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LOCALLY EKELAND'S VARIATIONAL PRINCIPLE AND SOME surjective mapping THEOREMS

Chinese Annals of Mathematics,Series B 1998年第3期19卷 273-280页

作者： ZHONG CHENGKUI ZHAO PEIHAO Lanzhou Univ Dept Math Lanzhou 730000 Peoples R China

This paper shows that if a Gateaux differentiable functional f has a finite lower bound(although it need not attain it),then,for everyε>0,there exists some point zεsuch that‖f′(zε)‖ε1+h(‖zε‖),where h:[0,∞)→[0,∞)is a continuous function such that∫∞011+h(r)dr=∞.Applications are given to extremum problem and some surjective mappings.

关键词： Variational principle Extremum problem Weak P.S.condition surjective mapping

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Gravitational Space-Time Curve Generation via Accelerated Charged Particles

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Journal of Modern Physics 2016年第9期7卷 863-874页

作者： Edward A. Walker Mathematics Department Florida Memorial University Miami Gardens USA

A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the part

关键词： Charged Particles Accelerated Particles Inertial Mass Gravitational Force Einstein’s Field Equations Space-Time Manifold Schwardchild Metric Stress Energy Tensor surjective mapping Lorentz Force

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