Volume 55 , issue 1$-$2 ( 2003 ) | back |

$r$-fuzzy strongly preopen sets in fuzzy topological spaces | 1$-$13 |

**Abstract**

We introduce $r$-fuzzy strongly preopen and $r$-fuzzy strongly preclosed sets in fuzzy topological spaces in the sense of the definition of Šostak~[8] and investigate some of their properties. Fuzzy strongly precontinuous, fuzzy strongly preopen and fuzzy strongly preclosed mappings between fuzzy topological spaces are defined. Their properties and the relationship between these mappings and other mappings introduced previously are investigated.

**Keywords:** $r$-fuzzy
strongly preopen, $r$-fuzzy strongly preclosed, fuzzy strong
precontinuity, fuzzy strongly preopen mapping, fuzzy strongly preclosed mapping.

**MSC:** 54A40

Spectral states of commutative l.m.c. algebras | 15$-$19 |

**Abstract**

We characterize the commutative locally multiplicative convex (l.m.c.) algebras in terms of the spectral states. We also give a characterization of spectral states in terms of commutative semisimple l.m.c. algebras. Further, with the help of radicals of l.m.c. algebras we give a necessary and a sufficient condition for an algebra to be commutative modulo its radical.

**Keywords:** Spectral states, probability
measure, l.m.c. algebra, commutative modulo.

**MSC:** 46J99, 46J15, 46K99

Optimality conditions and Toland's duality for a nonconvex minimization problem | 21$-$30 |

**Abstract**

This paper studies necessary and sufficinet conditions and provides a duality theory for a wide class of problems arising in nonconvex optimization, such as minimizing a difference of two convex functions subject to a convex vector constraint taking values in an ordered topological vector space. These results are then used to study a problem of nondifferentiable optimization.

**Keywords:** DC-minimization, optimality conditions,
Toland's duality, fractional programmation, partial order.

**MSC:** 49K27, 90C25

Generalized binomial law and regularly varying moments | 31$-$36 |

**Abstract**

In this paper we demonstrate a method for estimating asymptotic behavior of the regularly varying moments $E(K_\rho (X_n))$, $(n\toınfty)$ in the case of generalized Binomial Law. Here $K_\rho(x)$ is from the class of regularly varying functions in the sense of Karamata. We prove that $$ E(K_\rho(X_n))\sim K_\rho(E(X_n)), \ \rho>0, \ \ \ E(X_n)\toınfty \ \ \ (n\toınfty), $$ i.e., that the asymptotics of the first moment determines the behavior of all other moments.

**MSC:** 26A12

The lower and upper topologies as a bitopology | 37$-$52 |

**Abstract**

The importance of the theory of bitopological spaces is fully demonstrated by its natural relationship to the theory of ordered topological spaces. Using the parallels drawn by M. Canfell and T. McCallion between the theory of bitopological spaces and that of ordered topological spaces, we construct the dimension theory for ordered topological spaces and formulate and study the Baire-like properties of the latter spaces, thereby filling in the gap of the theory of ordered topological spaces. Further, based on these parallels, the relations between the separation axioms of ordered topological spaces and the corresponding bitopological spaces are established.

**Keywords:** $(l,u)$- and $(u,l)$-boundaries,
a hereditarily strong normally ordered space, $l$- and $u$-nowhere dense sets,
$l$- and $u$-first (second) category set, $l$- and $u$-Baire spaces.

**MSC:** 54F05, 54E55

Integral inequalities for maximal space-like submanifolds in the indefinite space form | 53$-$57 |

**Abstract**

In this note, we give two intrinsic integral inequalities for compact maximal space-like submanifolds in the indefinite space form and a sufficent and necessary condition for such submanifolds to be totally geodesic.

**Keywords:** Maximal space-like submanifold, indefinite space form,
flat normal bundle.

**MSC:** 53C40, 53C42, 53C50