We consider a phase space stability error control for numerical simulation of dynamical systems. Standard adaptive algorithm used to solve the linear systems perform well during the finite time of integration with fixed initial condition and performs poorly in three areas. To overcome the difficulties faced the Phase Space Error control criterion was introduced. A new error control was introduced by R. Vigneswaran and Tony Humbries which is generalization of the error control first proposed by some other researchers. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In earlier, it was analyzed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. Some theoretical results are obtained and numerical results are given.

In this article we construct a fuzzy topology on a non-empty set X called mixed semi-pre fuzzy topology from two given fuzzy topological spaces on X with the help of fuzzy semi-prequasi-neighborhood of a fuzzy point.

Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, ????(??) denotes the set of all fuzzy matrices of order ?? over the fuzzy unit interval [0,1]. Inaddi tion (???? (??), ??) dis called as fuzzy ?? -normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy ?? -normed linear space. The set of all fuzzy points in ???? (??) is denoted by ??*(????(??)). For a fuzzy ?? -normed linear space (???? (??), ??), we have |??(????)?? -??(????)?? | = ??(????,????)??. Besides ?? is a continuous function on ???? (??). That is, if ?????? ? ???? as ?? ? 8 then ??(?????? )?? ? ??(????)?? as ?? ? 8, where ?????? is a sequence in (???? (??), ??). Hence, ?? is always bounded on ????(??). Next we introduce the following result: Let ?????? , ?????? ? ?? * (????(??)) with ?????? and ?????? converge to ???? and ???? respectively as ?? ? 8. Then ?????? + ?????? converge to ???? + ???? as ?? ? 8. Furthermore, we are able to compare two different fuzzy ?? -norms with convergent sequence. The result states that for a fuzzy ?? -normed linear space (????(??), ??), we have ??(????)??1 = ????(???? )??2 , for some ?? > 0 and ???? ? ?? * (????(??)). If ?????? converges to ???? under fuzzy ??1 -norm then ?????? converges to ???? under fuzzy ??2 -norm. Moreover, if (???? (??), ??) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy ?? -normed linear space and its properties.

In this paper, we introduce the concept “Quotient bi-space” in bitopological spaces. In addition, we investigate the results related with quotient bi-space. Moreover, we have discussed the results related with pairwise regular and normal spaces in bitopological space. For a non-empty set X, we can define two topologies (these may be same or distinct topologies) t1 and t2 on X. Then, the triple (X, t1 , t2 ) is known as bitopological space. Let (X, t1 , t2 ) be bitopological space, (Y, s1 , s2 ) be trivial bitopological space and f : (X, t1 , t2 ) ? (Y, s1 , s2 ) be onto map. Then f is t1 t2 -continuous map. If ? = {G (s - open set in Y ) : f ^{-1} (G) is t1 t2 - open in X} then ? is a topology on Y . Moreover, if (Y, s, s) be a quotient bi-space of (X, t1 , t2) under f : (X, t1 , t2 ) ? (Y, s, s) and g : (Y, s, s) ? (Z, ?1 , ?2 ) be a map, then, gis s - continuous if and only if g ? f : (X, t1 , t2 ) ? (Z, ?1 , ?2 ) is t1 t2 -continuous. Let (X, t1 , t2) be bitopological space and A be t1 t2 - compact subset of pairwise Hausdorff space X. Then, A is t1 t2 - closed set. Finally, we have discussed the following : Let (X, t1 , t2 ) be bitopological space and t1 t2 -compact pairwise Hausdorff space. Then, the space (X, t1 , t2 ) is pairwise normal.

The cut-throat competition to be the best individual is coming out to be social issue in the hi-tech world. Society increases this pressure by imposing their expectation. Both these issues result into imbalance situations which results stress and depression in an individual. The life style adds more depression. The relation between depression due to social pressure and get relief from this needs medication is modelled in this research. The formulation is done with the help of system of nonlinear differential equations. The fraction (threshold) of individuals suffering from depression who needs medication is computed. The stress-free equilibrium and its stability are analysed.