Fuzzy system reliability analysis by the use of Tω (the weakest t-norm) on fuzzy number arithmetic operations

In general, the sup-min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability, where the reliability of each system component is represented by fuzzy numbers. It is well known that T_ω-based addition preserves the shape of L-R type fuzzy numbers. In this paper, we show T_ω-based multiplication also preserves the shape of L-R type fuzzy numbers. We then apply T_ω-based arithmetic operations to fuzzy system reliability analysis. In fact, we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for L-R type fuzzy system reliability, while others [Singer, Fuzzy Sets Syst. 34 (1990) 145; Cheng and Mon, Fuzzy Sets Syst. 56 (1993) 29; Chen, Fuzzy Sets Syst. 64 (1994) 31] have got the approximate solutions using sup-min convolution for evaluating fuzzy system reliability.     

On the definition of a Lachlan semilattice

We study the algorithmic properties of the semilattices introduced in 1972 by Lachlan in his work on recursively enumerable m-degrees, the so-called Lachlan semilattices. We show that in Lachlan’s definition the effectivity condition on the meet can be omitted in the sequence determining such a semilattice.     

Supervision of teachers based on adjusted arithmetic learning in special education

This article reports on 20 children's learning in arithmetic after teaching was adjusted to their conceptual development. The report covers periods from three months up to three terms in an ongoing intervention study of teachers and children in schools for the intellectually disabled and of remedial teaching in regular schools. The researcher classified each child's current counting scheme before and after each term. Recurrent supervision, aiming to facilitate the teachers’ modelling of their children's various conceptual levels and needs of learning, was conducted by the researcher. The teaching content in harmony with each child's ability was discussed with the teachers. This approach gives the teachers the opportunity to experience the children's own operational ways of solving problems. At the supervision meetings, the teachers theorized their practice together with the researcher, ending up with consistent models of the arithmetic of the child. So far, the children's and the teachers’ learning patterns are promising.     

On the number of registers needed to evaluate arithmetic expressions

The question of how many temporary storage registers are needed to evaluate compiled arithmetic and masking expressions is discussed. It is assumed that any combination of left-to-right, right-to-left, top-to-bottom, and bottom-to-top techniques may be used to evaluate an expression, but that no factoring or re-arranging of the expression may occur. On this basis, the maximum number of registers needed to evaluate nonparenthesized expressions isN+1, withN the number of dyadic operator precedence levels. For parenthesized expressions with a maximum ofK nested parenthetical subexpressions, the maximum number of registers needed is (K+1)N+1.     

A proof of Hamblin's algorithm for translation of arithmetic expressions from infix to postfix form

We show that a particular algorithm for the translation of arithmetic expressions in infix form into postfix form is correct. The method used is ad hoc, in contrast to other work in this field, and it is hoped that this in itself is of interest.Work carried out at the University of Leeds, England.     

AN ELEMENTARY SYSTEM AS AND ITS SEMI-COMPLETENESS AND DECIDABILITY

The author establishes an elementary system AS which contains functions +, ? and a constant 0 and then proves the semi-completeness and the decidability of AS, using the theory of systems of inequalities.     

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Statistical mechanics and the Gauss principle

In an essentially statistical approach to statistical mechanics, it is seen that the Gauss principle of the arithmetic mean may be taken as the starting point. The equations from which the subject can be built up are deduced from the Gauss principle of the arithmetic mean.     

Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions

A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is shown to be equivalent to an interval arithmetic global optimization algorithm which applies natural extension to the cord-slope form of Taylor's expansion. Several more efficient variants using other forms of interval arithmetic are explored. Extensions to rational functions are presented. Comparative computational experiences are reported.     

ORDINAL NUMBERS IN ARITHMETIC PROGRESSION

The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers (ß + δγ)γ<γ<>r, δ ≠ 0.