El Naschie’s cantorian strings and duality in Weyl–Dirac theory

In the weak-field approximation, some implications of duality in the Weyl–Dirac (WD) theory, using the Gregorash–Papini–Wood approach, are investigated. Any particle is in a permanent interaction with the ‘subquantic level’ (Madelung’s fluid) and, as a result of this interaction, the particle acquires the proper fluctuation curvature and the proper fluctuation energy, respectively. By fixing the fluctuations scale, the quantum fluid orders either by means of bright cnoidal oscillation modes inducing causality, or by means of dark cnoidal oscillation modes inducing acausality, and non-linear effects, respectively. The periodic mode is associated with the undulatory characteristic, and the solitonic one with the corpuscular one. By not fixing the fluctuations scale and keeping the symmetry, the quantum fluid orders like a two-dimensional (2D) lattice of vortices, so that the duality needs coherence. In the compatibility between quantum hydrodynamics in the Madelung’s representation and the wave mechanics, the self-gravitational field of the Weyl–Dirac type physical object is generated. El Naschie’s space–time implies, by means of transfinite heterotic string theory, the masses of nucleons, and, by the gravitational fractional quantum Hall effect, the dispersion of the wave-packet on the particle. The analysis of the fractal dimension of the physical object described by the WD theory shows that the waves, and corpuscle, respectively are 2D projections of a higher dimensional special string in El Naschie’s space–time (El Naschie’s string).     

Spacetime non-commutativity,generalized uncertainty principle and the fine structure constant

Quantum gravitational effects and spacetime non-commutativity should affect the value of the fine structure constant. In this paper, using generalized uncertainty principle, we calculate the modified fine structure constant in non-commutative spacetime.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

Fuzzy Dodecahedron topology and E-infinity spacetime as a model for quantum physics

The geometry of classical platonic solids and their generalization to four-dimensional fuzzy polytopes are considered. Subsequently it is shown how the so obtained relationships and the associated symmetry groups are related to high energy particle physics. In particular the topology of a fuzzy Dodecahedron and four-dimensional polytopes are used to give information about the elementary particles content of the standard model of high energy physics.     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature and the structure of generalized solutions for the constant mean curvature equation

As is well known, no manifold of constant mean curvature 1 can exist in a domain strictly containing the unit ball. In part 1 of this paper, we shall consider the problem of estimating absolute apriori bounds for a manifold of constant mean curvature 1 in a ball of radius less than 1 without imposing boundary conditions or bounds of any sort in the higher dimensional case. It is also well-known that we may reduce the capillary problem in the absence of gravity to the variational problem for the functional indicated in the beginning of 2.6.2. In case that a generalized solution for exists, we consider the sets P and N where this generalized solution takes the value and respectively. Using the method in part 1, we shall, in part 2, try to characterize the geometry of the boundaries of P and N in the interior of the domain and ask whether they are spherical caps or not. Received May 27, 1997 / Accepted October 3, 1997     

Fuzzy Dodecahedron topology and E-infinity spacetime as a model for quantum physics

The geometry of classical platonic solids and their generalization to four-dimensional fuzzy polytopes are considered. Subsequently it is shown how the so obtained relationships and the associated symmetry groups are related to high energy particle physics. In particular the topology of a fuzzy Dodecahedron and four-dimensional polytopes are used to give information about the elementary particles content of the standard model of high energy physics.     

On the equations for p-adic closed and open strings

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems. The text was submitted by the author in English.     

Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

This paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove similar results for the wave equation on the exterior of a very slowly rotating Kerr spacetime with recent work applying the vector-field method to the relativistic Vlasov equation.     

Arguments for the compactness and multiple connectivity of our cosmic spacetime

Some global topological as well as quantative arguments are given, indicating that our universe is most probably compact, multiply connected and without boundaries. The analysis leading to this tentative conclusion is based on a combination of Nash Euclidean embedding theorems, the local isomorphism theorem, cosmic crystallography and the theory of fractal-Cantorian spacetime. It is shown that the correct topological dimension D = 4 of space is derived from the Euclidean embedding of spacetime quanta when the corresponding manifold is assumed to be compact. This and other conclusions regarding multi-connectivity seems to reinforce the findings of relatively recent research results on topological cosmology by Luminet et al. (see Nature 425;9 Oct. 2003:593–95).     

The energy—Momentum tensor as the source of the gravitational field and the effective Riemannian spacetime

It is shown in this paper that the assumption of the matter energy—momentum tensor is the source of the gravitational field leads naturally to an effective Riemannian geometry of spacetime.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 538–542, September, 1995.     

A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov

Ladyzhenskaya & Solonnikov (1976) introduced a representationtheorem in ³, which contained an integral inequality involvinga multiplicative dimensionless constant. The existence of theconstant was established but not its magnitude which dependsonly on the shape of the domain. In this paper, we derive anupper bound for the optimal constant when the underlying domainis star shaped.     

The 26 dimensions of heterotic strings as a probabilistic standard deviation and expectation value of an infinite dimensional space

We give a short derivation showing that the 26 dimensions as indeed the 10 dimensions of heterotic string theory may be derived using classical probability theory as higher-order expectation and standard deviation of combined spaces within an infinite dimensional transfinite manifold.     

An exploratory study of a subspace iteration method as an alternative to the QR method for Floquet eigenanalysis

Floquet eigenanalysis requires a few dominant eigenvalues of the Floquet transition matrix (FTM). Although the QR method is used almost exclusively, it is expensive for such partial eigenanalysis; the operation counts and, thereby, the approximate machine-time grow cubically with the matrix order. Accordingly, for Floquet eigenanalysis, the Arnoldi-Saad method, a subspace iteration method, is investigated as an alternative to the QR method. The two methods are compared for machine-time efficiency and computational reliability, which is quantified by the condition numbers of the required eigenvalues and the residual errors of the corresponding eigenpairs. The Arnoldi-Saad method takes much less machine-time than the QR method with comparable computational reliability and offers promise for large-scale Floquet eigenanalysis (say, FTM order > 100).     

The measurement of human capital as an alternative method of job evaluation for purposes of remuneration

Central European Journal of Operations Research - Contemporary science and business practice in the field of remuneration management includes many concepts and models of the remuneration systems...