Projective Relativity: Present Status and Outlook

We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms and much more in the case of newer higher dimensional Kaluza–Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint.     

瓶罐玻璃行业企业绩效评价

笔者依据所能获得的资料———《瓶罐玻璃行业企业经济信息汇编》[6],建立了一套经济实用的瓶罐玻璃行业企业绩效评价指标体系;提供了一种科学评价方法和计算机化的评价技术;能适时地对该行业内各企业绩效进行评价和比较分析;及时获得各企业在全行业内竞争位次;及时发现影响企业绩效的主要因素;为高层管理者制定和调整企业管理决策提供了参考依据。     

Klein,Einstein, and Five-Dimensional Unification

I examine the changing attitudes of Oskar Klein (1894–1977) and Albert Einstein (1879–1955) toward the notion of extending general relativity by an extra dimension with the aim of encompassing electromagnetism and gravitation in a unified field theory. I show how Klein developed his model of five-dimensional unification with the goal of explaining the discreteness of atomic energy levels, and how Einstein later embraced that goal. By examining the correspondence between Klein and Einstein, some of which was relayed through Paul Ehrenfest (1880–1933), I speculate that Klein’s work helped motivate Einstein to explore deterministic five-dimensional theories as a potential alternative to probabilistic quantum mechanics. Finally, I consider the contributions of Wolfgang Pauli (1900–1958) to the subject and elucidate his role in convincing Klein and Einstein that their models were not viable. Paul Halpern is Professor of Physics at the University of the Sciences in Philadelphia. He currently is a member of the Executive Committee of the Forum on the History of Physics of the American Physical Society.     

The optimization of silica gel synthesis from chemical bottle waste using response surface methodology

To reduce the amount of hazardous chemical bottle waste in the environment, we report the optimization research of silica extraction in chemical bottle waste into silica gel. Alkali fusion and sol–gel process were utilised to prepare silica gel effectively. The alkali fusion process was carried out by adding sodium hydroxide to produce sodium silicate. Afterwards, silica gel was prepared by the sol–gel method using hydrochloric acid. Box-Behnken Design (BBD) was applied to Optimisation factors the poptimiseactors affecting the silica recovery. The factors that optimised mass ratio, particle size, and temperature. The optimum recovery of silica gel was obtained by SiO₂: NaOH mass ratio of 1:3, the particle size of 63–74 µm, and a temperature of 800 °C. The purity of silica gel optimum is 63.74% characterised using X-ray fluorescence. The structure of silica gel is the appearance of amorphous peaks at 2θ 20-30° characterised using an x-ray diffractogram. The silica gel surface was characterises using scanning electron microscopy-energy dispersive x-ray. It showed an irregular surface and characteristic showed that silica gel had a radius of 15.74 nm and a specific surface area of 297.08 m².     

Nordström,Ehrenfest, and the Role of Dimensionality in Physics

In the summer of 1916, Finnish physicist Gunnar Nordström (1881–1923) arrived in Leiden to carry out research with Paul Ehrenfest (1880–1933), Hendrik A. Lorentzs successor in the chair of theoretical physics. Nordström had recently published the first five-dimensional unified model of the universe, a theory that went virtually unnoticed by the physics community. Ehrenfests personal journals reveal that Nordströms visit coincided with a flowering of Ehrenfests own interest in dimensionality, which resulted in his well-known paper on the connection between the fundamental laws of physics and the three-dimensionality of space. I examine Nordströms and Ehrenfests collaboration and explore the relationship between their ideas and the Kaluza-Klein model of five-dimensional unification.Paul Halpern is Professor of Physics at the University of the Sciences in Philadelphia. He received a Guggenheim Fellowship in 2002 to study the history of dimensionality in science.     

Klein polyhedra for three extremal cubic forms

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms g _i, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms g ₁ – g ₄. They also computed the convergents for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms g ₅, – g ₇ and the adjoint form g ₇ ^* . Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincaré, Brun, Parusnikov, and Bryuno, are computed. The position of the convergents of the continued fractions with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Eulers and Poincarés algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 566–583.Original Russian Text Copyright © 2005 by V. I. Parusnikov.This revised version was published online in April 2005 with a corrected issue number.     

From the Cube to the Dyck and Klein Tessellations: Implications for the Structures of Zeolite-like Carbon and Boron Nitride Allotropes

Geometrical transformations can be described which have the effect of multiplying the numbers of vertices in a trivalent polyhedron by three, four, or seven. Tripling the cube by the so-called leapfrog transformation gives the truncated octahedron. Quadrupling the cube followed by identifying the square faces to give a genus 3 surface gives the Dyck surface of 12 octagons. Septupling the cube by the so-called capra transformation followed by identifying the square faces to give a genus 3 surface gives the Klein surface of 24 heptagons. These geometrical transformations relate to the construction of low-density zeolite-like structures for carbon and boron nitride allotropes based on a cubic lattice     

On the Klein–Gordon Equation in Higher Dimensions: Are Particle Masses Variable?

We consider ways to generalize the 4D Klein–Gordon equation of particle physics to higher dimensions. The most promising approach implies that the mass which appears in the 4D relation is a term in the source-free 5D relation. We check this explicitly for the case of exact solitonic and cosmological solutions of the Kaluza–Klein equations. In general, particle masses are variable; but are constant for the Schwarzschild and late-universe cases, in agreement with data from the solar system and astrophysics. Our results have significant implications for cosmology, and can easily be extended to 10D superstrings, 11D supergravity and higher dimensions.     

Improved oxygen delivery in a continuous-roller-bottle reactor

Variations to the original aeration system in a continuous roller bottle reactor of novel design have been tested and compared for optimal oxygen (O) delivery. Reactor operating parameters that affect O transfer are rotation rate, liquid-volume level, fresh-feed rate, and supplementary-aeration rate. Design modifications to enhance gas-liquid O transfer include the addition of wall baffles and center baffles. The number and location of each of these baffles are compared for their effect on k_La values in the reaction chamber. The liquid feed into the system has been modified to improve the axial liquid mixing and O transfer.     

A note on the reducibility of automorphisms of the Klein curve and the -invariant of mapping tori

We give a characterization for the reducibility of automorphisms of the genus 3 Klein curve in terms of the -invariant of finite order mapping tori.