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Comments on the Riemann conjecture and index theory on Cantorian fractal space-time  [PDF]
Carlos Castro,Jorge Mahecha
Physics , 2000,
Abstract: An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented. To substantiate this heuristic proposal we show using generalized index-theory arguments, corresponding to the (fractal) spectral dimensions of fractal branes living in Cantorian-fractal space-time, how the required $negative$ traces associated with those derivative operators naturally agree with the zeta function evaluated at the spectral dimensions. The $\zeta (0) = - 1/2$ plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made.
Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively  [PDF]
Mohamed S. El Naschie
Natural Science (NS) , 2016, DOI: 10.4236/ns.2016.812052
Abstract: In a one-dimension Mauldin-Williams Random Cantor Set Universe, the Sigalotti topological speed of light is?\"\" where \"\". It follows then that the corresponding topological acceleration must be a golden mean downscaling of c namely \"\". Since the maximal height in the one-dimensional universe must be \"\" where \"\" is the unit interval length and note that the topological mass (m) and topological dimension (D) where m = D = 5 are that of the largest unit sphere volume, we can conclude that the potential energy of classical mechanics \"\" translates to \"\". Remembering that the kinetic energy is \"\" , then by the same logic we see that \"\"?when m = 5 is replaced by \"\" for reasons which are explained in the main body of the present work. Adding both expressions together, we find Einstein’s maximal energy \"\". As a general conclusion, we note that within high energy cosmology, the sharp distinction between potential energy and kinetic energy of classical mechanics is blurred on the cosmic scale. Apart of being an original contribution, the article presents an almost complete bibliography on the Cantorian-fractal spacetime theory.
An Invitation to El Naschie’s Theory of Cantorian Space-Time and Dark Energy  [PDF]
Leila Marek-Crnjac, Jihuan He
International Journal of Astronomy and Astrophysics (IJAA) , 2013, DOI: 10.4236/ijaa.2013.34053

The paper is a condensed but accurate account of El Naschie’s theory of Cantorian space-time which was used by him to clarify some major problems in theoretical physics and cosmology. In particular El Naschie’s revision and completion of relativity theory and demystification of dark energy are destined to be two milestones in the history of theoretical physics.

Cantorian Fractal Space-Time Fluctuations in Turbulent Fluid Flows and the Kinetic Theory of Gases  [PDF]
A. M. Selvam
Physics , 1999,
Abstract: Fluid flows such as gases or liquids exhibit space-time fluctuations on all scales extending down to molecular scales. Such broadband continuum fluctuations characterise all dynamical systems in nature and are identified as selfsimilar fractals in the newly emerging multidisciplinary science of nonlinear dynamics and chaos. A cell dynamical system model has been developed by the author to quantify the fractal space-time fluctuations of atmospheric flows. The earth's atmosphere consists of a mixture of gases and obeys the gas laws as formulated in the kinetic theory of gases developed on probabilistic assumptions in 1859 by the physicist James Clerk Maxwell. An alternative theory using the concept of fractals and chaos is applied in this paper to derive these fundamental gas laws.
Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos  [PDF]
Mohamed S. El Naschie
Natural Science (NS) , 2016, DOI: 10.4236/ns.2016.83018
Abstract: The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.
Noncommutative Geometry, Negative Probabilities and Cantorian-Fractal Spacetime  [PDF]
Carlos Castro
Physics , 2000, DOI: 10.1016/S0960-0779(00)00196-X
Abstract: A straightforward explanation of the Young's two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometric associated with El Naschie's Cantorian-Fractal transfinite Spacetime continuum.
Fractal Strings as the Basis of Cantorian-Fractal Spacetime and the Fine Structure Constant  [PDF]
Carlos Castro
Physics , 2002,
Abstract: Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, $p$ prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature.
Fractal Time Series—A Tutorial Review
Ming Li
Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/157264
Abstract: Fractal time series substantially differs from conventional one in its statistic properties. For instance, it may have a heavy-tailed probability distribution function (PDF), a slowly decayed autocorrelation function (ACF), and a power spectrum function (PSD) of 1/ type. It may have the statistical dependence, either long-range dependence (LRD) or short-range dependence (SRD), and global or local self-similarity. This article will give a tutorial review about those concepts. Note that a conventional time series can be regarded as the solution to a differential equation of integer order with the excitation of white noise in mathematics. In engineering, such as mechanical engineering or electronics engineering, engineers may usually consider it as the output or response of a differential system or filter of integer order under the excitation of white noise. In this paper, a fractal time series is taken as the solution to a differential equation of fractional order or a response of a fractional system or a fractional filter driven with a white noise in the domain of stochastic processes.
Fantappie's group as an extension of special relativity on Cantorian space-time  [PDF]
G. Iovane,P. Giordano,E. Laserra
Physics , 2004, DOI: 10.1016/j.chaos.2004.04.019
Abstract: In this paper we will analyze the Fantappie group and its properties in connection with Cantorian space-time. Our attention will be focused on the possibility of extending special relativity. The cosmological consequences of such extension appear relevant, since thanks to the Fantappie group, the model of the Big Bang and that of stationary state become compatible. In particular, if we abandon the idea of the existence of only one time gauge, since we do not see the whole Universe but only a projection, the two models become compatible. In the end we will see the effects of the projective fractal geometry also on the galactic and extra-galactic dynamics.
On a Quantum Gravity Fractal Spacetime Equation: QRG ≃ HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion  [PDF]
Mohamed S. El Naschie
Journal of Modern Physics (JMP) , 2016, DOI: 10.4236/jmp.2016.78069
Abstract: The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.
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