The Riemann sums and the trapezoidal sums of functions defined on a bounded closed interval are well known as approximate sums of the Riemann integrals of the functions. In this paper the author represents the convergence rates of the Riemann sums and the trapezoidal sums as some limits of their expanded error terms. Let [a, b] be a bounded closed interval. We take an n-division ∆ of [a, b] def...

In a framework with two parties, deterministic voter preferences and geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts a party can win, given an electoral outcome (the " optimal gerrymandering rule "). As a corollary, we obtain that no districting rule can satisfy our axioms and treat p...

The idea of extending univariate quadrature formulae to cubature formulae that hold for spaces of polyharmonic functions is employed to obtain in a new way bivariate trapezoidal cubature rules. The notion of univariate monospline is extended to functions of two variables in terms of a solution of Poisson’s equation. This approach allows us to characterize the error of the trapezoidal cubature f...

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic func...

Let p and q be two points in a simple polygon Π. This chapter provides two rubberband algorithms for computing a shortest path between p and q that is contained in Π. The two algorithms use previously known results on triangular or trapezoidal decompositions of simple polygons, and have either O (n) or O (n log n) time complexity (where the super-linear time complexity is only due to preprocess...

Received 17September 2013; accepted 5 October2013 L Baba Abstract. Finding shortest paths in graphs has been the area of interest for many researchers. Shortest paths are one of the fundamental and most widely used concepts in networks. In this paper the authors present an algorithm to find an Intuitionistic Fuzzy Shortest Path (IFSP) in a directed graph in which the cost of every edges are rep...

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Systems with Applications 36 (2009) 6833-6842) is incorrect. The main aim of this paper is to prop...

Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These researches can be grouped into two classes: the Euclidean distance class and the non-Euclidean distance class. Most approximations in the Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. In this paper, we ...

In this paper we introduce an approach for solving multiattribute decision-making problems in which there are several decision-makers who individually and independently elicit their preferences. The preferences of each decision-maker are imprecise and represented by an imprecise additive multiattribute utility function. We allow for incomplete information on the component utility functions and ...

The three connected buildings of the former Suffolk Manufacturing Company in Lowell, known as Mills Nos. 5, 6 and 8, were built in 1862 (No. 5) and 1880 (Nos. 6 and 8), at a time when the city of Lowell was one of the predominant centers of the textile industry in the United States. Located within the Locks and Canals National Register Historic District, this impressive complex of mill building...