A multistep transversal linearization (MTL) method in non-linear dynamics through a Magnus characterization

A new form of a multi-step transversal linearization (MTL) method is developed and numerically explored in this study for a numeric-analytical integration of non-linear dynamical systems under deterministic excitations. As with other transversal linearization methods, the present version also requires that the linearized solution manifold transversally intersects the non-linear solution manifold at a chosen set of points or cross-section in the state space. However, a major point of departure of the present method is that it has the flexibility of treating non-linear damping and stiffness terms of the original system as damping and stiffness terms in the transversally linearized system, even though these linearized terms become explicit functions of time. From this perspective, the present development is closely related to the popular practice of tangent-space linearization adopted in finite element (FE) based solutions of non-linear problems in structural dynamics. The only difference is that the MTL method would require construction of transversal system matrices in lieu of the tangent system matrices needed within an FE framework. The resulting time-varying linearized system matrix is then treated as a Lie element using Magnus’ characterization [W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., VII (1954) 649–673] and the associated fundamental solution matrix (FSM) is obtained through repeated Lie-bracket operations (or nested commutators). An advantage of this approach is that the underlying exponential transformation could preserve certain intrinsic structural properties of the solution of the non-linear problem. Yet another advantage of the transversal linearization lies in the non-unique representation of the linearized vector field – an aspect that has been specifically exploited in this study to enhance the spectral stability of the proposed family of methods and thus contain the temporal propagation of local errors. A simple analysis of the formal orders of accuracy is provided within a finite dimensional framework. Only a limited numerical exploration of the method is presently provided for a couple of popularly known non-linear oscillators, viz. a hardening Duffing oscillator, which has a non-linear stiffness term, and the van der Pol oscillator, which is self-excited and has a non-linear damping term.

Tailoring texture of γ-Y2Si2O7 by strong magnetic field alignment and two-step sintering

In this study, a well-dispersed γ-Y2Si2O 7 ethanol-based suspension with 30 vol% solid loading was prepared by adding 1 dwb% polyethylene imine dispersant, which allows feeble magnetic γ-Y2Si2O7 particles with anisotropic magnetic susceptibility to rotate in a 12 T strong magnetic field during slip casting, resulting in the development of a strong texture in green bodies. Pressureless sintering gives rise to more pronounced grain growth in the textured sample than in the untextured sample prepared without the magnetic field due to the rapid migration of the grain boundaries of the well-oriented grains, which was revealed by constant-heating-rate sintering kinetics. It was found that the use of two-step sintering is very efficient not only for inhibiting the grain growth but also for enhancing the texture. This implies that controlled grain growth is crucial for enhancing texture development in γ-Y2Si2O7.

A theoretical model for the magnetic helicity of solar active regions

Active regions on the solar surface are known to possess magnetic helicity, which is predominantly negative in the northern hemisphere and positive in the southern hemisphere. Choudhuri et al. [Choudhuri, A.R. On the connection between mean field dynamo theory and flux tubes. Solar Phys. 215, 31–55, 2003] proposed that the magnetic helicity arises due to the wrapping up of the poloidal field of the convection zone around rising flux tubes which form active regions. Choudhuri [Choudhuri, A.R., Chatterjee, P., Nandy, D. Helicity of solar active regions from a dynamo model. ApJ 615, L57–L60, 2004] used this idea to calculate magnetic helicity from their solar dynamo model. Apart from getting broad agreements with observational data, they also predict that the hemispheric helicity rule may be violated at the beginning of a solar cycle. Chatterjee et al. [Chatterjee, P., Choudhuri, A.R., Petrovay, K. Development of twist in an emerging magnetic flux tube by poloidal field accretion. A&A 449, 781–789, 2006] study the penetration of the wrapped poloidal field into the rising flux tube due to turbulent diffusion using a simple 1-d model. They find that the extent of penetration of the wrapped field will depend on how weak the magnetic field inside the rising flux tube becomes before its emergence. They conclude that more detailed observational data will throw light on the physical conditions of flux tubes just before their emergence to the photosphere.

A novel single pot synthesis of binuclear copper(II) complexes of macrocyclic and macroacyclic compartmental ligands: Structures and magnetic properties

Two binuclear copper(II) complexes one (complex 1) with a macrocyclic ligand (H(2)L1) and other (complex 2) with a macroacyclic (end-off type) compartmental ligand (HL2) have been synthesized from single pot template synthesis involving copper(II) nitrate, 1,2diaminoethane, 4-methyl-2,6-diformylphenol, and sodium azide. Structure analysis of complex I reveals that there are actually two half molecules present in the asymmetric unit and so two complexes (molecule-I and molecule-II) are present in unit cell, although they show slight differences. The two Cu(II) centers are in distorted square pyramidal coordination environment with two endogenous phenoxo bridges provided by the phenolate of H(2)L1 I having Cu-Cu separations of 2.9133(10) angstrom and 2.9103(10) in the two molecules. In complex 2 the coordination environments around two Cu(II) centers are asymmetric, Cu1 is in distorted square pyramidal environment whereas, the coordination environment around Cu2 is distorted octahedral. The two Cu(II) centers in complex 2 are connected by two different kinds of bridges, one is endogenous phenoxo bridge provided by the phenolate of the ligand HL2 and the other is exogenous azido bridge (mu-(1),(l) type) with Cu-Cu distance of 3.032(10) angstrom. Variable temperature magnetic studies show that two Cu(II) centers in both the complexes are strongly antiferromagnetically coupled with J = -625 +/- 5 cm(-1) and J = -188.6 +/- 1cm(-1) for complex 1 and 2, respectively. (C) 2006 Elsevier B.V. All rights reserved.

Hydrothermal synthesis and structural characterization of novel alpha-Keggin unit-supported Cu(II)- and Mn(II)-bipyridine complexes from a tri-lacunary precursor

Blue [{Cu(2,2'-bipy)(2)}(2){alpha-SiW12O40}] (bipy = bipyridyl) (1) and pale yellow [Mn(2,2'-bipy)(3)](2)[alpha-SiW12O40] (2) have been synthesized hydrothermally and characterized by IR spectroscopy and single crystal X-ray structure analysis. In 1, the [alpha-SiW12O40](4-) ion acts as a bridge between the two [{Cu(2,2'-bipy)(2)](2+) moieties via coordination through the terminal oxygen atoms, while in 2, the [Mn(2,2'-bipy)(3)](2+) ion balances the charge on the polyoxo anion without forming any covalent bond. To the best of our knowledge, this is the first example of transition metal-mediated transformation of [alpha-SiW9O34](10-) to [alpha-SiW12O40](4-).

Anterior cruciate ligament reconstruction with special reference to magnetic resonance imaging of the postoperative outcome

Anterior cruciate ligament (ACL) tear is a common sports injury of the knee. Arthroscopic reconstruction using autogenous graft material is widely used for patients with ACL instability. The grafts most commonly used are the patellar and the hamstring tendons, by various fixation techniques. Although clinical evaluation and conventional radiography are routinely used in follow-up after ACL surgery, magnetic resonance imaging (MRI) plays an important role in the diagnosis of complications after ACL surgery. The aim of this thesis was to study the clinical outcome of patellar and hamstring tendon ACL reconstruction techniques. In addition, the postoperative appearance of the ACL graft was evaluated using several MRI sequences. Of the 175 patients who underwent an arthroscopically assisted ACL reconstruction, 99 patients were randomized into patellar tendon (n=51) or hamstring tendon (n=48) groups. In addition, 62 patients with hamstring graft ACL reconstruction were randomized into either cross-pin (n=31) or interference screw (n=31) fixation groups. Follow-up evaluation determined knee laxity, isokinetic muscle performance and several knee scores. Lateral and anteroposterior view radiographs were obtained. Several MRI sequences were obtained with a 1.5-T imager. The appearance and enhancement pattern of the graft and periligamentous tissue, and the location of bone tunnels were evaluated. After MRI, arthroscopy was performed on 14 symptomatic knees. The results revealed no significant differences in the 2-year outcome between the groups. In the hamstring tendon group, the average femoral and tibial bone tunnel diameter increased during 2 years follow-up by 33% and 23%, respectively. In the asymptomatic knees, the graft showed homogeneous and low signal intensity with periligamentous streaks of intermediate signal intensity on T2-weighted MR images. In the symptomatic knees, arthroscopy revealed 12 abnormal grafts and two meniscal tears, each with an intact graft. Among 3 lax grafts visible on arthroscopy, MRI showed an intact graft and improper bone tunnel placement. For diagnosing graft failure, all MRI findings combined gave a specificity of 90% and a sensitivity of 81%. In conclusion, all techniques appeared to improve patients' performance, and were therefore considered as good choices for ACL reconstruction. In follow-up, MRI permits direct evaluation of the ACL graft, the bone tunnels, and additional disorders of the knee. Bone tunnel enlargement and periligamentous tissue showing contrast enhancement were non-specific MRI findings that did not signify ACL deficiency. With an intact graft and optimal femoral bone tunnel placement, graft deficiency is unlikely, and the MRI examination should be carefully scrutinized for possible other causes for the patients symptoms.