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Apropos 1+2+3+4+5+...=<img src="http://***/gif.latex?-\frac{1}{12}"title="-\frac{1}{12}"/>: Mapping Infinity in Light of the Number Circle (or Cycle), in L. Euler’s Footsteps and with the Aid of Two Dimensional infinite Series, and Replacing Negative Infinity and Positive Infinity with Just Infinity

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Advances in Pure Mathematics 2017年第1期7卷 75-133页

作者： Leo Depuydt Egyptology and Assyriology Brown UniversityProvidenceRIUSA

The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.

关键词： Euler L. infinite Series infinite Series of infinite Infinity Geography of Negative Infinity Invalidity of Number Circle Veracity of Number Cycle Number Line Positive Infinity Two dimensional infinite Series Ramanujan Rational Human Intelligence Wallis J.

Theoretical Propulsion System for Fuel Cell Vehicles with infinite Cruising Range

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Journal of Traffic and Transportation Engineering 2018年第4期6卷 164-175页

作者： Katsutoshi Ono Department of Energy Science&Technology Kyoto UniversityKyoto 606-8501Japan

The realization of a clean automobile society would require electrically-powered propulsion systems in vehicles.In recent years,electric vehicles have attracted considerable attention from the perspective of utilizing electricity generated from natural sources,such as solar and wind power.The propulsion method considered in the present investigation differs from the conventional off-board energy scheme in a manner such that pure stoichiometric H2/O2 fuels for fuel cells are generated on-board during vehicle operation.In this method,energy conversion occurs by means of ESI-PSE(electrostatic-induction potential-superposed electrolysis).If a quasi-static process is assumed,the theoretical power requirement to produce pure stoichiometric H2/O2 fuels is only 17%of the total energy required owing to a new method for supplying power to the EC(electrolytic cell).If an ESI-PSE EC is combined with a fuel cell(FC)to form an energy cycle,a HREG(hydrogen redox electric power generator)that uses solid PEMs(polymer electrolyte membranes)for the EC as well as the FC can be realized.According to calculations based on data from the operational conditions of commercially available ECs and FCs,more than 70%of the power delivered from the FC can be extracted for driving a motor constantly while a car is in motion.Because of energy self-sustainability on the HREG side,the power control system should not have any power loss.This propulsion system will realize tough vehicles that can continue running at a top speed at long unlimited cruising range.

关键词： Fuel cell vehicle infinite cruising range water electrolysis electrostatic energy electrostatic induction

General Boundary Value Problems for Nonlinear Uniformly Elliptic Equations in Multiply Connected infinite Domains

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International Journal of Modern Nonlinear Theory and Application 2013年第3期2卷 170-175页

作者： Guochun Wen Yanhui Zhang Dechang Chen Mathematic Department Beijing Technology and Business University Beijing China School of Mathematical Sciences Peking University Beijing China Uniformed Services University of the Health Sciences Bethesda USA

This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the boundary T. The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.

关键词： Boundary Value Problems Nonlinear Elliptic Equations Multiply Connected infinite Domains

Expansion by Laguerre Function for Wave Diffraction around an infinite Cylinder

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Journal of Applied Mathematics and Physics 2015年第1期3卷 75-80页

作者： Mingdong Lv Hui Li Huilong Ren Xiaobo Chen College of Shipbuilding Engineering Harbin Engineering University Harbin China Bureau Veritas Research Department Singapore City Singapore

We consider a vertical circular cylinder on which the vertical variation of water diffraction waves is to be represented by a series of Laguerre functions ?using Laguerre Polynomials . The variation is assumed to be of the form ?with the integer n depending on the radius of cylinder. Generally, the integer n increases for a cylinder of larger diameter. The usual approximation by Laguerre functions is extended by introducing a scale parameter. The convergence of Laguerre series is then dependent on the value of the scale parameter s. The analytical and numerical computations of series coefficients are performed to study the number of series terms to keep the same accuracy. Indeed, the choice of integer n depends on the scale parameter. Furthermore, diffraction waves generated by a semi-sphere inside the cylinder are evaluated on the cylinder surface. It is shown that the approximation by Laguerre series for diffraction waves on the cylinder is effective. This work provides important information for the choice of the radius of control surface in the domain decomposition method for solving hydrodynamic problems of body-wave interaction.

关键词： Expansion by Laguerre Function for Wave Diffraction around an infinite Cylinder

THE h-p VERSION OF THE infinite ELEMENT METHOD AND ITS APPLICATION

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1998年第2期7卷 125-132页

作者：洪莉陈金如

The paper investigates the h-p version of the infinite element method. An exponential conver gence can be abtained by a small amount of computing work.

关键词： Finite element method infinite element method h-p version.

Optimality of Distributed Control for <i>n ×n</i>Hyperbolic Systems with an infinite Number of Variables

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Advances in Pure Mathematics 2013年第6期3卷 598-608页

作者： Ahlam Hasan Qamlo Department of Mathematics Faculty of Applied sciences Umm AL-Qura University Makkah Saudi Arabia

In this paper, we study the existence of solutions for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients. The necessary and sufficient conditions for optimality of the distributed control with constraints are obtained and the set of inequalities that defining the optimal control of these systems are also obtained.

关键词： Hyperbolic Operators Cooperative Systems Spaces With an infinite Number of Variables Existence and Uniqueness of Solutions Distributed Control Problems Control Constraints Variable Coefficients

On a Problem of an infinite Plate with a Curvilinear Hole inside the Unit Circle

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Applied Mathematics 2015年第1期6卷 206-220页

作者： F. S. Bayones B. M. Alharbi Department of Mathematics Faculty of Science Taif University Taif KSA

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

关键词： Complex Variable Method An infinite Plate Curvilinear Hole Conformal Mapping Goursat Functions

On the Total Dynamic Response of Soil-Structure Interaction System in Time Domain Using Elastodynamic infinite Elements with Scaling Modified Bessel Shape Functions

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American Journal of Computational Mathematics 2013年第2期3卷 104-109页

作者： Konstantin Kazakov Structural Mechanic Department VSU “L. Karavelov” Sofia Bulgaria

This paper is devoted to a new approach—the dynamic response of Soil-Structure System (SSS), the far field of which is discretized by decay or mapped elastodynamic infinite elements, based on scaling modified Bessel shape functions are to be calculated. These elements are appropriate for Soil-Structure Interaction problems, solved in time or frequency domain and can be treated as a new form of the recently proposed elastodynamic infinite elements with united shape functions (EIEUSF) infinite elements. Here the time domain form of the equations of motion is demonstrated and used in the numerical example. In the paper only the formulation of 2D horizontal type infinite elements (HIE) is used, but by similar techniques 2D vertical (VIE) and 2D corner (CIE) infinite elements can also be added. Continuity along the artificial boundary (the line between finite and infinite elements) is discussed as well and the application of the proposed elastodynamical infinite elements in the Finite element method is explained in brief. A numerical example shows the computational efficiency and accuracy of the proposed infinite elements, based on scaling Bessel shape functions.

关键词： Soil-Structure Interaction Wave Propagation infinite Elements Finite Element Method Bessel Functions Duhamel Integral

Boundary Control Problem of infinite Order Distributed Hyperbolic Systems Involving Time Lags

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Intelligent Control and Automation 2012年第3期3卷 211-221页

作者： Bahaa Gaber Mohamed Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef Egypt

Various optimal boundary control problems for linear infinite order distributed hyperbolic systems involving constant time lags are considered. Constraints on controls are imposed. Necessary and sufficient optimality ... 详细信息

关键词： Boundary Control (n × n) Hyperbolic Systems Time Lags Distributed Control Problems Neumann Conditions Existence and Uniqueness of Solutions infinite Order Operator