Abstract:
This tutorial review is dedicated to the work of the outstanding Egyptian theoretical physicist and engineering scientist Prof. Mohamed El Naschie. Every physics student knows the well-known Einstein’s mass-energy equation, E=mc^{2}, but unfortunately for physics, few know El Naschie’s modification, E(O)=mc^{2}/22, and El Naschie’s dark energy equation E(D)=mc^{2}(21/22) although this new insight has truly far reaching implications. This paper gives a short tutorial review of El Naschie’s fractal-Cantorian space-time as well as dark energy. Emphasis is put on the fundamental concept of Cantor set, fractal dimensions, zero set, empty set, and Casimir effect.

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.

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.

Abstract:
In recent times, Discrete Space Time Architectures are being considered, in the context of Quantum Gravity, Quantum Super Strings, Dark Energy and so on. We show that such a scheme is intimately tied up with a varying $G$ cosmology, which again explains otherwise inexplicable observations like the anomalous accelerations of Pioneer space crafts as also considerations involving the Zero Point Field, Random Electrodynamics and the derivation of Quantum Mechanical effects therefrom as also intertial mass.

Abstract:
The exact solution for dynamic of conform-flat space homogeneous since dynamic equation is given. Conform mode of space metric changing in Global time theory has negative energy density. Swap of energy to this mode from another ones lead to increasing of Universe homogeneity although probability of this swap from local objects is negligibly small. Conform mode is corresponding to "dark energy" in observation astronomy.

Abstract:
Theoretical and experimental studies of special processes in linear electric circuits have proved the principle of physical reality of imaginary numbers discovered 500 years ago. This principle of physical reality of imaginary numbers has allowed astrophysics to prove the existence of invisible parallel universes and explain the phenomena of dark matter, dark energy and dark space associated therewith. Physical entities corresponding to imaginary, complex and hypercomplex numbers in the other exact sciences together form an invisible and still unknown world that is to be learned by science of the future.

Abstract:
Agegraphic dark energy, has been recently proposed, based on the so-called Karolyhazy uncertainty relation, which arises from quantum mechanics together with general relativity. In the first part of the article we study the original agegraphic dark energy model by including the interaction between agegraphic dark energy and pressureless (dark) matter. The phase space analysis was made and the critical points were found, one of which is the attractor corresponding to an accelerated expanding Universe. Recent observations of near supernova show that the acceleration of Universe decreases. This phenomenon is called the transient acceleration. In the second part of Article we consider the 3-component Universe composed of a scalar field, interacting with the dark matter on the agegraphic dark energy background. We show that the transient acceleration appears in frame of such a model. The obtained results agree with the observations.

Abstract:
The accelerated expansion of our universe results from properties of dark matter particles deduced from Space-Time Quantization. This theory accounts for all possible elementary particles by considering a quantum of length a in addition to c and h. It appears that dark matter particles allow for fusion and fission processes. The resulting equilibrium enables the cosmic dark matter gas to produce dark energy in an adaptive way. It keeps the combined matter-energy density at a constant level, even when space is expanding. This accounts for the cosmological constant Λ and the accelerated expansion of space without requiring any negative pressure. The Big Bang is related to G, c, h and a. It started with a “primeval photon” and led to the cosmic matter-antimatter asymmetry as well as inflation.

Abstract:
A wide-field imager in space could make remarkable progress in two very different frontiers of astronomy: dark energy and extra-solar planets. Embedding such an imager on a much larger and more complicated DE mission would be a poor science-approach under any circumstances and is a prescription for disaster in the present fiscal climate. The 2010 Decadal Committee must not lead the lemming stampede that is driving toward a DE mega-mission, but should stand clearly in its path.

Abstract:
We study the phase space asymptotics of the so called Veneziano ghost dark energy models. Models where the ghost field's energy density: i) $\rho_{ghost}\propto H$, and ii) $\rho_{ghost}\propto H+H^2$, are investigated. Both, cases with and without additional non-gravitational interaction between cold dark matter and ghost dark energy, are subject to scrutiny. We pay special attention to the choice of phase space variables leading to bounded and compact phase space so that no critical point of physical interest is missing. A rich asymptotic structure is revealed: depending on the kind of non-minimal coupling critical points associated with radiation dominance, matter dominance, cold dark matter/ghost dark energy scaling, and ghost dark energy dominance, are found. Past and future attractors, as well as saddle equilibrium points, are identified in the corresponding phase spaces.