Exact Random Walk Distributions using Noncommutative Geometry

Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $N$ on a two-dimensional square lattice for large $N$, taking into account finite size contributions.Comment: Latex, 3 pages, 1 figure, to be published in J. Phys. A : Math. Ge

Symmetry of Quantum Torus with Crossed Product Algebra

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function. An invariant function under the group action is constructed as a variant of the classical theta function. Then our main issue, the crossed product algebra representation of quantum torus with complex structure under the symplectic group is analyzed as a quantum version of orbifolding. We perform this analysis with Manin's so-called model II quantum theta function approach. The symplectic group Sp(2n,Z) satisfies the consistency condition of crossed product algebra representation. However, only a subgroup of Sp(2n,Z) satisfies the consistency condition for orbifolding of quantum torus.Comment: LaTeX 17pages, changes in section 3 on crossed product algebr

Experimental determination of grain density function of AZ91/SiC composite with different mass fractions of SiC and undercoolings using heterogeneous nucleation model

The grain density, Nv, in the solid state after solidification of AZ91/SiC composite is a function of maximum undercooling, ΔT, of a liquid alloy. This type of function depends on the characteristics of heterogeneous nucleation sites and number of SiC present in the alloy. The aim of this paper was selection of parameters for the model describing the relationship between the grain density of primary phase and undercooling. This model in connection with model of crystallisation, which is based on chemical elements diffusion and grain interface kinetics, can be used to predict casting quality and its microstructure. Nucleation models have parameters, which exact values are usually not known and sometimes even their physical meaning is under discussion. Those parameters can be obtained after mathematical analysis of the experimental data. The composites with 0, 1, 2, 3 and 4wt.% of SiC particles were prepared. The AZ91 alloy was a matrix of the composite reinforcement SiC particles. This composite was cast to prepare four different thickness plates.They were taken from the region near to the thermocouple, to analyze the undercooling for different composites and thickness plates and its influence on the grain size. The microstructure and thermal analysis gave set of values that connect mass fraction of SiC particles, and undercooling with grain size. These values were used to approximate nucleation model adjustment parameters. Obtained model can be very useful in modelling composites microstructure

Vacuum Compatibility of 3D-Printed Materials

The fabrication fidelity and vacuum properties are tested for currently available 3D-printed materials including polyamide, glass, acrylic, and sterling silver. The silver was the only material found to be suitable to ultrahigh vacuum environments due to outgassing and sublimation observed in other materials.Comment: 3 pages, 2 figures, preprin

Fundamental Studies Relating to the Mechanical Behavior of Solid Propellants, Rocket Grains and Rocket Motors

During the past three years, the mechanical testing of solid propellants, solid rocket grains, and solid rocket motors under idealized conditions has been receiving increased attention. Today it is not uncommon to see a multitude of new techniques and analyses being investigated. One may expect to see dummy propellant prepared with glass bead filler to observe its dilatation to rupture; to ink circles, rectangular g rids at various critical areas on a grain surface, and to observe the distortion of these grids as a result of thermal cycling and/or slump; to subj e ct rectangular parallel-opipedal-shaped specimens to both torsion and biaxial tension as well as hydrostatic compression and parallel-plate tension; to apply theories of large elastic strain, and non-linear viscoelasticity; to search for an isotropic failure criterion as well as a crack propagation criterion. In short the mechanics of propellant behavior from small deformation all the way to fracture initiation and propagation has become quite sophisticated. Gradually the results of this testing and their thinking are being integrated in a logical scheme of analysis which is being passed along to the engineer and being used in predicting performance of rocket motors. This particular program will pertain to four areas: 1) The characterization of polyurethane propellant behavior out to fracture initiation in terms of large strain theory. 2) The development of a failure criterion and crack propagation criteria for said materials. 3) The generation, where possible, of macroscopic mechanical parameters in terms of molecular parameters. 4} The solution of certain stress problems, in both linear and non-linear theory, which are prerequisite to engineering applications. As such it is part of a continuing research study of structural integrity problems in solid propellant rocket motors being conducted under the general direction of Dr. M. L. Williams in the Guggenheim Aeronautical Laboratory. This preliminary report is intended as an interim working document to be circulated for the purpose of stimulating discussion

Diffractive energy spreading and its semiclassical limit

We consider driven systems where the driving induces jumps in energy space: (1) particles pulsed by a step potential; (2) particles in a box with a moving wall; (3) particles in a ring driven by an electro-motive-force. In all these cases the route towards quantum-classical correspondence is highly non-trivial. Some insight is gained by observing that the dynamics in energy space, where $n$ is the level index, is essentially the same as that of Bloch electrons in a tight binding model, where $n$ is the site index. The mean level spacing is like a constant electric field and the driving induces long range hopping 1/(n-m).Comment: 19 pages, 11 figs, published version with some improved figure

Fundamental Studies Relating to the Mechanical Behavior of Solid Propellants, Rocket Grains and Rocket Motors

The former reports provided considerable information about foam and continuum rubbers under three types of tensile loading (i.e. uniaxial, strip-biaxial and homogeneous-biaxial tension). Since continuum rubbers are almost incompressible it is extremely difficult to determine the strain energy function beyond the linear term. On the other hand the highly dilatable foam rubber enables one to determine the functional form of the strain energy valid up to higher order terms. Special attention is being paid to foam rubber, since it represents .the limiting case of completely dewetted propellant. The present report will (i) furnish the method of determination of strain energy function and the associated constitutive stress-strain law for large deformations out to fracture and (ii) present the triaxial tensile test data needed to double check item (i)

A phased array-based method for damage detection and localization in thin plates

A method for damage localization based on the phased array idea has been developed. Four arrays oftransducers are used to perform a beam-forming procedure. Each array consists of nine transducersplaced along a line, which are able to excite and register elastic waves. The A0 Lamb wave mode hasbeen chosen for the localization method. The arrays are placed in such a way that the angulardifference between them is 458 and the rotation point is the middle transducer, which is common for allthe arrays. The idea has been tested on a square aluminium plate modeled by the Spectral Element Method. Two types of damage were considered, namely distributed damage, which was modeled asstiffness reduction, and cracks, modeled as separation of nodes between selected spectral elements.The plate is excited by a wave packet. The whole array system is placed in the middle of the plate.Each linear phased array in the system acts independently and produces maps of a scanned fieldbased on the beam-forming procedure. These maps are made of time signals (transferred to spacedomain) that represent the difference between the damaged plate signals and those from the intactplate. An algorithm was developed to join all four maps. The final map is modified by proposed signal processing algorithm to indicate the damaged area of the plate more precisely. The problem fordamage localization was investigated and exemplary maps confirming the effectiveness of theproposed system were obtained. It was also shown that the response of the introduced configurationremoves the ambiguity of damage localization normally present when a linear phased array is utilized.The investigation is based exclusively on numerical data

Theory of a magnetic microscope with nanometer resolution

We propose a theory for a type of apertureless scanning near field microscopy that is intended to allow the measurement of magnetism on a nanometer length scale. A scanning probe, for example a scanning tunneling microscope (STM) tip, is used to scan a magnetic substrate while a laser is focused on it. The electric field between the tip and substrate is enhanced in such a way that the circular polarization due to the Kerr effect, which is normally of order 0.1% is increased by up to two orders of magnitude for the case of a Ag or W tip and an Fe sample. Apart from this there is a large background of circular polarization which is non-magnetic in origin. This circular polarization is produced by light scattered from the STM tip and substrate. A detailed retarded calculation for this light-in-light-out experiment is presented.Comment: 17 pages, 8 figure