Harmonic Spinors on a Family of Einstein Manifolds

The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and $P^2(C)$ with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of $C^2$ and extend to $P^2(C)$ as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Patodi-Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.Comment: Updated to match the published versio

Asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness with same boundary conditions

In the present paper, asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness resting on Winkler elastic foundation is studied by using boundary characteristic orthonormal polynomials in Rayleigh-Ritz method. Convergence of the results is tested and comparison is made with results already available in the existing literature. Numerical results for the first ten frequencies for various values of parameters describing width of annular plate, thickness profile, material orthotropy and foundation constant for all three possible combinations of clamped, simply supported and free edge conditions are shown and discussed. It is found that (a) higher elastic property in circumferential direction leads to higher stiffness against lateral vibration; (b) Lateral vibration characteristics of F-F plates is more sensitive towards parametric changes in material orthotropy and foundation stiffness thanC-C and S-S plates; (c) Effect of quadratical thickness variation on fundamental frequency is more significant in cases of C-C and S-S plates than that of F-F plates. Thickness profile which is convex relative to plate center-line tends to result in higher stiffness of annular plates against lateral vibration than the one which is concave and (d) Fundamental mode of vibration of C-C and S-S plates is axisymmetrical while that of F-F plates is asymmetrical

Divergence of the correlation length for critical planar FK percolation with 1≤q≤41\le q\le4 via parafermionic observables

Parafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p,q)=(p_c(q),q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when $1\le q\le 4$. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster-weight $q>0$.Comment: 26 page

The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone

Correct identification of the Bravais lattice of a crystal is an important step in structure solution. Niggli reduction is a commonly used technique. We investigate the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G6 by algebraic analysis and organized random probing of regions near 1- through 8-fold boundary polytope intersections. We limit consideration of boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of 0 length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to 0 or without neighboring probe points are eliminated. 216 boundary polytopes are found: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. All primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provides a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helps to illuminate some of the complexities in Bravais lattice identification. The classification is intended to help in organizing database searches and in understanding which lattice symmetries are "close" to a given experimentally determined cell

Initial development of ice crystal ice accretion at conditions related to turbofan operation at high Altitude

Ice accretion on external surfaces of aircraft is a widely recognised problem, but more recently identified problem of ice crystal ice accretion within aero-engine compressors during flight through deep convection systems also represents a significant hazard and forms the motivation for the present work. The experimental studies targeting solid phase ice accretion are very limited due to the high wind tunnel facilities operational cost and safety concern for in-flight icing testing, which requires flight through severe weather conditions. In this study, a small wind tunnel was established to simulate some of the conditions relevant to aircraft engine icing from ice crystals and explore the application of a model for the initiation of ice accretion. In this facility, liquid nitrogen was used to freeze liquid water droplets generated using an ultrasonic nozzle. The liquid nitrogen section reduces the droplet temperature to less than -40 �C and maintains this temperature for su�cient time to ensure complete freezing occurs. The particle diameters were controlled by the air and water pressure delivered to the ultrasonic nozzle and particle diameters around 50 �m were generated. The ice water content was also measured experimentally and it was found to be around 0:42 g/m3. A temperature controller was developed to keep the specimen surface temperature essentially constant and four specimen surface temperatures were tested: -9, -5, 0, and 5 degrees �C. The wind tunnel duct had a diameter of 70mm and was operated at the relatively low flow speed of 6:5 m/s. A cylinder with diameter of 10mm and flat plate surface with length of 3:6 cm and a leading edge diameter of 3mm were used as the test specimens. A microscope video camera was used to visualise a small area on the specimen surface of 9x�9mm and record the initiation of the accretion process. The experimental data were analysed using image processing techniques, and di�erent locations around the centre line of the test specimens in the vicinity of the stagnation point were investigated. Two regions with different roughness were used on both specimens with an average roughness (Ra) for the smooth side of 0:5 �m and 1:0 �m for the rough side, but no effect of the surface roughness was observed in the experimental accretion results for these conditions. The mathematical model for accretion initiation which was developed considers the aerodynamic, adhesive, and friction force a�ecting the particles in contact with the surface. The model indicates that ice accretion can occur at subfreezing conditions in the stagnation region and this effect was observed in the present experiments. The model also indicates that accretion is less likely to occur as the temperature increases due to reductions in the coe�cient of friction. Such an effect was also observed in the experiments: accretion occurred most rapidly in the -9 degrees �C case but virtually no accretion was registered in the 0 degrees �C and 5 degrees �C cases. Although the mathematical model suggested the accretion could also initiate on a flat plate with a laminar boundary layer, this was not observed experimentally. The lack of the accretion in the laminar boundary layer configuration is attributed to the �finite leading edge diameter on which substantial ice accretion was observed. The rate of accretion development on the leading edge of the flat plate was comparable to that on the large diameter cylinder specimen which is not consistent with the trends suggested by the mathematical model. The new wind tunnel duct conditions can be controlled and solid ice particles of a uniform shape and known size distribution can be produced. The development of the new facility and the force-balance model has established useful tools which can be further enhanced in future ice accretion studies

Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.Comment: 38 pages, 5 figure; Added referenc

Stability and Vortex Shedding of Bluff Body Arrays

The primary purpose of this study was to develop an understanding of the stability of laminar flow through bluff body arrays, and investigate the nature of the unsteady vortex shedding regime that follows. The flow was numerically investigated using a specially developed multi-domain spectral element solver. Important criteria in the solver development were flexibility, efficiency, and accuracy. Flexibility was critical to the functionality of the code, as arrays of varying geometry were investigated. Efficiency with a high degree of accuracy was also of primary importance, with the code implemented to run efficiently on today's massively parallel architectures. Numerical two-dimensional stability analysis of the flow in several configurations of inline and staggered array geometries was performed. The growth rate, eigenfunction, and frequency of the disturbances were determined. The critical Reynolds number for flow transition in each case was identified and compared to that of flow over a single body. Based on the solutions of the laminar flow, a one-dimensional analytical analysis was performed on selected velocity profiles in the wake region. The results of this analysis were used to guide the interpretation of the two dimensional results and formulate a general theory of stability of inline and staggered bluff body arrays. The nature of the flow in the unsteady regime following the onset of instability was examined for an inline and a staggered arrangement. Particular attention was focused on the vortex shedding which was visualized and quantified through computation of the flow swirl, a quantity which identifies regions of rotary motion. The conditions required for the generation of leading edge vortex shedding were identified and discussed. Finally, a third geometry related to the inline and staggered arrays was considered. Flow solution data for this geometry is presented and its suitability as a model for louvered arrays was discussed.Air Conditioning and Refrigeration Project 11