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  • Author: J. K. Kohli x
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Abstract

A new class of functions called ‘quasi cl-supercontinuous functions’ is introduced. Basic properties of quasi cl-supercontinuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the mathematical literature is elaborated. The notion of quasi cl-supercontinuity, in general, is independent of continuity but coincides with cl-supercontinuity (≡ clopen continuity) (Applied General Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772), a significantly strong form of continuity, if range is a regular space. The class of quasi cl-supercontinuous functions properly contains each of the classes of (i) quasi perfectly continuous functions and (ii) almost cl-supercontinuous functions; and is strictly contained in the class of quasi z-supercontinuous functions. Moreover, it is shown that if X is sum connected (e.g. connected or locally connected) and Y is Hausdorff, then the function space Lq(X, Y) of all quasi cl-supercontinuous functions as well as the function space Lδ(X, Y) of all almost cl-supercontinuous functions from X to Y is closed in YX in the topology of pointwise convergence.

Abstract

In this paper, closedness of certain classes of functions in YX in the topology of uniform convergence is observed. In particular, we show that the function spaces SC(X, Y) of quasi continuous (≡semi-continuous) functions, Cα(X, Y) of α-continuous functions and L(X, Y) of cl-supercontinuous functions are closed in YX in the topology of uniform convergence.

Abstract

Three variants of R-weakly commuting mappings in the realm of uniform spaces are defined. Examples are included to reflect upon the distinctiveness of the notions so defined. Common fixed point theorems concerning these variants of R-weakly commuting mappings in the framework of uniform spaces are obtained. This generalizes several known results in the literature including those of Granas and Dugundji [], Tarafdar [] and others. Moreover, as a bi-product we obtain several common fixed point theorems which are well in contrast with a common fixed point theorem of Jungck [] who proved the same for commuting mappings.

Abstract

The notion of almost cl-supercontinuity (≡ almost clopen continuity) of functions (Acta Math. Hungar. 107 (2005), 193–206; Applied Gen. Topology 10 (1) (2009), 1–12) is extended to the realm of multifunctions. Basic properties of upper (lower) almost cl-supercontinuous multifunctions are studied and their place in the hierarchy of strong variants of continuity of multifunctions is discussed. Examples are included to reflect upon the distinctiveness of upper (lower) almost cl-supercontinuity of multifunctions from that of other strong variants of continuity of multifunctions which already exist in the literature.

Abstract

A new class of functions called ‘R cl-supercontinuous functions’ is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of R cl-supercontinuous functions properly contains the class of cl-supercontinuous (≡ clopen continuous) functions (Applied Gen. Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772) and is strictly contained in the class of Rδ-supercontinuous functions which in its turn, is properly contained in the class of R-supercontinuous functions (Demonstratio Math. 43(3) (2010), 703–723).

Abstract

A new class of functions called ‘Rδ-supercontinuous functions’ is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity which already exist in the literature is elaborated. The class of Rδ-supercontinuous functions (Math. Bohem., to appear) properly contains the class of Rz-supercontinuous functions which in its turn properly contains the class of Rcl- supercontinuous functions (Demonstratio Math. 46(1) (2013), 229-244) and so includes all Rcl-supercontinuous (≡clopen continuous) functions (Applied Gen. Topol. 8(2) (2007), 293-300; Indian J. Pure Appl. Math. 14(6) (1983), 767-772) and is properly contained in the class of R-supercontinuous functions (Demonstratio Math. 43(3) (2010), 703-723).

The acetylation of proteins in biological systems is largely catalyzed by specific acetyl transferases utilizing acetyl CoA as the acetyl donor. The enzymatic acetylation of proteins independent of acetyl CoA was unknown until we discovered a unique membrane-bound enzyme in mammalian cells catalyzing the transfer of acetyl groups from polyphenolic peracetates (PAs) to certain enzyme proteins, resulting in the modulation of their catalytic activities. Since for the enzyme, acetyl derivatives of several classes of polyphenols such as coumarins, flavones, chromones, and xanthones were found to be acetyl donors, the enzyme was termed as acetoxy drug: protein transacetylase (TAase). TAase was found to be ubiquitously present in tissues of several animal species and a variety of animal cells. Liver microsomal cytochrome P-450 (CYP), NADPH-cytochrome c reductase and cytosolic glutathione S-transferase (GST) were found to be the targets for TAase-catalyzed acetylation by the model acetoxy drug 7,8-diacetoxy-4-methylcoumarin (DAMC). Accordingly, the catalytic activities of CYP-linked, mixed function oxidases (MFOs) and GST were irreversibly inhibited while the reductase was remarkably activated. In this report, we have reviewed the details concerning purification and characterization of TAase and the protein acetylation by DAMC. Quantitative structure–activity relationship (QSAR) studies concerning the specificities of various PAs to liver microsomal TAase and TAase-related biological effects have also been reviewed.