. They give conclusive evidence that the so-called essentially adequate theory of quaternionic holomorphy is true. All considered ℍ -holomorphic functions are built from complex holomorphic ones in accordance with the general constructing rule defined earlier. At that combinations of ℍ -holomorphic functions are built from the simplest basis (irreducible) ℍ -holomorphic functions by using the usual rules of quaternionic algebraic operations.
Category: Functions and Analysis

Division by Zero Calculus and Laplace Transform

Authors: Saburou Saitoh
Comments: 13 Pages.

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.
Category: Functions and Analysis

Neuro-Amorphic Function

Authors: Egger Mielberg
Comments: 15 Pages.

As a good try, we took liberty for formula derivation that would allow describing many physical phenomena. In Nature, we face many situations when a single magnitude is a reason for drastic changes in the whole physical process. A single flexible instrument for describing processes of any kind would help a lot. We propose a function for generating mathematical models for a process behavior. We introduce special parameters that will help researchers find an acceptable solution for their tasks. The Dynamics coefficient, as well as dynamic function, is crucial for graph change. It can be used for the dynamic corrections of the whole physical process.
Category: Functions and Analysis

RLC Circuits and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 7 Pages.

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.
Category: Functions and Analysis

Representations of the Division by Zero Calculus by Means of Mean Values

Authors: Saburou Saitoh
Comments: 14 Pages.

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.
Category: Functions and Analysis

A Method to Find the Global Optimum of a Function

Authors: Joseph Valles
Comments: 1 Page.

This is a method to find the global optimum value of a function. It uses a generalization of the min function applied to the values of a function. I provide a method to find the x value of the global optimum.
Category: Functions and Analysis

A Method to Smooth Functions

Authors: Joseph Valles
Comments: 2 Pages.

This is a method to analytically smooth functions. It involves taking a multiple integral from $x-\delta$ to $x+\delta$. The smoothing amount is denoted by $\delta$.
Category: Functions and Analysis

Mirror Images and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 13 Pages. 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.

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.
Category: Functions and Analysis

[A] Short Proof of Generalized Cauchy's Residue Theorem

Authors: Federico Pagano
Comments: (Note: Corrections on 1st page are made by viXra Admin to conform with scholarly norm)

We can derive the Cauchy's residue theorem (its general form) just by direct integration of a Taylorseries “without” making any radius go to zero, even without the limit circumference idea take place.
Category: Functions and Analysis

[A] Corollary [from a Theorem Proposed by the Author]

Authors: Federico Pagano
Comments: (Note: Corrections on 1st page are made by viXra Admin to conform with scholarly norm)

From [a] theorem [proposed by the author], we can derive the generalized Dirichlet integral for anynatural value when the whole integrand is raised to the n-th power.
Category: Functions and Analysis

Quantum Permutations and Quantum Reflections

Authors: Teo Banica
Comments: 200 Pages.

The permutation group $S_N$ has a free analogue $S_N^+$, which is non-classical and infinite at $N\geq4$. We review here the known basic facts on $S_N^+$, with emphasis on algebraic and probabilistic aspects. We discuss as well the structure of the closed subgroups $G\subset S_N^+$, with particular attention to the quantum reflection groups.
Category: Functions and Analysis

New Principles of Differential Equations Ⅱ

Authors: Hong Lai Zhu
Comments: 36 Pages.

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.
Category: Functions and Analysis

Liouville-Type Theorems Outside Small Exceptional Sets for Functions of Finite Order

Authors: Bulat N. Khabibullin
Comments: 6 Pages. in Russian

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.
Category: Functions and Analysis

A Conjecture On Some ds Periods On The Complex Plane

Authors: Saburou Saitoh
Comments: 3 Pages. 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.

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.
Category: Functions and Analysis

Exponential Function as a Polynomial of Non-Integer and Negative Exponents: Fractional Derivative of a Constant

Authors: Jesus Sanchez
Comments: 23 Pages.

In this paper, it will be shown a generalization of the series of the exponential function using non-integer and negative exponents for the correspondent polynomial such as: e^x=⋯-Γ(7/2)/π x^(-7/2)+Γ(5/2)/π x^(-5/2)-Γ(3/2)/π x^(-3/2)+Γ(1/2)/π x^(-1/2)+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯ Being Γ(z) the gamma function. The series with negative exponents seems to diverge. But it will be demonstrated that they can be calculated using the following integral. This integral converges and therefore, it can be solved, leading for the exact result for the exponential function. e^x=x^(-1/2)/π ∫_0^∞〖(t^(-1/2) e^(-t))/(1+t/x) dt〗+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯ A generalization with complex exponents for the series will be shown in the paper. This expansion of the exponential function will lead to an alternative definition of the fractional derivative of a constant. It will be shown that there are infinite solutions for the fractional derivative of a constant. But the only one that at the same time keeps the derivative of the exponential function being itself again, is the following: (d^(1/2) C)/(dx^(1/2) )=C/π ∫_0^∞〖(t^(-1/2) e^(-t) x^(-1/2))/(1+t/x) dt〗 Again, a generalization for fractional derivatives of zth grade (being z a general complex number) of a constant will be shown.
Category: Functions and Analysis

An Application of the Cauchy Problem in a Semi-Empirical Context

Authors: Pardis Tabaee Damavandi
Comments: 3 Pages.

The Cauchy problem has been applied to a simple case of linear function in the form y=f(x) expressing wavelength values previously acquired through computer simulations and satisfying the observed empirical initial condition that the extrapolated value of the C-N torsion of 14.10 μm from the Spitzer Telescope spectrum should be sustained.
Category: Functions and Analysis

On Evaluation of an Improper Real Integral Involving a Logarithmic Function

Authors: Henry Wong
Comments: 4 Pages.

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.
Category: Functions and Analysis

Global Stability for a System of Parabolic Conservation Laws Arising from a Keller-Segel Type Chemotaxis Model

Authors: Zefu Feng, Jiao Xu, Kun Zhao, Changjiang Zhu
Comments: 32 Pages.

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.
Category: Functions and Analysis

Time-periodic Solution to the Compressible Viscoelastic Flows in Periodic Domain

Authors: Zefu Feng, Jiao Xu, Kun Zhao
Comments: 19 Pages.

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.
Category: Functions and Analysis

Initial-boundary Value Problems for a System of Parabolic Conservation Laws Arising From a Keller-segel Type Chemotaxis Model

Authors: Zefu Feng, Jiao Xu, Kun Zhao, Changjiang Zhu
Comments: 22 Pages.

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.
Category: Functions and Analysis

A Sequence of Elementary Integrals Related to Integrals Studied by Glaisher that Contain Trigonometric and Hyperbolic Functions

Authors: Martin Nicholson
Comments: 6 Pages.

We generalize several integrals studied by Glaisher. These ideas are then applied to obtain an analog of an integral due to Ismail and Valent.
Category: Functions and Analysis

A Solution to the (Hyper)invariant Subspace Problem

Authors: Manuel Norman
Comments: 6 Pages.

In this paper we will give an affirmative solution to the (hyper)invariant subspace problem for complex, separable, infinite dimensional reflexive Banach spaces. Our method of proof is based on an extension of Lomonosov Theorem proved in [4]: we will show that every nonscalar operator belongs to the class $\Delta(X)$ defined in [5], which will imply that every nonscalar $T \in B(X)$ has a nontrivial (hyper)invariant subspace. In the last section, we will discuss the relationship between our proof and the general case of the problem for Banach spaces (in particular, nonreflexive ones).
Category: Functions and Analysis

The Concept of a String

Authors: Theophilus Agama
Comments: 6 Pages.

In this short note we introduce and develop the concept of a string. We examine the various elementary properties of a string. Further, we relate the concept of the string to the concept of continuity of a function. In fact we prove that the two are loosely connected.
Category: Functions and Analysis

Limited Polynomials

Authors: Theophilus Agama
Comments: 10 Pages.

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.
Category: Functions and Analysis

Deriving Formula for Volume of Spheres in "Higher Dimensions"

Authors: isreal Morawo
Comments: 4 Pages.

In this short paper, I will expresss and prove the volume of n-dimensional Spherical balls. The evaluation corely is expressed via multivariate calculus, factorials and special function.
Category: Functions and Analysis

Differential Quotients and Division by Zero

Authors: Saburou Saitoh
Comments: 8 Pages. The title of the paper will be very fine with the result.

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 $\tan (\pi/2)=0$.
Category: Functions and Analysis

Approximation of Harmonic Series

Authors: Aryan Phadke
Comments: 9 Pages.

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.
Category: Functions and Analysis

Majorization in the Framework of 2-Convex Systems

Authors: George Precupescu
Comments: Pages. The v1 version is a very early draft, expect many changes/corrections on next versions

We define a 2-convex system by the restrictions $x_{1} + x_{2} + \ldots + x_{n} = ns$, $e(x_{1}) + e(x_{2}) + \ldots + e(x_{n}) = nk$, $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$ where $e:I \to \RR$ it's a strictely convex function. We study the compacity/connexity of $A_S$ (the solution's set) and also the variation intervals for $x_k$. Next we define a majorization relation on $A_S$ by $x\preccurlyeq_p y$ $\stackrel{def}{\Leftrightarrow}$ $L_k(x) \leq L_k(y) \ \ \forall 1 \leq k \leq p-1$ and $R_k(x) \leq R_k(y) \ \ \forall p+2 \leq k \leq n$ (for fixed $1 \leq p \leq n-1$) where $L_k(x) = x_1 + \ldots + x_k$, $R_k(x) = x_k + \ldots + x_n$. 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 relatively to $e$. As a consequence, we obtain a more general version for the equal variable method of V. Cartoaje.
Category: Functions and Analysis

First Steps of Vector Differential Calculus

Authors: Eckhard Hitzer
Comments: 31 Pages. Proofs for all common formulas of vector differential calculus in an elementary step by step fashion.

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.
Category: Functions and Analysis

Nothing is Unstable?

Authors: Daniel Thomas Hayes
Comments: 1 Page.

A note on the instability of nothing.
Category: Functions and Analysis

The String Method

Authors: Daniel Thomas Hayes
Comments: 1 Page.

A method for finding exact solutions of differential equations is proposed.
Category: Functions and Analysis

Newton's Limit Operator Has no Sense

Authors: Dmitri Martila
Comments: 2 Pages.

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''.
Category: Functions and Analysis

Quantum Isometries and Noncommutative Geometry

Authors: Teo Banica
Comments: 200 Pages.

The free complex sphere $S^{N-1}_{\mathbb C,+}$ is the noncommutative manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. Certain submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, related to the quantum groups, are known to have Riemannian features, including an integration functional. We review here the known facts on the subject.
Category: Functions and Analysis

Dirichlet Eta Function Negative Integer Formula

Authors: Isaac Mor
Comments: 26 Pages. Dirichlet Eta Function Negative Integer Formula without using the zeta function in the form of a polynomial of a single indeterminate.

f(x)=1n−2n+3n−4n+5n−6n+...+xn=???
Category: Functions and Analysis

An Asymptotic Study of Nonlinear Instability to Langmuir Circulation in Stratified Shallow Layers

Authors: Daniel Thomas Hayes
Comments: 30 Pages.

The CL equations governing instability to Langmuir circulation (LC) are solved by three approximate methods, viz: a small-l asymptotic expansion where l is the spanwise wavenumber, a power series method and a Galerkin method. Interest is focussed on the CL2 instability mechanism to LC and how it is influenced by stratification throughout the layer in which LC live. Results are provided to illustrate the CL2 instability and how it is aected by nonlinearities.
Category: Functions and Analysis

On the Riemann Hypothesis

Authors: Daniel Thomas Hayes
Comments: 2 Pages.

A proposed proof of the Riemann hypothesis.
Category: Functions and Analysis

Log-Trigonometric Integrals and Elliptic Functions

Authors: Martin Nicholson
Comments: 10 Pages.

A class of log-trigonometric integrals are evaluated in terms of elliptic functions.
Category: Functions and Analysis

On Attractivity for $\psi$-Hilfer Fractional Differential Equations Systems

Authors: J. Vanterler da C. Sousa; Donal O'Regan, E. Capelas de Oliveira
Comments: 18 Pages.

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.
Category: Functions and Analysis

Faedo-Galerkin Approximation of Mild Solutions of Nonlocal Fractional Functional Dierential Equations

Authors: J. Vanterler da C. Sousa; Michal Feckan, E. Capelas de Oliveira
Comments: 23 Pages.

The existence, uniqueness and convergence of approximation of mild solutions for a class of nonlocal fractional functional differential equations in Hilbert separable space, will be investigated. To this end, the Gronwall inequality and Faedo-Galerkin approximation will be used.
Category: Functions and Analysis

Attractivity for Differential Equations Systems of Fractional Order

Authors: J. Vanterler da C. Sousa; Mouffak Benchohra; Gaston M. N'Guerekata.
Comments: 15 Pages.

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.
Category: Functions and Analysis

Why Quasi-Interpolation onto Manifold has Order 4

Authors: Markus Sprecher
Comments: 6 Pages.

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 the shortest point projection $P$. We show that for $m\geq 3$ we will have approximation order $4$ and why higher approximation order can not be expected when the control points are constructed as the Projections of the filtered samples using a fixed mask.
Category: Functions and Analysis

Filter Exhaustiveness and Filter Limit Theorems for K-Triangular Lattice Group-Valued Set Functions

Authors: Antonio Boccuto, Xenofon Dimitriou
Comments: 13 Pages.

We give some limit theorems for sequences of lattice group-valued k-triangular set functions, in the setting of filter convergence, and some results about their equivalence. We use the concept of filter exhaustiveness to get uniform (s)-boundedness, uniform continuity and uniform regularity of a suitable subsequence of the given sequence, whose indexes belong to the involved filter.
Category: Functions and Analysis

Relative Uniform Convergence of a Sequence of Functions at a Point and Korovkin-Type Approximation Theorems

Authors: Kamil Demirci, Antonio Boccuto, Sevda Yildiz e Fadime Dirik
Comments: 10 Pages.

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.
Category: Functions and Analysis

On Matrix Methods of Convergence of Order Alpha in L-Groups

Authors: Antonio Boccuto, Pratulananda Das
Comments: 13 Pages.

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.
Category: Functions and Analysis

A Note on Laguerre Original Ode and Polynomials (1879)

Authors: Mohamed E. Hassani
Comments: 4 Pages; 22 References.

In the present note a critical discussion of two ODEs and two polynomials that have been wrongly attributed to the French mathematician Edmond Nicolas Laguerre (1834-1886) is provided. It is shown that Laguerre had nothing to do with such a wrong attribution and the actual discoverer was the Russian mathematician Nikolay Yacovlevich Sonine (1849-1915).
Category: Functions and Analysis

Differential Equations for Conic Section. Revision 5

Authors: Viktor Strohm
Comments: 9 Pages.

: The movement of a point along an ellipse under the action of a generalized force is studied. A well-known differential equation of second-order curves with respect to the focus is derived. Similar arguments are made for the differential equation of second-order curves with respect to the center. Received constant linear velocity for the motion along the ellipse. A comparison is made with the constant of Kepler's third law.
Category: Functions and Analysis

Division by Zero Calculus in Ford Circles

Authors: Saburou Saitoh
Comments: 10 Pages. I am writing a book on the division by zero calculus and for its purpose, I am collecting the related materials. Please kindly give me your kind information.

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.
Category: Functions and Analysis

Ankur Tiwari's Great Discovery of the Division by Zero $1/0 = \tan (\pi/2) = 0$ on $ 2011$

Authors: Saburou Saitoh, Yoshinori Saitoh
Comments: 10 Pages. How will be the paper of the history of division by zero? Please kindly give kind suggestions and comments.

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.
Category: Functions and Analysis

A Fourier Derivative Collocation Method for the Solution of the Navier-Stokes Problem

Authors: Daniel Thomas Hayes
Comments: 6 Pages.

A proposed solution to the millennium problem on the existence and smoothnessof the Navier--Stokes equations.
Category: Functions and Analysis

Extended Proof of the Collatz Conjecture with Quasi-Induction

Authors: Hans Rieder
Comments: 4 Pages.

In this manuscript, we present an extended proof of the Collatzconjecture, based on the novel approach of quasi-induction and a detailed analysis of the shrinking rate. The logarithmic approach plays a central role in demonstrating that the sequence continuously shrinks on average and eventually reaches the number 1. In addition, numerical computations on GitHub are mentioned to support these theoretical results.
Category: Functions and Analysis

Chirality (Electroweak Interaction) Using Geometric (Real Clifford) Algebra Cl(3,0)

Authors: Jesús Sánchez
Comments: 45 Pages.

In this paper, we obtain the left and the right-handed (chirality) representation of the wavefunction using Geometric (real Clifford) Algebra Cl 3,0. We will use both the Chiral representation and the Pauli/Dirac representation. Also, a summary of how all the interactions can be calculated and represented using Geometric (real Clifford) Algebra is shown.
Category: Functions and Analysis

A Proof of the Kakeya Maximal Function Conjecture Via Big Bush Argument

Authors: Johan Aspegren
Comments: 7 Pages.

In this paper we reduce the Kakeya maximal function conjecture to the tube sets of unit measure. We show that the Kakeya maximal function is essentially monotonic. So by adding tubes we can reduce the conjecture to the case of unit measure tube set if we allow the technicality that there are possibly two tubes on the same direction. Then we proof the Kakeya maximal function conjecture from our lemma.
Category: Functions and Analysis

A Proof of the Kakeya Maximal Function Conjecture Via Big Bush Argument

Authors: Johan Aspegren
Comments: 6 Pages.

In this paper we reduce the Kakeya maximal function conjecture to the tube sets of unit measure. We show that the Kakeya maximal function is essentially monotonic. So by adding tubes we can reduce the conjecture to the case of unit measure tube set if we allow the technicality that there are possibly two tubes on the same direction. Then we proof the Kakeya maximal function conjecture from our lemma.
Category: Functions and Analysis

A New Numerical Interpretation of the Concept of Exponentory (Θ Notation)

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

In this paper I show a possible change in the theory of series beyond product. Instead of a resolution Bottom-to-Top we will see a necessary application of the method for exponents that is a process Top-to-Bottom. That implies a change in the numerical results in a same proposition of a series.
Category: Functions and Analysis

On Navier-Stokes Equations

Authors: Dmitri Martila
Comments: 3 Pages.

All classical systems must be Galilean invariant,but Navier--Stokes equations are not. The solutionis the correct derivation of Navier--Stokes equations.
Category: Functions and Analysis

The Infinite Series on the Lviv Scottish Book is Bounded

Authors: Amine Oufaska
Comments: 2 Pages.

In this article we prove that the infinite series on the Lviv Scottish book is bounded , consequently it is convergent.
Category: Functions and Analysis

The Infinite Series 1+2+3+4+⋯ is Strictly Divergent

Authors: Amisha Oufaska
Comments: 2 Pages.

In this paper , I prove that the infinite series 1+2+3+4+⋯ is strictly divergent or simply 1+2+3+4+⋯=+∞ applying an argument by contradiction .
Category: Functions and Analysis

Integrability of Continuous Functions in 2 Dimensions

Authors: Hans Detlef Hüttenbach
Comments: 16 Pages. Several misprints corrected.

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.
Category: Functions and Analysis

Nontrivial Zeros of the Riemann Zeta Function

Authors: James C. Austin
Comments: 7 Pages.

The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by adding an extra unbounded term to the traditional definition, extending its validity to Rez>0. This is then analysed in both halves of the critical strip (0<Rez<1/2, 1/2<Rez<1 ). A contradiction is obtained when it is assumed that zeta(z)=0 in either of these halves.
Category: Functions and Analysis

Nontrivial Zeros of the Riemann Zeta Function

Authors: James C Austin
Comments: 7 Pages. This is version 2 of this document.

The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by solving an integral form of the zeta function for the real parts and showing that a ratio of divergent terms can only be finite and nonzero, as required, when the real parts are exactly 1/2.
Category: Functions and Analysis

Approximation by Power Series of Functions

Authors: Andrej Liptaj
Comments: 14 Pages.

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.
Category: Functions and Analysis

Approximating Roots and π Using Pythagorean Triples

Authors: Joseph Bakhos
Comments: 7 Pages. Published December 18, 2022: Applied Mathematical Sciences, Vol. 16, 2022, no. 12, 665-677 doi: 10.12988/ams.2022.917217 Link: http://www.m-hikari.com/ams/ams-2022/ams-9-12-2022/p/bakhosAMS9-12-2022.pdf

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.
Category: Functions and Analysis

Hyperoperator Analysis

Authors: Dmitrii V. Guryanov
Comments: 36 Pages.

The purpose of this article as a continuation of development of the Multiplical concept is to give an answer to the earlier raised question of why the place of the operator in the function y = e↗x is taken by the operator - a power tower with left associativity, and not with the generally accepted right associativity (the Tetration). Answering on this question required to conduct an hyperoperator analyze. The hyperoperator nature is considered, definition is made and an alternative way of its development is proposed in the present analysis.
Category: Functions and Analysis

Hyperoperator Analysis

Authors: Dmitrii V. Guryanov
Comments: 35 Pages.

The purpose of this article as a continuation of development of the Multiplical concept is to give an answer to the earlier raised question of why the place of the operator in the function y = e↗x is taken by the operator - a power tower with left associativity, and not with the generally accepted right associativity (the Tetration). Answering on this question required to conduct an hyperoperator analyze. The hyperoperator nature is considered, definition is made and an alternative way of its development is proposed in the present analysis.
Category: Functions and Analysis

Singular Properties

Authors: Dmitrii V. Guryanov
Comments: 59 Pages.

The purpose of this article as a continuation of development of the multiplical topic is to find a solution for operation of differentiation and factorization of a function with points of interruption, points where function turns to zero. The solution which allows restoring the original function as result of reverse operation of integration and factorial-multiplication of previously obtained derivative and factor-derivative respectively and with an appropriate selection of an arbitrary multiplier B or addend C, respectively. As the result of the work madenew classes of function properties are introduced as function point properties.
Category: Functions and Analysis

Multiplical Concept

Authors: Dmitrii V. Guryanov
Comments: 48 Pages.

The purpose of this article is to introduce and to describe a concept of math calculus"Multiplical". To my total surprise I have found that currently such a concept does not existamong set of math definitions in its direct and explicit form. Nevertheless there are number ofareas of its practical use, where this concept would be suitable and potentially would benaturally used in its direct and explicit form, especially, in statistics, finance and economyresearches and analysis and many other areas. Moreover from my perspective this conceptperfectly fits into the coherent system of standard mathematical concepts and operators andshould take its rightful place there. In this article also other topics are considered and someinteresting conclusions are made.
Category: Functions and Analysis

Multiplical Concept

Authors: Dmitri V. Guryanov
Comments: 45 Pages. in English and in Russian

The purpose of this article is to introduce and to describe a concept of math calculus “Multiplical”. To my total surprise I have found that currently such a concept does not exist among set of math definitions in its direct and explicit form. Nevertheless there are number of areas of its practical use, where this concept would be suitable and potentially would be naturally used in its direct and explicit form, especially, in statistics, finance and economy researches and analysis and many other areas. Moreover from my perspective this concept perfectly fits into the coherent system of standard mathematical concepts and operators and should take its rightful place there. In this article also other topics are considered and some interesting conclusions are made.
Category: Functions and Analysis

Multiplical Concept

Authors: Dmitri V. Guryanov
Comments: 23 Pages. in Russian

The purpose of this article is to introduce and to describe a concept of math calculus “Multiplical”. To my total surprise I have found that currently such a concept does not exist among set of math definitions in its direct and explicit form. Nevertheless there are number of areas of its practical use, where this concept would be suitable and potentially would be naturally used in its direct and explicit form, especially, in statistics, finance and economy researches and analysis and many other areas. Moreover from my perspective this concept perfectly fits into the coherent system of standard mathematical concepts and operators and should take its rightful place there. In this article also other topics are considered and some interesting conclusions are made.
Category: Functions and Analysis

A Proof of the Line Like Kakeya Maximal Function Conjecture

Authors: Johan Aspegren
Comments: 5 Pages.

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.
Category: Functions and Analysis

A Proof of the Kakeya Maximal Function Conjecture from a Special Case

Authors: Johan Aspegren
Comments: 6 Pages.

First in this paper we will prove the Kakeya maximal function conjecture in a special case when tube intersections behave like line intersections. This paper highlights how different tube intersections can be than line intersections. However, we show that the general case can be deducted from the line like case.
Category: Functions and Analysis

The Generating Function Technique and Algebraic Ordinary Differential Equations

Authors: Robert Lloyd Jackson
Comments: 8 Pages. contact at rljacksonmd@gmail.com

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.
Category: Functions and Analysis

The Generating Function Technique and Algebraic Ordinary Differential Equations

Authors: Robert Lloyd Jackson
Comments: 7 Pages. contact at rljacksonmd@gmail.com

In the past, theorems have shown implementing a (former) 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 Laurent series comprised of product ring-based generating functions 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.
Category: Functions and Analysis

The Generating Function Technique and Algebraic Ordinary Differential Equations

Authors: Robert Lloyd Jackson
Comments: 8 Pages. contact at rljacksonmd@gmail.com

In the past, theorems have shown implementing a (former) 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 Laurent series comprised of product ring-based generating functions 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.
Category: Functions and Analysis

Matrix Exponential Computational Algorithm

Authors: Kenneth C. Johnson
Comments: 11 Pages.

A numerical algorithm for the matrix exponential is developed, based on the scale-and-square method applied to a Padé approximant for small-norm matrices.
Category: Functions and Analysis

The Solution of the Invariant Subspace Problem Part I. Complex Hilbert space

Authors: Jaykov Foukzon
Comments: 101 Pages.

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory TG.It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the existence of inaccessible cardinals [1].In this paper we look at a set theory NC_{∞^{#}}^{#}, based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [2]-[5].In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule.We present a new approach to the In this paper we deal with set theory INC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule.We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace.Non-conservative extension based on set theory NC_{∞}^{} of the model theoretical nonstandard analysis[6] is considered
Category: Functions and Analysis

The Local Product and Local Product Space

Authors: Theophilus Agama
Comments: 6 Pages.

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 $\mathbf{e}(q)=e^{2\pi iq}$.
Category: Functions and Analysis

Analysis of the Collatz Conjecture Through the Methods of Sequence Theory

Authors: Ruslan Enikeev
Comments: 23 Pages.

A solution is proposed for the so-called Collatz conjecture, also known as the "3��+1 problem". The idea of the proof involves representing the algorithm's operations through an infinite sequence as a formal object. Hypotheses regarding the convergence or divergence are considered within the framework of sequence theory corresponds to hypotheses about the output values of the algorithm.
Category: Functions and Analysis

Collatz Conjecture Solution Through the Convergence Study

Authors: Ruslan Enikeev
Comments: 15 Pages.

We propose a solution to the so-called Collatz conjecture problem. The iterations of the algorithm are represented through the sequence which convergence analysis is supposed to confirm the conjecture.
Category: Functions and Analysis

Collatz Conjecture Solution Through the Convergence Study

Authors: Ruslan Enikeev
Comments: 13 Pages.

We propose a solution to the so-called Collatz conjecture problem. The iterations of the algorithm are represented through the sequence which convergence analysis is supposed to confirm the conjecture.
Category: Functions and Analysis

Collatz Conjecture Solution Approach Through the Series Convergence Study

Authors: Ruslan Enikeev
Comments: 7 Pages.

We propose a solution approach to the so-called Collatz conjecture problem. The iterations of the algorithm are represented through the series which convergence analysis is supposed to confirm the conjecture.
Category: Functions and Analysis

Set Theory NC_{∞^{}}^{} Based on Bivalent Infinitary Logic with Restricted Modus Ponens Rule. Basic Real Analysis on External Non-Archimedean Field ℝ_{c}^{}. Basic Complecs Analysis on External Field ℂ_{c}^{}=ℝ_{c}^{}[i].

Authors: Jaykov Foukzon
Comments: 132 Pages.

In this paper we deal with set theory NC_{∞}^{} based on gyper infinitary logic with Restricted Modus Ponens Rule [1]-[3].The main goal of this paper is to present basic analysis on external non Archimedean field ℝ_{c}^{}.The non Archimedean external field ℝ_{c}^{} consist of Cauchy hyperreals.The non-Archimedean external field ℝ_{c}^{#}≠┊^{∗}ℝ┊ is obtained as generalized Cauchy completion of non-Archimedean field ℚ^{#} or ^{∗}ℚ.In order to obtain such completion we deal with external hyper infinite Cauchy sequences {x_{n}}_{n∈ℕ^{#}},{x_{n}}_{n∈┊^{∗}ℕ┊}.We have emphasised that such external Cauchy sequences defined external hyperreal numbers in natural way. Basic Analysis on External Non-Archimedean Field ℝ_{c}^{#} is considered.
Category: Functions and Analysis

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 147 Pages. Remedied a few errata. Grammar and punctuation updates.

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.
Category: Functions and Analysis

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 147 Pages. Remedied a few errata. Grammar and punctuation updates.

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.
Category: Functions and Analysis

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis

Authors: Jonathan W. Tooker
Comments: 147 Pages. Remedied a few errata. Grammar and punctuation updates.

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.
Category: Functions and Analysis

Rotation Without Imaginary Numbers, Transcendental Functions, or Infinite Sums

Authors: Joseph Bakhos
Comments: Published in: Journal of Advances in Mathematics and Computer Science, Volume 38, Issue 6, Pages 33-38. DOI: 10.9734/jamcs/2023/v38i61766 Published: 27 March 2023. May be viewed at this site: https://journaljamcs.com/index.php/JAMCS/article/view/1766

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.
Category: Functions and Analysis

Kouider Function have Basis a

Authors: Kouider Mohammed Ridha
Comments: 4 Pages.

Josephus function is a new numerical function which presented by Kouider (2019,[1]) by studying Joseph's problem . In this paper we interesting in study of its derived function and some of its related properties for Josephus function. From this point of view we saw that we can define a more comprehensive function than the Josephus function. And we called it the Kouider function with basis ض. We have also studied some of its related properties with proof as well.
Category: Functions and Analysis

Kouider Function have Basis a

Authors: Kouider Mohammed Ridha
Comments: 4 Pages.

Josephus function is a new numerical function which presented by Kouider (2019,[1]) by studying Joseph's problem . In this paper we interesting in study of its derived function and some of its related properties for Josephus function. From this point of view we saw that we can define a more comprehensive function than the Josephus function. And we called it the Kouider function with basis a. We have also studied some of its related properties with proof as well.
Category: Functions and Analysis

Kouider Function have Basis a

Authors: Kouider Mohammed Ridha
Comments: 4 Pages.

Josephus function is a new numerical function which presented by Kouider (2019,[1]) by studying Joseph's problem . In this paper we interesting in study of its derived function and some of its related properties for Josephus function. From this point of view we saw that we can define a more comprehensive function than the Josephus function. And we called it the Kouider function with basis a. We have also studied some of its related properties with proof as well
Category: Functions and Analysis

Orbital Rotation Bulk Intersection Theory - ORBIT

Authors: Seamus McCelt
Comments: 9 Pages.

The universe does NOT have an instruction manual to use on itself. The way things work can only be very simple and automatic. Spiral arms and or galactic bars are no exception, they are formed by 4 simple mechanisms: Group mass, Orbit lag, Star lag, and Triple focus.
Category: Functions and Analysis

A General Definition of Means and Corresponding Inequalities

Authors: Pranjal Jain
Comments: 18 Pages.

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.
Category: Functions and Analysis

Topological Stationarity and Precompactness of Probability Measures

Authors: Yu-Lin Chou
Comments: 6 Pages. Two minor changes: an example added; a sentence modified.

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.
Category: Functions and Analysis

Distribution of Integrals of Wiener Paths

Authors: Yu-Lin Chou
Comments: 3 Pages. expository article; the first version is somewhat misleading.

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 $C([0,1], \R)$.
Category: Functions and Analysis

Processing of the ℍ Holomorphic Functions

Authors: Michael Parfenov
Comments: 16 Pages.

To automate cumbersome, error-prone and tedious manual procedures of calculations with quaternionic holomorphic (ℍ -holomorphic) functions we have developed and present here a special programmes pack in the Wolfram Mathematica® programming language. By using this pack a lot of examples of ℍ -holomorphic functions is processed. They give conclusive evidence that the so-called essentially adequate theory of quaternionic holomorphy is true. All considered ℍ -holomorphic functions are built from complex holomorphic ones in accordance with the general constructing rule defined earlier. At that combinations of ℍ -holomorphic functions are built from the simplest basis (irreducible) ℍ -holomorphic functions by using the usual rules of quaternionic algebraic operations.
Category: Functions and Analysis

Processing of the H-Holomorphic Functions

Authors: Michael Parfenov
Comments: 16 Pages.

To automate cumbersome, error-prone and tedious manual procedures of calculations with quaternionic holomorphic (ℍ -holomorphic) functions we have developed and present here a special programmes pack in the Wolfram Mathematica® programming language. By using this pack a lot of examples of ℍ -holomorphic functions is processed. They give conclusive evidence that the so-called essentially adequate theory of quaternionic holomorphy is true. All considered ℍ -holomorphic functions are built from complex holomorphic ones in accordance with the general constructing rule defined earlier. At that combinations of ℍ -holomorphic functions are built from the simplest basis (irreducible) ℍ -holomorphic functions by using the usual rules of quaternionic algebraic operations.
Category: Functions and Analysis

Quantum Permutations and Quantum Reflections

Authors: Teo Banica
Comments: 300 Pages.

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.
Category: Functions and Analysis

Liouville-Type Theorems Outside Small Sets

Authors: Bulat N. Khabibullin
Comments: Pages.

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.
Category: Functions and Analysis

Exponential Function as a Polynomial of Non-Integer and Negative Exponents: Newton Binomial Theorem and Generalization Fractional Derivative of a Constant

Authors: Jesús Sánchez
Comments: 28 Pages.

In this paper, it will be shown a generalization of the series of the exponential function using non-integer and negative exponents for the correspondent polynomial such as:e^x=⋯-Γ(7/2)/π x^(-7/2)+Γ(5/2)/π x^(-5/2)-Γ(3/2)/π x^(-3/2)+Γ(1/2)/π x^(-1/2)+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯Being Γ(z) the gamma function. The series with negative exponents seems to diverge. But it will be demonstrated that they can be calculated using the following integral. This integral converges and therefore, it can be solved, leading for the exact result for the exponential function.e^x=x^(-1/2)/π ∫_0^∞〖(t^(-1/2) e^(-t))/(1+t/x) dt〗+1/Γ(3/2) x^(1/2)+1/Γ(5/2) x^(3/2)+1/Γ(7/2) x^(5/2)+⋯A generalization with complex exponents for the series will be shown in the paper.The Newton binomial theorem is also generalized this way (being w a free complex parameter):(m+1)^s=∑_(k=-∞@k=integer)^∞〖m^(k+w) Γ(s+1)/Γ(k+w+1)Γ(s-k-w+1) ==⋯+m^(-2+w) Γ(s+1)/Γ(-1+w)Γ(s+3-w) +m^(-1+w) Γ(s+1)/Γ(w)Γ(s+2-w) 〗+m^w Γ(s+1)/Γ(w+1)Γ(s+1-w) +m^(1+w) Γ(s+1)/Γ(w+2)Γ(s-w) +m^(2+w) Γ(s+1)/Γ(w+3)Γ(s-1-w) +⋯.It will be also shown that there are infinite solutions for the fractional derivative of a constant. But the only one that at the same time keeps the derivative of the exponential function being itself again, is the following:(d^(1/2) C)/(dx^(1/2) )=C/π ∫_0^∞〖(t^(-1/2) e^(-t) x^(-1/2))/(1+t/x) dt〗Again, a generalization for fractional derivatives of zth grade (being z a general complex number) of a constant will be shown.
Category: Functions and Analysis

A Sequence of Elementary Integrals Related to Integrals Studied by Glaisher that Contain Trigonometric and Hyperbolic Functions

Authors: Martin Nicholson
Comments: 9 Pages. A new theorem, and a new section with a proof, and 5 new references are added. Discussion section is updated. Discussion of some open questions is added.

We generalize several integrals studied by Glaisher. These ideas are then applied to obtain an analog of an integral due to Ismail and Valent.
Category: Functions and Analysis

Quasinilpotent Operators on Separable Hilbert Spaces Have Nontrivial Invariant Subspaces

Authors: Manuel Norman
Comments: 9 Pages. The previous version contained an error in Remark 2.1. This new version deals with the case of quasinilpotent operators on separable Hilbert spaces.

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.
Category: Functions and Analysis

Majorization in the Framework of 2-Convex Systems

Authors: George Precupescu
Comments: 30 Pages. This is an English version (the original v1 was in Romanian)

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
Category: Functions and Analysis

Affine Noncommutative Geometry

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $mathbb R^N,mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres $S^{N-1}_mathbb R,S^{N-1}_mathbb C$ do have free analogues $S^{N-1}_{mathbb R,+},S^{N-1}_{mathbb C,+}$. There are many examples of real algebraic submanifolds $Xsubset S^{N-1}_{mathbb R,+},S^{N-1}_{mathbb C,+}$, some of which are of Riemannian flavor, coming with a Haar integration functional $int:C(X)tomathbb C$, that we will study here. We will mostly focus on free geometry, but we will discuss as well some related geometries, called easy, completing the picture formed by the 4 main geometries, namely real/complex, classical/free.
Category: Functions and Analysis

Quantum Isometries and Noncommutative Geometry

Authors: Teo Banica
Comments: 300 Pages.

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 $tr:C(X)\to\mathbb C$.
Category: Functions and Analysis

Quantum Isometries and Noncommutative Geometry

Authors: Teo Banica
Comments: 200 Pages.

The free complex sphere $S^{N-1}_{\mathbb C,+}$ is the noncommutative manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. Certain submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, related to the quantum groups, are known to have Riemannian features, including an integration functional. We review here the known facts on the subject.
Category: Functions and Analysis

Quantum Isometries and Noncommutative Geometry

Authors: Teo Banica
Comments: 200 Pages.

The free complex sphere $S^{N-1}_{\mathbb C,+}$ is the noncommutative manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. Certain submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, related to the quantum groups, are known to have Riemannian features, including an integration functional. We review here the known facts on the subject.
Category: Functions and Analysis

Dirichlet Eta Function Negative Integer Formula

Authors: Isaac Mor
Comments: 26 Pages. Dirichlet Eta Function Negative Integer Formula without using the zeta function in the form of a polynomial of a single indeterminate.

f(x)=1n−2n+3n−4n+5n−6n+...+xn=??? Pages 22 & 23 were fixed! but everything else is still good :)
Category: Functions and Analysis

Dirichlet Eta Function Negative Integer Formula

Authors: Isaac Mor
Comments: 26 Pages. Dirichlet Eta Function Negative Integer Formula without using the zeta function in the form of a polynomial of a single indeterminate.

f(x)=1n−2n+3n−4n+5n−6n+...+xn=???
Category: Functions and Analysis

An Asymptotic Study of Nonlinear Instability to Langmuir Circulation in Stratified Shallow Layers

Authors: Daniel Thomas Hayes
Comments: 29 Pages.

The CL equations governing instability to Langmuir circulation (LC) are solved by three approximate methods, viz: a small-l asymptotic expansion where l is the spanwise wavenumber, a power series method, and a Galerkin method. Interest is focused on the CL2 instability mechanism to LC and how it is influenced by stratification throughout the layer in which LC live. Some results are provided to illustrate the CL2 instability and how it is affected by nonlinearities.
Category: Functions and Analysis

On the Riemann Hypothesis

Authors: Daniel Thomas Hayes
Comments: 2 Pages.

A proposed proof of the Riemann hypothesis.
Category: Functions and Analysis

On the Riemann Hypothesis

Authors: Daniel Thomas Hayes
Comments: 3 Pages.

A proposed proof of the Riemann hypothesis.
Category: Functions and Analysis

Why Quasi-Interpolation onto Manifold has Order 4

Authors: M. Sprecher
Comments: 7 Pages. typos corrected and reference added

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.
Category: Functions and Analysis