Generalised quaternion methods in conformal geometry

A new approach to Penrose's twistor algebra is given. It is based on the use of a generalised quaternion algebra for the translation of statements in projective five-space into equivalent statements in twistor (conformal spinor) space. The formalism leads toSO(4, 2)-covariant formulations of the Pauli-Kofink and Fierz relations among Dirac bilinears, and generalisations of these relations.

Geometry of the de-Sitter universe

By making use of the fact that the de-Sitter metric corresponds to a hyperquadric in a five-dimensional flat space, it is shown that the three Robertson-Walker metrics for empty spacetime and positive cosmological constant, corresponding to 3-space of positive, negative and zero curvative, are geometrically equivalent. The 3-spaces correspond to intersections of the hyperquadric by hyperplanes, and the time-like geodesics perpendicular to them correspond to intersections by planes, in all three cases.

A note on fractal dimensions of biomedical waveforms

In this paper, we study performance of Katz method of computing fractal dimension of waveforms, and its estimation accuracy is compared with Higuchi's method. The study is performed on four synthetic parametric fractal waveforms for which true fractal dimensions can be calculated, and real sleep electroencephalogram. The dependence of Katz's fractal dimension on amplitude, frequency and sampling frequency of waveforms is noted. Even though the Higuchi's method has given more accurate estimation of fractal dimensions, the study suggests that the results of Katz's based fractal dimension analysis of biomedical waveforms have to be carefully interpreted.

Inter-helical Interactions in Membrane Proteins: Analysis Based on the Local Backbone Geometry and the Side Chain Interactions

The availability of a significant number of the Structures of helical membrane proteins has prompted us to investigate the mode of helix-helix packing. In the present study, we have considered a dataset of alpha-helical membrane proteins representing Structures solved from all the known superfamilies. We have described the geometry of all the helical residues in terms of local coordinate axis at the backbone level. Significant inter-helical interactions have been considered as contacts by weighing the number of atom-atom contacts, including all the side-chain atoms. Such a definition of local axis and the contact criterion has allowed us to investigate the inter-helical interaction in a systematic and quantitative manner. We show that a single parameter (designated as alpha), which is derived from the parameters representing the Mutual orientation of local axes, is able to accurately Capture the details of helix-helix interaction. The analysis has been carried Out by dividing the dataset into parallel, anti-parallel, and perpendicular orientation of helices. The study indicates that a specific range of alpha value is preferred for interactions among the anti-parallel helices. Such a preference is also seen among interacting residues of parallel helices, however to a lesser extent. No such preference is seen in the case of perpendicular helices, the contacts that arise mainly due to the interaction Of Surface helices with the end of the trans-membrane helices. The Study Supports the prevailing view that the anti-parallel helices are well packed. However, the interactions between helices of parallel orientation are non-trivial. The packing in alpha-helical membrane proteins, which is systematically and rigorously investigated in this study, may prove to be useful in modeling of helical membrane proteins.

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.

Strong motion compatible source geometry

Strong motion array records are analyzed in this paper to identify and map the source zone of four past earthquakes. The source is represented as a sequence of double couples evolving as ramp functions, triggering at different instants, distributed in a region yet to be mapped. The known surface level ground motion time histories are treated as responses to the unknown double couples on the fault surface. The location, orientation, magnitude, and risetime of the double couples are found by minimizing the mean square error between analytical solution and instrumental data. Numerical results are presented for Chi-Chi, Imperial Valley, San Fernando, and Uttarakashi earthquakes. Results obtained are in good agreement with field investigations and those obtained from conventional finite fault source inversions.

Enhanced Ion Anisotropy by Nonconventional Coordination Geometry: Single-Chain Magnet Behavior for a [{(FeL)-L-II}(2){Nb-IV(CN)(8)}] Helical Chain Compound Designed with Heptacoordinate Fe-II

Nonconventional heptacoordination in combination with efficient magnetic exchange coupling is shown to yield a 1-D heteronuclear {(FeNbIV)-Nb-II} compound with remarkable magnetic features when compared to other Fe(II)-based single chain magnets (SCM). Cyano-bridged heterometallic {3d-4d} and {3d-5d} chains are formed upon assembling Fe(II) bearing a pentadentate macrocycle as the blocking ligand with octacyano metallates, [M(CN)(8)](4-) (M = Nb-IV, Mo-IV, W-IV.) X-ray diffraction (single-crystal and powder) measurements reveal that the [{(H2O)Fe(L-1)}{M(CN)(8)}{Fe(L-1)}](infinity) architectures consist of isomorphous 1-D polymeric structures based on the alternation of {Fe(L-1)}(2+) and {M(CN)(8)}(4-) units (L-1 stands for the pentadentate macrocycle). Analysis of the magnetic susceptibility behavior revealed cyano-bridged {Fe-Nb} exchange interaction to be antiferromagnetic with J = -20 cm(-1) deduced from fitting an Ising model taking into account the noncollinear spin arrangement. For this ferrimagnetic chain a slow relaxation of its magnetization is observed at low temperature revealing a SCM behavior with Delta/k(B) = 74 K and tau(0) = 4.6 x 10(-11) s. The M versus H behavior exhibits a hysteresis loop with a coercive field of 4 kOe at 1 K and reveals at 380 mK magnetic avalanche processes, i.e., abrupt reversals in magnetization as H is varied. The origin of these characteristics is attributed to the combination of efficient {Fe-Nb} exchange interaction and significant anisotropy of the {Fe(L-1)) unit. High field EPR and magnetization experiments have revealed for the parent compound [Fe(L-1)(H2O)(2)]Cl-2 a negative zero field splitting parameter of D approximate to -17 cm(-1). The crystal structure, magnetic behavior, and Mossbauer data for [Fe(L-1)(H2O)(2)]Cl-2 are also reported.

Enhanced Ion Anisotropy by Nonconventional Coordination Geometry: Single-Chain Magnet Behavior for a [{(FeL)-L-II}(2){Nb-IV(CN)(8)}] Helical Chain Compound Designed with Heptacoordinate Fe-II

Nonconventional heptacoordination in combination with efficient magnetic exchange coupling is shown to yield a 1-D heteronuclear {(FeNbIV)-Nb-II} compound with remarkable magnetic features when compared to other Fe(II)-based single chain magnets (SCM). Cyano-bridged heterometallic {3d-4d} and {3d-5d} chains are formed upon assembling Fe(II) bearing a pentadentate macrocycle as the blocking ligand with octacyano metallates, [M(CN)(8)](4-) (M = Nb-IV, Mo-IV, W-IV.) X-ray diffraction (single-crystal and powder) measurements reveal that the [{(H2O)Fe(L-1)}{M(CN)(8)}{Fe(L-1)}](infinity) architectures consist of isomorphous 1-D polymeric structures based on the alternation of {Fe(L-1)}(2+) and {M(CN)(8)}(4-) units (L-1 stands for the pentadentate macrocycle). Analysis of the magnetic susceptibility behavior revealed cyano-bridged {Fe-Nb} exchange interaction to be antiferromagnetic with J = -20 cm(-1) deduced from fitting an Ising model taking into account the noncollinear spin arrangement. For this ferrimagnetic chain a slow relaxation of its magnetization is observed at low temperature revealing a SCM behavior with Delta/k(B) = 74 K and tau(0) = 4.6 x 10(-11) s. The M versus H behavior exhibits a hysteresis loop with a coercive field of 4 kOe at 1 K and reveals at 380 mK magnetic avalanche processes, i.e., abrupt reversals in magnetization as H is varied. The origin of these characteristics is attributed to the combination of efficient {Fe-Nb} exchange interaction and significant anisotropy of the {Fe(L-1)) unit. High field EPR and magnetization experiments have revealed for the parent compound [Fe(L-1)(H2O)(2)]Cl-2 a negative zero field splitting parameter of D approximate to -17 cm(-1). The crystal structure, magnetic behavior, and Mossbauer data for [Fe(L-1)(H2O)(2)]Cl-2 are also reported.

Geometry changes induced by negative hyperconjugative interactions involving carbonyl and thiocarbonyl groups

MNDO geometry optimizations have been carried out on a series of acyclic and cyclic unsymmetrically disubstituted carbonyl and thiocarbonyl compounds. The C=X unit shows a consistent and often sizeable tilt towards one of the substituents, following the order O > Snot, vert, similarN > C > B. Reference ab initio calculations and available experimental results support the MNDO results. The effect, which is particularly dramatic in small rings, is attributed primarily to favorable negative hyperconjugative interaction between the lone pair on X and a low lying adjacent σ* orbital. Such an interaction can lead to highly distorted structures, including perhaps to a planar molecule with an inverted sp2 carbon center.

Cross-field gas breakdown in a coaxial cylinder geometry

Sparking potentials in a coaxial cylinder geometry in oxygen and dry air were measured in crossed electric and magnetic fields. From the data effective collision frequencies were calculated using the equivalent pressure concept. It is shown that the equivalent pressure concept holds good for deriving the effective collision frequencies in non-uniform electric fields.

Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim

A key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim, particularly when free wake models are used. Trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero. An appropriate initial estimate of the trim state is needed for successful helicopter trim. This study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations (set of initial guess points from which the nonlinear equations converge). It is illustrated that the three main rotor pitch controls of collective pitch, longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution. Trajectories of the Newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed. It is found that the basins of attraction can have fractal boundaries. (C) 2010 Elsevier Ltd. All rights reserved.

A standard reference frame for the description of Nucleic Acid Base-Pair Geometry

A common point of reference is needed to describe the three-dimensional arrangements of bases and base-pairs in nucleic acid structures. The different standards used in computer programs created for this purpose give rise to con¯icting interpretations of the same structure.1 For example, parts of a structure that appear ``normal'' according to one computational scheme may be highly unusual according to another and vice versa. It is thus dif®cult to carry out comprehensive comparisons of nucleic acid structures and to pinpoint unique conformational features in individual structures

HELANAL: A program to characterise helix geometry in proteins,

A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C-alpha atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.

Nature of Cl center dot center dot center dot Cl Intermolecular Interactions via Experimental and Theoretical Charge Density Analysis:Correlation of Polar Flattening Effects with Geometry

The experimental charge density distribution in three compounds, 2-chloro-3-quinolinyl methanol, 2-chloro-3-hydroxypyridine, and 2-chloro-3-chloromethyl-8-methylquinoline, has been obtained using high-resolution X-ray diffraction data collected at 100 K based on the aspherical multipole modeling of electron density. These compounds represent type I (cis), type I (trans), and type II geometries, respectively, as defined for short Cl center dot center dot center dot Cl interactions. The experimental results are compared with the theoretical charge densities using theoretical structure factors obtained from a periodic quantum calculation at the B3LYP/6-31G** level. The topological features derived from the Bader's theory of atoms in molecules (AIM) approach unequivocally suggest that both cis and trans type I geometries show decreased repulsion, whereas type II geometry is attractive based on the nature of polar flattening of the electron density around the Cl atom.