The algebra and geometry of SU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real Linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of 'multiplying' two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed as a generalization of that for SU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.

Helical structures: The geometry of protein helices and nanotubes

In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit. A plane hexagonal lattice wrapped round a cylinder provides a useful starting point for describing the helical conformations of protein molecules, for investigating the geometrical properties of carbon nanotubes, and for certain types of dense packings of equal spheres.

Encapsulation of cobalt phthalocyanine in zeolite-Y: Evidence for nonplanar geometry

Cobalt (11) phthalocyanine (CoPc) molecules have been encapsulated within the supercage of zeolite-Y. The square-planar complex, being larger than the almost spherical cage, is forced to adopt a distorted geometry on encapsulation. A comparative spectroscopic and magnetic investigation of CoPc encapsulated in zeolite-Y and in the unencapsulated state is reported. These results supported by molecular modeling have been used to understand the nature and extent of the loss of planarity of CoPc on encapsulation. The encapsulated molecule is shown to be the trans-diprotonated species in which the center of inversion is lost due to distortions required to accommodate the square complex within the zeolite. Encapsulation also leads to an enhancement of the magnetic moment of the CoPc. This is shown to be a consequence of the nonplanar geometry of the encapsulated molecule resulting in an excited high-spin state being thermally accessible.

Simultaneous Geometry Optimization and Material Selection for Truss Structures

In this work, we explore simultaneous design and material selection by posing it as an optimization problem. The underlying principles for our approach are Ashby's material selection procedure and structural optimization. For the simplicity and ease of initial implementation of the general procedure, truss structures under static load are considered in this work in view of maximum stiffness, minimum weight/cost and safety against failure. Along the lines of Ashby's material indices, a new design index is derived for trusses. This helps in choosing the most suitable material for any design of a truss. Using this, both the design space and material database are searched simultaneously using optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous even though the material selection is an inherently discrete problem.

Online Handwriting Recognition of Tamil Script Using Fractal Geometry

We present a fractal coding method to recognize online handwritten Tamil characters and propose a novel technique to increase the efficiency in terms of time while coding and decoding. This technique exploits the redundancy in data, thereby achieving better compression and usage of lesser memory. It also reduces the encoding time and causes little distortion during reconstruction. Experiments have been conducted to use these fractal codes to classify the online handwritten Tamil characters from the IWFHR 2006 competition dataset. In one approach, we use fractal coding and decoding process. A recognition accuracy of 90% has been achieved by using DTW for distortion evaluation during classification and encoding processes as compared to 78% using nearest neighbor classifier. In other experiments, we use the fractal code, fractal dimensions and features derived from fractal codes as features in separate classifiers. While the fractal code is successful as a feature, the other two features are not able to capture the wide within-class variations.

Changing resonator geometry to boost sound power decouples size and song frequency in a small insect

Despite their small size, some insects, such as crickets, can produce high amplitude mating songs by rubbing their wings together. By exploiting structural resonance for sound radiation, crickets broadcast species-specific songs at a sharply tuned frequency. Such songs enhance the range of signal transmission, contain information about the signaler's quality, and allow mate choice. The production of pure tones requires elaborate structural mechanisms that control and sustain resonance at the species-specific frequency. Tree crickets differ sharply from this scheme. Although they use a resonant system to produce sound, tree crickets can produce high amplitude songs at different frequencies, varying by as much as an octave. Based on an investigation of the driving mechanism and the resonant system, using laser Doppler vibrometry and finite element modeling, we show that it is the distinctive geometry of the crickets' forewings (the resonant system) that is responsible for their capacity to vary frequency. The long, enlarged wings enable the production of high amplitude songs; however, as a mechanical consequence of the high aspect ratio, the resonant structures have multiple resonant modes that are similar in frequency. The drive produced by the singing apparatus cannot, therefore, be locked to a single frequency, and different resonant modes can easily be engaged, allowing individual males to vary the carrier frequency of their songs. Such flexibility in sound production, decoupling body size and song frequency, has important implications for conventional views of mate choice, and offers inspiration for the design of miniature, multifrequency, resonant acoustic radiators.

Influence of stack geometry and resonator length on the performance of thermoacoustic engine

Thermoacoustic engines convert heat energy into high amplitude sound waves, which is used to drive thermoacoustic refrigerator or pulse tube cryocoolers by replacing the mechanical pistons such as compressors. The increasing interest in thermoacoustic technology is of its potentiality of no exotic materials, low cost and high reliability compared to vapor compression refrigeration systems. The experimental setup has been built based on the linear thermoacoustic model and some simple design parameters. The engines produce acoustic energy at the temperature difference of 325-450 K imposed along the stack of the system. This work illustrates the influence of stack parameters such as plate thickness (PT) and plate spacing (PS) with resonator length on the performance of thermoacoustic engine, which are measured in terms of onset temperature difference, resonance frequency and pressure amplitude using air as a working fluid. The results obtained from the experiments are in good agreement with the theoretical results from DeltaEc. (C) 2012 Elsevier Ltd. All rights reserved.

Influence of the Blade Hub Geometry on the Performance of Low-Head Axial Flow Turbines

The influence of geometric parameters, such as blade profile and hub geometry on axial flow turbines for micro hydro application remains poorly characterized. This paper first introduces a holistic theoretical model for studying the hydraulic phenomenon resulting from geometric modification to the blades. It then describes modification carried out on two runner stages, of which one has untwisted blades and the other has twisted blades obtained by modifying the inlet hub. The experimental results showed that the performance of the untwisted blade runner was satisfactory with a maximum efficiency of 68%. However, positive effects of twisted blades were clearly evident with an efficiency rise of more than 2%. This study also looks into the possible limitations of the model and suggests the extension of the experimental work and the use of computational tools to conduct a progressive validation of all experimental findings, especially on the flow physics within the hub region and the slip phenomena. The paper finally underlines the importance of developing a standardization philosophy for axial flow turbines specific for micro hydro requirements. DOI:10.1061/(ASCE)EY.1943-7897.0000060. (C) 2012 American Society of Civil Engineers.

Geometry of static balancing during posture transition

The interaction between the digital human model (DHM) and environment typically occurs in two distinct modes; one, when the DHM maintains contacts with the environment using its self weight, wherein associated reaction forces at the interface due to gravity are unidirectional; two, when the DHM applies both tension and compression on the environment through anchoring. For static balancing in first mode of interaction, it is sufficient to maintain the projection of the centre of mass (COM) inside the convex region induced by the weight supporting segments of the body on a horizontal plane. In DHM, static balancing is required while performing specified tasks such as reach, manipulation and locomotion; otherwise the simulations would not be realistic. This paper establishes the geometric relationships that must be satisfied for maintaining static balance while altering the support configurations for a given posture and altering the posture for a given support condition. For a given location of the COM for a system supported by multiple point contacts, the conditions for simultaneous withdrawal of a specified set of contacts have been determined in terms of the convex hulls of the subsets of the points of contact. When the projection of COM must move beyond the existing support for performing some task, new supports must be enabled for maintaining static balance. This support seeking behavior could also manifest while planning for reduction of support stresses. Feasibility of such a support depends upon the availability of necessary features in the environment. Geometric conditions necessary for selection of new support on horizontal,inclined and vertical surfaces within the workspace of the DHM for such dynamic scenario have been derived. The concepts developed are demonstrated using the cases of sit-to-stand posture transition for manipulation of COM within the convex supporting polygon, and statically stable walking gaits for support seeking within the kinematic capabilities of the DHM. The theory developed helps in making the DHM realize appropriate behaviors in diverse scenarios autonomously.