A Study of Fuzzy Sequence Spaces

The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces FSJθT, FS0JθT and FS∞JθT on a fuzzy real number F defined by a compact operator T. We investigate algebraic properties like linearity, solidness and monotinicity with some important examples. Further, we also analyze closedness of the subspace and inclusion relations on the said spaces.

Completeness of Sequence Spaces Generated by an Orlicz Function

In this paper, we discuss about completeness property of Orlicz sequence spaces defined by an Orlicz function. Orlicz sequence space is generalization of p-summable sequence space, for every which is also an Orlicz sequence space. Based on the property of convergence sequence on norm space, we define $\Phi$-convergence sequence on Orlicz sequence space. Moreover, we define $\Phi$-Cauchy sequence and $\Phi$-complete on Orlicz sequence space. In this paper, we show the relationship between the (ordinary) convergent sequence, $\Phi$-convergent and $\Phi$-Cauchy sequences. Finally, it will also be shown that Orlicz sequence space is Banach space and $\Phi$-complete space.

Sequences, Nets, and Filters of Fuzzy Soft Multi Sets in Fuzzy Soft Multi Topological Spaces

In this chapter, the authors introduce a new sequence of fuzzy soft multi sets in fuzzy soft multi topological spaces and their basic properties are studied. The concepts of subsequence, convergence sequence and cluster fuzzy soft multi sets of fuzzy soft multi sets are proposed. Actually Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar to each other than to those in other groups (clusters). It is a main task of exploratory data mining and a common technique for statistical data analysis used in many fields including machine learning, pattern recognition, image analysis, information retrieval and bioinformatics. Here the authors define the notions of net and filter and establish the correspondence between net convergence and filter convergence in fuzzy soft multi topological spaces.

Analyzing ion distributions around DNA

We present a new method for analyzing ion, or molecule, distributions around helical nucleic acids and illustrate the approach by analyzing data derived from molecular dynamics simulations. The analysis is based on the use of curvilinear helicoidal coordinates and leads to highly localized ion densities compared to those obtained by simply superposing molecular dynamics snapshots in Cartesian space. The results identify highly populated and sequence-dependent regions where ions strongly interact with the nucleic and are coupled to its conformational fluctuations. The data from this approach is presented as ion populations or ion densities (in units of molarity) and can be analyzed in radial, angular and longitudinal coordinates using 1D or 2D graphics. It is also possible to regenerate 3D densities in Cartesian space. This approach makes it easy to understand and compare ion distributions and also allows the calculation of average ion populations in any desired zone surrounding a nucleic acid without requiring references to its constituent atoms. The method is illustrated using microsecond molecular dynamics simulations for two different DNA oligomers in the presence of 0.15 M potassium chloride. We discuss the results in terms of convergence, sequence-specific ion binding and coupling with DNA conformation.