A Note On Asymptotic Smoothness Of The Extensions Of Zadeh

The concept of asymptotic smooth transformation was introduced by J. Hale [10]. It is a very important property for a transformation between complete metric spaces to have a global attractor. This property has also consequences on asymptotic stability of attractors. In our work we study the conditions under which the Zadeh's extension of a continuous map f : R n → R n is asymptotically smooth in the complete metric space JF(R n) of normal fuzzy sets with the induced Hausdorff metric d ∞ (see Kloeden and Diamond [8]).212141153Barros, L.C., Bassanezi, R.C., Tonelli, P.A., On the continuity of Zadeh's extension (1997) Proceedings Seventh IFSA World Congress, 2, pp. 3-8. , PragueBarros, L.C., Bassanezi, R.C., Tonelli, P.A., Fuzzy modeling in populations dynamics (2000) Ecological Modeling, 128, pp. 27-33Brumley, W.E., On the asymptotic behavior of solutions of differential difference equations of neutral type (1970) J. of Differential Equations, 7, pp. 175-188Cabrelli, C.A., Forte, B., Molter, U., Vrscay, E., Iterated Fuzzy Sets Systems: A new approach to the inverse for fractals and other sets (1992) J. of Math. Anal, and Appl., 171, pp. 79-100Cooperman, G., (1978) α-Condensing Maps and Dissipative Processes, , Ph. D. Thesis, Brown University, Providence, R. IDiamond, P., Chaos in iterated fuzzy systems (1994) J. of Mathematical Analysis and Applications, 184, pp. 472-484Diamond, P., Time Dependent Differential Inclusions, Cocycle Attractors and Fuzzy Differential Equations (1999) IEEE Trans. on Fuzzy Systems, 7, pp. 734-740Diamond, P., Kloeden, P., (1994) Metric Spaces of Fuzzy Sets: Theory and Applications, , World Scientific PubFriedmann, M., Ma, M., Kandel, A., Numerical solutions of fuzzy differential and integral equations (1999) Fuzzy Sets and Systems, 106, pp. 35-48Hale, J.K., Asymptotic Behavior of Dissipative Systems (1988) Math. Surveys and Monographs, 25. , American Mathematical Society, ProvidenceHüllermeier, E., An Approach to Modeling and Simulation of Uncertain Dynamical Systems (1997) J. Uncertainty, Fuzziness, Know Ledge-Bases Syst., 5, pp. 117-137Kloeden, P.E., Fuzzy dynamical systems (1982) Fuzzy Sets and Systems, 7, pp. 275-296Kloeden, P.E., Chaotic iterations of fuzzy sets (1991) Fuzzy Sets and Systems, 42, pp. 37-42Nguyen, H.T., A note on thé extension principle for fuzzy sets (1978) J. Math. Anal. Appl., 64, pp. 369-380Puri, M.L., Ralescu, D.A., Fuzzy Random Variables (1986) J. of Mathematical Analysis and Applications, 114, pp. 409-422Roman-Flores, H., Barros, L.C., Bassanezzi, R., A note on Zadeh's Extensions (2001) Fuzzy Sets and Systems, 117, pp. 327-331Roman-Flores, H., On the Compactness of E(X) (1998) Appl. Math. Lett., 11, pp. 13-17Zadeh, L.A., Fuzzy sets (1965) Inform. Control, 8, pp. 338-35

Angiotensin-converting enzyme inhibitor protects against cisplatin nephrotoxicity by modulating kinin B1 receptor expression and aminopeptidase P activity in mice

Cisplatin is a highly effective chemotherapeutic agent. However, its use is limited by nephrotoxicity. Enalapril is an angiotensin I-converting enzyme inhibitor used for the treatment of hypertension, mainly through the reduction of angiotensin II formation, but also through the increase of kinins half-life. Kinin B1 receptor is associated with inflammation and migration of immune cells into the injured tissue. We have previously shown that the deletion or blockage of kinin B1 and B2 receptors can attenuate cisplatin nephrotoxicity. In this study, we tested enalapril treatment as a tool to prevent cisplatin nephrotoxicity. Male C57Bl/6 mice were divided into 3 groups: control group; cisplatin (20 mg/kg i.p) group; and enalapril (1.5 mg;kg i.p) + cisplatin group. The animals were treated with a single dose of cisplatin and euthanized after 96 h. Enalapril was able to attenuate cisplatin-induced increase in creatinine and urea, and to reduce tubular injury and upregulation of apoptosis-related genes, as well as inflammatory cytokines in circulation and kidney. The upregulation of B1 receptor was blocked in enalapril + cisplatin group. Carboxypeptidase M expression, which generates B1 receptor agonists, is blunted by cisplatin + enalapril treatment. The activity of aminopeptidase P, a secondary key enzyme able to degrade kinins, is restored by enalapril treatment. These findings were confirmed in mouse renal epithelial tubular cells, in which enalaprilat (5 μM) was capable of decreasing tubular injury and inflammatory markers. We treated mouse renal epithelial tubular cells with cisplatin (100 μM), cisplatin+enalaprilat and cisplatin+enalaprilat+apstatin (10 μM). The results showed that cisplatin alone decreases cell viability, cisplatin plus enalaprilat is able to restore cell viability, and cisplatin plus enalaprilat and apstatin decreases cell viability. In the present study, we demonstrated that enalapril prevents cisplatin nephrotoxicity mainly by preventing the upregulation of B1 receptor and carboxypeptidase M and the increased concentrations of kinin peptides through aminopeptidase activity restoration