Fractional Quantum Mechanics

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the L\'evy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the L\'evy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schr\"odinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.Comment: 27 page

Spring-Block Model Reveals Region-Like Structures

A mechanical spring-block model is used for realizing an objective space partition of settlements from a geographic territory in region-like structures. The method is based on the relaxation-dynamics of the spring-block system and reveals in a hierarchical manner region-like entities at different spatial scales. It takes into account in an elegant manner both the spatiality of the elements and the connectivity relations among them. Spatiality is taken into account by using the geographic coordinates of the settlements, and by detecting the neighbors with the help of a Delaunay triangulation. Connectivity between neighboring settlements are quantified using a Pearson-like correlation for the relative variation of a relevant socio-economic parameter (population size, GDP, tax payed per inhabitant, etc.). The method is implemented in an interactive JAVA application and it is applied with success for an artificially generated society and for the case of USA, Hungary and Transylvania

Dynamic Hybrid Modeling of the Vertical Z Axis in a High-Speed Machining Center: Towards Virtual Machining

This paper describes the modeling and simulation of the Z axis of a five axis machining center for high-speed milling. The axis consists of a mechanical structure: machine head and electro-mandrel, a CNC system interfaced with the feed drive, and a pneumatic system to compensate for the weight of the vertical machine head. These subsystems were studied and modeled by means of: (1) finite element method modeling of the mechanical structure; (2) a concentrated parameter model of the kinematics of the axis; (3) a set of algebraic and logical relations to represent the loop CNC-Z feed drive; (4) an equation set to represent the functioning of the pneumatic system; and (5) a specific analytical model of the friction phenomena occurring between sliding and rotating mechanical components. These modeled subsystems were integrated to represent the dynamic behavior of the entire Z axis. The model was translated in a computer simulation package and the validation of the model was made possible by comparing the outputs of simulation runs with the records of experimental tests on the machining center. The firm which promoted and financed the research now has a virtual tool to design improved machine-tool versions with respect to present models, designed by traditional tools

FROM TRADITIONAL TO VIRTUAL DESIGN OF MACHINE-TOOLS: A LONG WAY TO GO... PART 1: PROBLEM IDENTIFICATION AND MODEL VALIDATION

This paper reports the final results of research devoted to providing a package of tools for the virtual design of machining centers for an italian manufacturer involved until today with traditional design procedures. The general thematics of virtual design have been realistically defined and focalised on the topics of particular interest for the firm; these constraints are analysed and the chosen solutions discussed. The paper also reports on the experimental validation of some complex parameters of the models of the machining centers already set up

Dynamic hybrid modeling of the vertical z-axis in an HSM machining centre towards virtula machining

The paper describes the dynamical real-time simulation of the Z axis of a five axis machining center for high speed milling. The axis consists of a mechanical structure: machine head and electro-mandrel, a CNC Control System provided with feed drives and a Pneumatic System to compensate the weight of the entire vertical machine head. These three sub-systems have been studied and modeled by means of: - FEM modeling of the mechanical structure; - an equation set to represent the main functions of the CNC; - an equation set to represent the functioning of the Pneumatic System. These different modeling sub-systems have been integrated to obtain the entire actual dynamical behavior of the Z axis. A particular analysis was developed to represent the friction phenomena by a specific analytical model. Experimental activity was developed to test and validate the different modeling sub-systems, and other experimental tests were performed on the machining center to compare simulation outputs with experimental responses