In this note I examine conditions under which a probability relation on a set of events is lexicographic. Chipman (1971) discussed the question of the minimal ordinal CT such that there exists an order-preserving function from an ordered set A (in his; paper A is fR with any order on it) to lRa with the lexicographic order on it. Since there always exists an ordinal /? such that there is an ord...

This paper investigates the local convergence of the Lagrange{SQP{Newton method applied to an optimal control problem governed by a phase eld equation with distributed control. The phase eld equation is a system of two semilinear parabolic diierential equations. Stability analysis of optimization problems and regularity results for parabolic diierential equations are used to proof convergence o...

In this paper, we first construct the function space of (L, M)-fuzzy Q-convergence spaces to show Cartesian-closedness category M)-QC spaces. Secondly, introduce several subcategories M)-QC, including M)-KQC Kent spaces, M)-LQC Q-limit and M)-PQC pretopological investigate their relationships.

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...

We consider solutions of a system of reenement equations written in the form as = X 2Z a()(2 ?) where the vector of functions = (1 ; : : : ; r) T is in (L p (R)) r and a is a nitely supported sequence of rr matrices called the reenement mask. Associated with the mask a is a linear operator Q a deened on (L p (R)) r by Q a f := P 2Z a()f(2 ?). This paper is concerned with the convergence of the ...

An inequality between the number of coverings in the ordered set J(Con L) of join irreducible congruences on a lattice L and the size of L is given. Using this inequality it is shown that this ordered set can be computed in time O(n 2 log 2 n), where n = jLj. This paper is motivated by the problem of eeciently calculating and representing the congruence lattice Con L of a nite lattice L. Of cou...

The monotone circuit problem Q MC is shown to be complete for xed point logic IFP under quantiier free reductions. Enhancing the circuits with an oracle Q leads to a problem complete for IFP(Q). By contrast, if L is any extension of FO with generalized quantiiers, one can always nd a Q such that IFP(Q) is not contained in L(Q). For L = FO(Q MC), we have L IFP but L(Q) < IFP(Q). The adjunction o...

We prove the following theorem. Suppose that M is a trim DFA on the Boolean alphabet 0, 1. The language L(M) is well-ordered by the lex-icographic order < ℓ iff whenever the non sink states q, q.0 are in the same strong component, then q.1 is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a < ℓ-descending sequence of words....