An analogue of the logistic map in two dimensions

This is a sequel to our earlier work on the modulated logistic map. Here, we first show that the map comes under the universality class of Feigenbaum. We then give evidence for the fact that our model can generate strange attractors in the unit square for an uncountable number of parameter values in the range μ∞<μ<1. Numerical plots of the attractor for several values of μ are given and the self-similar structure is explicity shown in one case. The fractal and information dimensions of the attractors for many values of μ are shown to be greater than one and the variation in their structure is analysed using the two Lyapunov exponents of the system. Our results suggest that the map can be considered as an analogue of the logistic map in two dimensions and may be useful in describing certain higher dimensional chaotic phenomena.

On Chaos and Fractals in General Topological Spaces

The present study on chaos and fractals in general topological spaces. Chaos theory originated with the work of Edward Lorenz. The phenomenon which changes order into disorder is known as chaos. Theory of fractals has its origin with the frame work of Benoit Mandelbrot in 1977. Fractals are irregular objects. In this study different properties of topological entropy in chaos spaces are studied, which also include hyper spaces. Topological entropy is a measures to determine the complexity of the space, and compare different chaos spaces. The concept of fractals can’t be extended to general topological space fast it involves Hausdorff dimensions. The relations between hausdorff dimension and packing dimension. Regular sets in Metric spaces using packing measures, regular sets were defined in IR” using Hausdorff measures. In this study some properties of self similar sets and partial self similar sets. We can associate a directed graph to each partial selfsimilar set. Dimension properties of partial self similar sets are studied using this graph. Introduce superself similar sets as a generalization of self similar sets and also prove that chaotic self similar self are dense in hyper space. The study concludes some relationships between different kinds of dimension and fractals. By defining regular sets through packing dimension in the same way as regular sets defined by K. Falconer through Hausdorff dimension, and different properties of regular sets also.

A New Fast Fractal Modeling Approach for the Detection of Microcalcifications in Mammograms

In this paper, a novel fast method for modeling mammograms by deterministic fractal coding approach to detect the presence of microcalcifications, which are early signs of breast cancer, is presented. The modeled mammogram obtained using fractal encoding method is visually similar to the original image containing microcalcifications, and therefore, when it is taken out from the original mammogram, the presence of microcalcifications can be enhanced. The limitation of fractal image modeling is the tremendous time required for encoding. In the present work, instead of searching for a matching domain in the entire domain pool of the image, three methods based on mean and variance, dynamic range of the image blocks, and mass center features are used. This reduced the encoding time by a factor of 3, 89, and 13, respectively, in the three methods with respect to the conventional fractal image coding method with quad tree partitioning. The mammograms obtained from The Mammographic Image Analysis Society database (ground truth available) gave a total detection score of 87.6%, 87.6%, 90.5%, and 87.6%, for the conventional and the proposed three methods, respectively.

Design and study of self-assembled functional organic and hybrid systems for biological applications

The focus of self-assembly as a strategy for the synthesis has been confined largely to molecules, because of the importance of manipulating the structure of matter at the molecular scale. We have investigated the influence of temperature and pH, in addition to the concentration of the capping agent used for the formation of the nano-bio conjugates. For example, the formation of the narrower size distribution of the nanoparticles was observed with the increase in the concentration of the protein, which supports the fact that γ-globulin acts both as a controller of nucleation as well as stabiliser. As analyzed through various photophysical, biophysical and microscopic techniques such as TEM, AFM, C-AFM, SEM, DLS, OPM, CD and FTIR, we observed that the initial photoactivation of γ-globulin at pH 12 for 3 h resulted in small protein fibres of ca. Further irradiation for 24 h, led to the formation of selfassembled long fibres of the protein of ca. 5-6 nm and observation of surface plasmon resonance band at around 520 nm with the concomitant quenching of luminescence intensity at 680 nm. The observation of light triggered self-assembly of the protein and its effect on controlling the fate of the anchored nanoparticles can be compared with the naturally occurring process such as photomorphogenesis.Furthermore,our approach offers a way to understand the role played by the self-assembly of the protein in ordering and knock out of the metal nanoparticles and also in the design of nano-biohybrid materials for medicinal and optoelectronic applications. Investigation of the potential applications of NIR absorbing and water soluble squaraine dyes 1-3 for protein labeling and anti-amyloid agents forms the subject matter of the third chapter of the thesis. The study of their interactions with various proteins revealed that 1-3 showed unique interactions towards serum albumins as well as lysozyme. 69%, 71% and 49% in the absorption spectra as well as significant quenching in the fluorescence intensity of the dyes 1-3, respectively. Half-reciprocal analysis of the absorption data and isothermal titration calorimetric (ITC) analysis of the titration experiments gave a 1:1 stoichiometry for the complexes formed between the lysozyme and squaraine dyes with association constants (Kass) in the range 104-105 M-1. We have determined the changes in the free energy (ΔG) for the complex formation and the values are found to be -30.78, -32.31 and -28.58 kJmol-1, respectively for the dyes 1, 2 and 3. Furthermore, we have observed a strong induced CD (ICD) signal corresponding to the squaraine chromophore in the case of the halogenated squaraine dyes 2 and 3 at 636 and 637 nm confirming the complex formation in these cases. To understand the nature of interaction of the squaraine dyes 1-3 with lysozyme, we have investigated the interaction of dyes 1-3 with different amino acids. These results indicated that the dyes 1-3 showed significant interactions with cysteine and glutamic acid which are present in the side chains of lysozyme. In addition the temperature dependent studies have revealed that the interaction of the dye and the lysozyme are irreversible. Furthermore, we have investigated the interactions of these NIR dyes 1-3 with β- amyloid fibres derived from lysozyme to evaluate their potential as inhibitors of this biologically important protein aggregation. These β-amyloid fibrils were insoluble protein aggregates that have been associated with a range of neurodegenerative diseases, including Huntington, Alzheimer’s, Parkinson’s, and Creutzfeldt-Jakob diseases. We have synthesized amyloid fibres from lysozyme through its incubation in acidic solution below pH 4 and by allowing to form amyloid fibres at elevated temperature. To quantify the binding affinities of the squaraine dyes 1-3 with β-amyloids, we have carried out the isothermal titration calorimetric (ITC) measurements. The association constants were determined and are found to be 1.2 × 105, 3.6× 105 and 3.2 × 105 M-1 for the dyes, 1-3, respectively. To gain more insights into the amyloid inhibiting nature of the squaraine dyes under investigations, we have carried out thioflavin assay, CD, isothermal titration calorimetry and microscopic analysis. The addition of the dyes 1-3 (5μM) led to the complete quenching in the apparent thioflavin fluorescence, thereby indicating the destabilization of β-amyloid fibres in the presence of the squaraine dyes. Further, the inhibition of the amyloid fibres by the squaraine dyes 1-3, has been evidenced though the DLS, TEM AFM and SAED, wherein we observed the complete destabilization of the amyloid fibre and transformation of the fibre into spherical particles of ca. These results demonstrate the fact that the squaraine dyes 1-3 can act as protein labeling agents as well as the inhibitors of the protein amyloidogenesis. The last chapter of the thesis describes the synthesis and investigation of selfassembly as well as bio-imaging aspects of a few novel tetraphenylethene conjugates 4-6.Expectedly, these conjugates showed significant solvatochromism and exhibited a hypsochromic shift (negative solvatochromism) as the solvent polarity increased, and these observations were justified though theoretical studies employing the B3LYP/6-31g method. We have investigated the self-assembly properties of these D-A conjugates though variation in the percentage of water in acetonitrile solution due to the formation of nanoaggregates. Further the contour map of the observed fluorescence intensity as a function of the fluorescence excitation and emission wavelength confirmed the formation of J-type aggregates in these cases. To have a better understanding of the type of self-assemblies formed from the TPE conjugates 4-6, we have carried out the morphological analysis through various microscopic techniques such as DLS, SEM and TEM. 70%, we observed rod shape architectures having ~ 780 nm in diameter and ~ 12 μM in length as evidenced through TEM and SEM analysis. We have made similar observations with the dodecyl conjugate 5 at ca. 70% and 50% water/acetonitrile mixtures, the aggregates formed from 4 and 5 were found to be highly crystalline and such structures were transformed to amorphous nature as the water fraction was increased to 99%. To evaluate the potential of the conjugate as bio-imaging agents, we have carried out their in vitro cytotoxicity and cellular uptake studies though MTT assay, flow cytometric and confocal laser scanning microscopic techniques. Thus nanoparticle of these conjugates which exhibited efficient emission, large stoke shift, good stability, biocompatibility and excellent cellular imaging properties can have potential applications for tracking cells as well as in cell-based therapies. In summary we have synthesized novel functional organic chromophores and have studied systematic investigation of self-assembly of these synthetic and biological building blocks under a variety of conditions. The investigation of interaction of water soluble NIR squaraine dyes with lysozyme indicates that these dyes can act as the protein labeling agents and the efficiency of inhibition of β-amyloid indicate, thereby their potential as anti-amyloid agents.