Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy. DOI: 10.1115/1.4002516]

Synthesis of 2-(5-((5-(4-chlorophenyl)furan 2-yl)methylene)-4-oxo-2-thioxo-thiazolidi n-3-yl)acetic acid derivatives and evaluation of their cytotoxicity and induction of apoptosis in human leukemia cells

In order to explore the anticancer effect associated with the thiazolidinone framework, several 2-(5-((5-(4-chlorophenyl)furan-2-yl)methylene)-4-oxo-2-thioxothiazolidin-3-yl)acetic acid derivatives 5(a-1) were synthesized. Variation in the functional group at C-terminal of the thiazolidinone led to set of compounds bearing amide moiety. Their chemical structures were confirmed by H-1 NMR, IR and Mass Spectra analysis. These thiazolidinone compounds containing furan moiety exhibits moderate to strong antiproliferative activity in a cell cycle stage-dependent and dose dependent manner in two different human leukemia cell lines. The importance of the electron donating groups on thiazolidinone moiety was confirmed by MTT and Trypan blue assays and it was concluded that the 4th position of the substituted aryl ring plays a dominant role for its anticancer property. Among the synthesized compounds, 5e and 5f have shown potent anticancer activity on both the cell lines tested. To rationalize the role of electron donating group in the induction of cytotoxicity we have chosen two molecules (5e and 5k) having different electron donating group at different positions. LDH assay, Flow cytometric analysis and DNA fragmentation suggest that 5e is more cytotoxic and able to induce the apoptosis. (C) 2009 Elsevier Ltd. All rights reserved.

Phase Characterization and Study of Molecular Order of a Three-Ring Mesogen by (13)C NMR in Smectic C and Nematic Phases

Molecules exhibiting a thermotropic liquid-crystalline property have acquired significant importance due to their sensitivity to external stimuli such as temperature, mechanical forces, and electric and magnetic fields. As a result, several novel mesogens have been synthesized by the introduction of various functional groups in the vicinity of the aromatic core as well as in the side chains and their properties have been studied. In the present study, we report three-ring mesogens with hydroxyl groups at one terminal. These mesogens were synthesized by a multistep route, and structural characterization was accomplished by spectral techniques. The mesophase properties were studied by hot-stage optical polarizing microscopy, differential scanning calorimetry, and small-angle X-ray scattering. An enantiotropic nematic phase was noticed for lower homologues, while an additional smectic C phase was found for higher homologues. Solid-state high-resolution natural abundance (13)C NMR studies of a typical mesogen in the solid phase and in the mesophases have been carried out. The (13)C NMR spectrum of the mesogen in the smectic C and nematic phases indicated spontaneous alignment of the molecule in the magnetic field. By utilizing the two-dimensional separated local field (SLF) NMR experiment known as SAMPI4, (13)C-(1)H dipolar couplings have been obtained, which were utilized to determine the orientational order parameters of the mesogen.

Measurement of Large Dipolar Couplings of a Liquid Crystal with Terminal Phenyl Rings and Estimation of the Order Parameters

NMR spectroscopy is a powerful means of studying liquid-crystalline systems at atomic resolutions. Of the many parameters that can provide information on the dynamics and order of the systems, H-1-C-13 dipolar couplings are an important means of obtaining such information. Depending on the details of the molecular structure and the magnitude of the order parameters, the dipolar couplings can vary over a wide range of values. Thus the method employed to estimate the dipolar couplings should be capable of estimating both large and small dipolar couplings at the same time. For this purpose, we consider here a two-dimensional NMR experiment that works similar to the insensitive nuclei enhanced by polarization transfer (INEPT) experiment in solution. With the incorporation of a modification proposed earlier for experiments with low radio frequency power, the scheme is observed to enable a wide range of dipolar couplings to be estimated at the same time. We utilized this approach to obtain dipolar couplings in a liquid crystal with phenyl rings attached to either end of the molecule, and estimated its local order parameters.

Evaluation and Minimization of Low-Order Harmonic Torque in Low-Switching-Frequency Inverter-Fed Induction Motor Drives

A low-order harmonic pulsating torque is a major concern in high-power drives, high-speed drives, and motor drives operating in an overmodulation region. This paper attempts to minimize the low-order harmonic torques in induction motor drives, operated at a low pulse number (i.e., a low ratio of switching frequency to fundamental frequency), through a frequency domain (FD) approach as well as a synchronous reference frame (SRF) based approach. This paper first investigates FD-based approximate elimination of harmonic torque as suggested by classical works. This is then extended into a procedure for minimization of low-order pulsating torque components in the FD, which is independent of machine parameters and mechanical load. Furthermore, an SRF-based optimal pulse width modulation (PWM) method is proposed to minimize the low-order harmonic torques, considering the motor parameters and load torque. The two optimal methods are evaluated and compared with sine-triangle (ST) PWM and selective harmonic elimination (SHE) PWM through simulations and experimental studies on a 3.7-kW induction motor drive. The SRF-based optimal PWM results in marginally better performance than the FD-based one. However, the selection of optimal switching angle for any modulation index (M) takes much longer in case of SRF than in case of the FD-based approach. The FD-based optimal solutions can be used as good starting solutions and/or to reasonably restrict the search space for optimal solutions in the SRF-based approach. Both of the FD-based and SRF-based optimal PWM methods reduce the low-order pulsating torque significantly, compared to ST PWM and SHE PWM, as shown by the simulation and experimental results.

Beyond fractional derivatives: local approximation of other convolution integrals

Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs O(t(2)) computations owing to the repeated evaluation of integrals over intervals that grow like t. Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled in finite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives (Singh & Chatterjee 2006 Nonlinear Dyn. 45, 183-206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with O(t) computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e. g. in stability analyses.

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Synthesis and characterization of mono- and bis-D-pi-A cryptand derivatives for second-order nonlinear optics and its modulation with different metal ion inputs

Two new classes of mono- and bis-D-pi-A cryptand derivatives with a flexible and a rigid cryptand core have been synthesized. The linear and nonlinear optical properties of these molecules are probed. The three dimensional cavity of the cryptand moiety has been utilized to modulate the SHG intensity to different extents in solution with metal ion inputs such as Ni-II,Cu-II,Zn-II, and Cd-II. We also report that decomplexation events can be used to reversibly modulate their NLO responses.

A fractional order proportional integral controller for path following and trajectory tracking of miniature air vehicles

In this paper, a fractional order proportional-integral controller is developed for a miniature air vehicle for rectilinear path following and trajectory tracking. The controller is implemented by constructing a vector field surrounding the path to be followed, which is then used to generate course commands for the miniature air vehicle. The fractional order proportional-integral controller is simulated using the fundamentals of fractional calculus, and the results for this controller are compared with those obtained for a proportional controller and a proportional integral controller. In order to analyze the performance of the controllers, four performance metrics, namely (maximum) overshoot, control effort, settling time and integral of the timed absolute error cost, have been selected. A comparison of the nominal as well as the robust performances of these controllers indicates that the fractional order proportional-integral controller exhibits the best performance in terms of ITAE while showing comparable performances in all other aspects.

solation and characterization of monodeamidated derivatives of bovine pancreatic Ribonuclease A

The isolation and characterization of the initial intermediates formed during the irreversible acid denaturation of enzyme Ribonuclease A are described. The products obtained when RNase A is maintained in 0.5 M HCl at 30° for periods up to 20 h have been analyzed by ion-exchange chromatography on Amberlite XE-64. Four distinct components were found to elute earlier to RNase A; these have been designated RNase Aa2, Aa1c, Aa1b, and Aa1a in order of their elution. With the exception of RNase Aa2, the other components are nearly as active as RNase A. Polyacrylamide gel electrophoresis at near-neutral pH indicated that RNase Aa1a, Aa1b, and Aa1c are monodeamidated derivatives of RNase A; RNase Aa2 contains, in addition, a small amount of a dideamidated component. RNase Aa2, which has 75% enzymic activity as compared to RNase A, consists of dideamidated and higher deamidated derivatives of RNase A. Except for differences in the proteolytic susceptibilities at an elevated temperature or acidic pH, the monodeamidated derivatives were found to have very nearly the same enzymic activity and the compact folded structure as the native enzyme. Fingerprint analyses of the tryptic peptides of monodeamidated derivatives have shown that the deamidations are restricted to an amide cluster in the region 67–74 of the polypeptide chain. The initial acid-catalyzed deamidation occurs in and around the 65–72 disulfide loop giving rise to at least three distinct monodeamidated derivatives of RNase A without an appreciable change in the catalytic activity and conformation of the ribonuclease molecule. Significance of this specific deamidation occurring in highly acidic conditions, and the biological implications of the physiological deamidation reactions of proteins are discussed.

The modes of binding methyl-alpha (and beta)-D-glucopyranosides and some of their derivatives to concanavalin A--a theoretical approach

The probable modes of binding of Methyl--alpha (and beta)-D-glucopyranosides and some of their derivatives to concanavalin A have been proposed from theoretical studies. Theory predicts that beta-MeGlcP can bind to ConA in three different modes whereas alpha-MeGlcP can bind only in one mode. beta-MeGlcP in its most favourable mode of binding differs from alpha-MeGlcP in its alignment in the active-site of the lectin where it binds in a flipped or inverted orientation. Methyl substitution at the C-2 atom of the alpha-MeGlcP does not significantly affect the possible orientations of the sugar in the active-site of the lectin. Methyl substitution at C-3 or C-4, however, affects the allowed orientations drastically leading to the poor inhibiting power of Methyl-3-O-methyl-alpha-D-glucopyranoside and the inactivity of Methyl-4-O-methyl-alpha-D-glycopyranoside. These studies suggest that the increased activity of the alpha-MeGlcP over beta-MeGlcP may be due to the possibility of formation of better hydrogen bonds and to hydrophobic interactions rather than to steric factors as suggested by earlier workers. These models explain the available NMR and other binding studies.

Synthesis and biological evaluation of novel 1-(4-methoxyphenethyl)-1H-benzimidazole-5-carboxylic acid derivatives and their precursors as antileukemic agents

We report here the synthesis and preliminary evaluation of novel 1-(4-methoxyphenethyl)-1H-benzimidazole-5-carboxylic acid derivatives 6(a–k) and their precursors 5(a–k) as potential chemotherapeutic agents. In each case, the structures of the compounds were determined by FTIR, 1H NMR and mass spectroscopy. Among the synthesized molecules, methyl 1-(4-methoxyphenethyl)-2-(4-fluoro-3-nitrophenyl)-1H-benzimidazole-5-carboxylate (5a) induced maximum cell death in leukemic cells with an IC50 value of 3 μM. Using FACS analysis we show that the compound 5a induces S/G2 cell cycle arrest, which was further supported by the observed down regulation of CDK2, Cyclin B1 and PCNA. The observed downregulation of proapoptotic proteins, upregulation of antiapoptotic proteins, cleavage of PARP and elevated levels of DNA strand breaks indicated the activation of apoptosis by 5a. These results suggest that 5a could be a potent anti-leukemic agent.

Oxidation of spiroketones with DDQ - synthesis of tropone derivatives and DDHQ diesters

Oxidation of spiroketones 3a–f with DDQ in dry benzene gave tropone derivatives 4a–f and DDHQ esters 5a–f (cis -cis isomer 6a–f, (cis -cis isomer 7a–f). While the aryl substituted spirokeone 17a gave a 2:1 mixture of 19a and the corresponding trans -trans isomer, the aryl substituted spiroketones 17b–d gave exclusively trans-trans isomers 19b–d. Heating acid chloride of acid 9c with DDHQ resulted in compounds 4a and 7a, thus confirming the structures assigned. Mechanism of formation of these compounds has been rationalised. A detailed study of 2D 1H-1H COSY, 1H-13C COSY, HMBC and 2D NOESY of compound 7d led to complete assignment of 1H and 13C NMR signals and its solution conformation.

An investigation of first-order transition across charge ordered and ferromagnetic phases in Gd0.5Sr0.5MnO3 single crystals by magnetic and magnetotransport studies

Gadolinium strontium manganite single crystals of the composition Gd0.5Sr0.5MnO3 were grown using the optical float zone method. We report here the magnetic and magnetotransport properties of these crystals. A large magnetoresistance similar to 10(9)% was observed at 45 K under the application of a 110 kOe field. We have observed notable thermomagnetic anomalies such as open hysteresis loops across the broadened first-order transition between the charge order insulator and the ferromagnetic metallic phase while traversing the magnetic field-temperature (H-T) plane isothermally or isomagnetically. In order to discern the cause of these observed anomalies, the H-T phase diagram for Gd0.5Sr0.5MnO3 is formulated using the magnetization-field (M-H), magnetization-temperature (M-T) and resistance-temperature (R-T) measurements. The temperature dependence of the critical field (i.e. H-up, the field required for transformation to the ferromagnetic metallic phase) is non-monotonic. We note that the non-monotonic variation of the supercooling limit is anomalous according to the classical concepts of the first-order phase transition. Accordingly, H-up values below similar to 20 K are unsuitable to represent the supercooling limit. It is possible that the nature of the metastable states responsible for the observed open hysteresis loops is different from that of the supercooled ones.

Synthesis of liquid crystalline benzothiazole base derivatives: A study of their optical and electrical properties

Two new donor-acceptor type liquid crystalline semiconductors based on benzothiazole have been synthesized. Their structural, photophysical and electronic properties were investigated using X-ray diffraction, atomic force microscopy, cyclic voltammetry, UV-Vis, photoluminescence, and Raman spectroscopy. The liquid crystalline behaviour of the molecules was thoroughly examined by differential scanning calorimetry (DSC) and optical polarizing microscope. The DSC and thermogravimetric analysis (TGA) show that these materials posses excellent thermal stability and have decomposition temperatures in excess of 300 degrees C. Beyond 160 degrees C both molecules show a smectic A liquid crystalline phase that exists till about 240 degrees C. Field-effect transistors were fabricated by vacuum evaporating the semiconductor layer using standard bottom gate/top contact geometry. The devices exhibit p-channel behaviour with hole mobilities of 10(-2) cm(2)/Vs. (C) 2009 Elsevier B.V. All rights reserved.