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.

Fractional damping: Stochastic origin and finite approximations

Fractional-order derivatives appear in various engineering applications including models for viscoelastic damping. Damping behavior of materials, if modeled using linear, constant coefficient differential equations, cannot include the long memory that fractional-order derivatives require. However, sufficiently great rnicrostructural disorder can lead, statistically, to macroscopic behavior well approximated by fractional order derivatives. The idea has appeared in the physics literature, but may interest an engineering audience. This idea in turn leads to an infinite-dimensional system without memory; a routine Galerkin projection on that infinite-dimensional system leads to a finite dimensional system of ordinary differential equations (ODEs) (integer order) that matches the fractional-order behavior over user-specifiable, but finite, frequency ranges. For extreme frequencies (small or large), the approximation is poor. This is unavoidable, and users interested in such extremes or in the fundamental aspects of true fractional derivatives must take note of it. However, mismatch in extreme frequencies outside the range of interest for a particular model of a real material may have little engineering impact.

A fast time-domain algorithm for the assessment of tissue blood flow in laser-Doppler flowmetry

In this study, we derive a fast, novel time-domain algorithm to compute the nth-order moment of the power spectral density of the photoelectric current as measured in laser-Doppler flowmetry (LDF). It is well established that in the LDF literature these moments are closely related to fundamental physiological parameters, i.e. concentration of moving erythrocytes and blood flow. In particular, we take advantage of the link between moments in the Fourier domain and fractional derivatives in the temporal domain. Using Parseval's theorem, we establish an exact analytical equivalence between the time-domain expression and the conventional frequency-domain counterpart. Moreover, we demonstrate the appropriateness of estimating the zeroth-, first- and second-order moments using Monte Carlo simulations. Finally, we briefly discuss the feasibility of implementing the proposed algorithm in hardware.

Identification of dynamical systems with fractional derivative damping models using inverse sensitivity analysis

The problem of identifying parameters of time invariant linear dynamical systems with fractional derivative damping models, based on a spatially incomplete set of measured frequency response functions and experimentally determined eigensolutions, is considered. Methods based on inverse sensitivity analysis of damped eigensolutions and frequency response functions are developed. It is shown that the eigensensitivity method requires the development of derivatives of solutions of an asymmetric generalized eigenvalue problem. Both the first and second order inverse sensitivity analyses are considered. The study demonstrates the successful performance of the identification algorithms developed based on synthetic data on one, two and a 33 degrees of freedom vibrating systems with fractional dampers. Limited studies have also been conducted by combining finite element modeling with experimental data on accelerances measured in laboratory conditions on a system consisting of two steel beams rigidly joined together by a rubber hose. The method based on sensitivity of frequency response functions is shown to be more efficient than the eigensensitivity based method in identifying system parameters, especially for large scale systems.

Novel derivatives of spirohydantoin induce growth inhibition followed by apoptosis in leukemia cells

Hydantoin derivatives possess a variety of biochemical and pharmacological properties and consequently are used to treat many human diseases. However, there are only few studies focusing on their potential as cancer therapeutic agents. In the present study, we have examined anticancer properties of two novel spirohydantoin compounds, 8-(3,4-difluorobenzyl)-1'-(pent-4-enyl)-8-azaspiro[bicyclo[3.2.1] octane-3,4'-imidazolidine]-2',5'-dione (DFH) and 8-(3,4-dichlorobenzyl)-1'-(pent-4-enyl)-8-azaspiro[bicyclo[3.2.1]octane-3,4'-imidazolidine]-2',5'-dione (DCH). Both the compounds exhibited dose- and time-dependent cytotoxic effect on human leukemic cell lines, K562, Reh, CEM and 8ES. Incorporation of tritiated thymidine ([H-3) thymidine) in conjunction with cell cycle analysis suggested that DFH and DCH inhibited the growth of leukemic cells. Downregulation of PCNA and p-histone H3 further confirm that the growth inhibition could be at the level of DNA replication. Flow cytometric analysis indicated the accumulation of cells at subG1 phase suggesting induction of apoptosis, which was further confirmed and quantified both by fluorescence-activated cell sorting (FACS) and confocal microscopy following annexin V-FITC/propidium iodide (PI) staining. Mechanistically, our data support the induction of apoptosis by activation of the mitochondrial pathway. Results supporting such a model include, elevated levels of p53, and BAD, decreased level of BCL2, activation and cleavage of caspase 9, activation of procaspase 3, poly (ADP-ribosyl) polymerase (PARP) cleavage, downregulation of Ku70, Ku80 and DNA fragmentation. Based on these results we discuss the mechanism of apoptosis induced by DFH and its implications in leukemia therapy. (C) 2008 Elsevier Inc. All rights reserved.

A New Synthesis Of Dibenzo[Aj]Xanthene Derivatives

Oxidation of mu-H/alkylbisnaphthols (4a-g) gives the 14-substituted dibenzo[aj]xanthenes (5a-g) as the sole product while that of mu-arylbisnaphthols (4h-j) gives the xanthenes (5h-j) along with the corresponding spironaphthalenones (1h-j). A probable mechanism for the formation of the products has been suggested.

Solution of a System of Generalized Abel Integral Equations Using Fractional Calculus

A method is presented for obtaining useful closed form solution of a system of generalized Abel integral equations by using the ideas of fractional integral operators and their applications. This system appears in solving certain mixed boundary value problems arising in the classical theory of elasticity.

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.

Synthesis of Unnatural C-2 Amino Acid Nucleosides Using NIS-Mediated Ring Opening of 1,2-Cyclopropane Carboxylated Sugar Derivatives

We have developed a general and efficient method for the stereoselective construction of pyrimidine-based pyranosyl C-2 amino acid nucleosides using NIS-mediated ring opening of 1,2-cyclopropanated sugar derivatives. This methodology has been successfully extended to the synthesis of furanosyl nucleosides, Which have potential applications in the development of novel, nontoxic antifungal therapeutics.

13C NMR studies on trifluoroacetyl derivatives of 2-acetamido-2-deoxyhexoses

The effect of trifluoroacetylation on the 13C chemical shifts of 2-acetamido-2-deoxyhexoses was examined. Studies of the 2-acetamido derivatives of glucose, galactose and mannose established that no regular trend in the 13C shifts occurred on trifluoroacetylation. This was in marked contrast to the results obtained for the 1H chemical shifts.

Conformational analysis of hydroxyproline and its protected derivatives by 1H NMR

From a computer simulation of the 270 MHz 1H NMR spectra of hydroxyproline (Hyp) and its protected derivatives, precise values of ring vicinal coupling constants were obtained. These couplings were related to ring torsional angles, using a Karplus type analysis. From the NMR analysis it was observed that the pyrrolidine ring possesses a unique and highly homogeneous conformation (Cγ-exo form). Temperature dependence studies on protected dipeptides suggest that the pyrrolidine ring conformation is independent of backbone conformation. An unusual X-Hyp, β-turn was observed for Boc-Aib-Hyp-NHMe.

Solid-State Acyl Migration In Salicylamides - An X-Ray Study Of The Ortho-Propionyl And Normal-Propionyl Derivatives

In recent years there has been an upsurge of interest in the study of organic reactions in the solid state. It is now realised that the crystalline matrix provides an extra-ordinary spatial control on the initiation and progress of these reactions. Electronic and dipolar effects which are important in solution are replaced by structural and geometric effects in solids. These 'spatial' or 'topochemical' aspects are important in understanding the mechanistic details of the reaction. In our laboratory, the thermally induced acyl migration in salicylamides from 0- to N- position in the solid state has been under study (Scheme 1). The structures of the acetyl and benzoyl derivatives (Ia,IIa, Ib and IIb) have been reported.

Synthesis and in vitro cytotoxic evaluation of novel diazaspiro bicyclo hydantoin derivatives in human leukemia cells: A SAR study

To study the structure activity relationship (SAR) on the cytotoxic activity and probe the structural requirement for the potent antitumor activity, a series of novel diazaspiro bicyclo hydantoin derivatives were designed and synthesized. Their structures were confirmed by H-1 NMR, LCMS and IR analyses. The antiproliferative effect of these compounds were determined against human leukemia, K562 (chronic myelogenous leukemia) and CEM (T-cell leukemia) cells using trypan blue and MTT assay, and the SAR associated with the position of N-terminal substituents in diazaspiro bicyclo hydantoin have also been discussed. It has been observed that these compounds displayed strong, moderate and weak cytotoxic activities. Interestingly, compounds having electron withdrawing groups at third and fourth position of the phenyl ring displayed selectively cytotoxic activities to both the cell lines tested with IC50 value lower than 50 mu M. In addition, the cytotoxic activities of the compounds 7(a-o) bearing the substituents at N-3 position of diazaspiro bicyclo hydantoin increases in the order alkene > ester > ether and plays an important role in determining their antitumor activities. The position and number of substituents in benzyl group attached to N-8 of diazaspiro bicyclo hydantoin nucleus interacted selectively with specific targets leading to the difference of biochemical and pharmacological effects.

A PPP AND CNDO Study of Dithiocarbonic Antonic Derivatives

The application of the CNDO and PPP-CI methods to N,N-dimethyl dithiocarbamate, O-methyl dithiocarbonate (methyl xanthate) and methyl trithiocarbonate ions for the elucidation of electronic structure and electronic spectra is described. The CNDO/2 calculations have been used to obtain the one centre core integrals of the ionic compounds required in calculating the pi electronic spectra of these molecules using the PPP method. The calculated spectra are in good agreement with the experiment. The atomic charge densities determined for alkyl xanthate, dithiocarbamate and trithiocarbonate ions support the earlier qualitative predictions regarding electronic structure from spectroscopic and other studies.

Fractional power dependence of rate on activation energy for reactions with very low internal barriers

The dynamics of reactions with low internal barriers are studied both analytically and numerically for two different models. Exact expressions for the average rate,kI, are obtained by solving the associated first passage time problems. Both the average rate constant, kI, and the numerically calculated long-time rate constant, kL, show a fractional power law dependence on the barrier height for very low barriers. The crossover of the reaction dynamics from low to high barrier is investigated.