Dimension on Discrete Spaces

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from the author upon request. Preprint SU-GP-93/1-1. To appear in "International Journal of Theoretical Physics

Quantum Theory and Galois Fields

We discuss the motivation and main results of a quantum theory over a Galois field (GFQT). The goal of the paper is to describe main ideas of GFQT in a simplest possible way and to give clear and simple arguments that GFQT is a more natural quantum theory than the standard one. The paper has been prepared as a presentation to the ICSSUR' 2005 conference (Besancon, France, May 2-6, 2005).Comment: Latex, 24 pages, 1 figur

Dimerization and low-dimensional magnetism in nanocrystalline TiO2 semiconductors doped by Fe and Co

The report is devoted to an analysis of the structural and magnetic state of the nanocrystalline diluted magnetic semiconductors based on TiO2 doped with Fe and Co atoms. Structural and magnetic characterization of samples was carried out using X-ray diffraction (XRD) analysis, transmission electron microscopy (TEM), X-ray absorption spectroscopy (XAS), electron paramagnetic resonance (EPR) spectroscopy, SQUID magnetometry, and the density functional theory (DFT) calculations. Analysis of the experimental data suggests the presence of non-interacting paramagnetic Fe3+ and Co2+ ions in the high-spin state and negative exchange interactions between them. The important conclusions is that the distribution of dopants in the TiO2 matrix, even at low concentrations of 3d-metal dopant (less than one percent), is not random, but the 3d ions localization and dimerization is observed both on the surface and in the nanoparticles core. Thus, in the paper the quantum mechanical model for describing the magnetic properties of TiO2:(Fe, Co) was suggested. The model operates only with two parameters: paramagnetic contribution of non-interacting 3d-ions and dimers having different exchange interactions between 3d magnetic carriers. © Published under licence by IOP Publishing Ltd

Unconventional magnetism of non-uniform distribution of Co in TiO2 nanoparticles

High-resolution transmission electron microscopy (HRTEM), X-ray diffraction (XRD) analysis, electron paramagnetic resonance (EPR), X-ray absorption spectroscopy (XAS), magnetic methods, and density-functional theory (DFT) calculations were applied for the investigations of Co-doped anatase TiO2 nanoparticles (∼20 nm). It was found that high-spin Co2+ ions prefer to occupy the interstitial positions in the TiO2 lattice which are the most energetically favourable in compare to the substitutional those. A quantum mechanical model which operates mainly on two types of Co2+ – Co2+ dimers with different negative exchange interactions and the non-interacting paramagnetic Co2+ ions provides a satisfactorily description of magnetic properties for the TiO2:Co system. © 2020 Elsevier B.V.Russian Foundation for Basic Research. Ministry of Science and Higher Education of the Russian Federatio

Density functional theory for nearest-neighbor exclusion lattice gasses in two and three dimensions

To speak about fundamental measure theory obliges to mention dimensional crossover. This feature, inherent to the systems themselves, was incorporated in the theory almost from the beginning. Although at first it was thought to be a consistency check for the theory, it rapidly became its fundamental pillar, thus becoming the only density functional theory which possesses such a property. It is straightforward that dimensional crossover connects, for instance, the parallel hard cube system (three-dimensional) with that of squares (two-dimensional) and rods (one-dimensional). We show here that there are many more connections which can be established in this way. Through them we deduce from the functional for parallel hard (hyper)cubes in the simple (hyper)cubic lattice the corresponding functionals for the nearest-neighbor exclusion lattice gases in the square, triangular, simple cubic, face-centered cubic, and body-centered cubic lattices. As an application, the bulk phase diagram for all these systems is obtained.Comment: 13 pages, 13 figures; needs revtex