It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special funct...

A “scoring rule” is an assignment of a value to every tuple (of varying sizes). This paper is concerned with the issue of how to modify a scoring rule to apply to the case where weights are assigned to the importance of each argument. We give an explicit formula for incorporating weights that can be applied no matter what the underlying scoring rule is. The formula is surprisingly simple, in th...

The effects of cutting edge shapes on the phenomena of rubbing and ploughing in nano-abrasive machining were investigated. The shapes under investigation include flat, spherical and trapezoidal shapes. The tool with the flat end geometry shows a fast initiation of ploughing, because it has the largest surface area to engage more atoms. It doesn’t show rubbing phenomenon at the (initial) depth o...

The main aim of this paper is to give an axiomatic characterization of the interval Baker-Thompson rule.

A generalized trapezoid rule is derived. Various error bounds for this rule are established. © 2005 Elsevier Ltd. All rights reserved. MSC: 26D15; 65D30

An extension of the standard rationing model is introduced. Agents are not only identified by their respective claims on some amount of a scarce resource, but also by some exogenous ex-ante conditions (characteristics), different from claims (e.g., endowments, entitlements, wealth, obligations, assets). Inequalities in the ex-ante conditions induce compensations between agents which influence t...

We study the behavior of rules for the adjudication of conflicting claims when there are a large number of claimants with small claims. We model such situations by replicating some basic problem. We show that under replication, the random arrival rule (O’Neill, 1982) behaves like the proportional rule, the rule that is the most often recommended in this context. Also, under replication, the min...

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals I (1)[g] = ∫ b a g(x) x−t dx and I (2)[g] = ∫ b a g(x) (x−t)2 dx . These integrals are not defined in the regular sense; I (1)[g] is defined in the sense ofCauchy PrincipalValuewhile I (2)[g] is defined in the sense of Hadamard Finite Part. With h = (b − a)/n, n = 1, 2, . . ., and t = a+kh for...

This paper investigates existence and uniqueness results for the first order fuzzy integro-differential equations. Then numerical results and error bound based on the left rectangular quadrature rule, trapezoidal rule and a hybrid of them are obtained. Finally an example is given to illustrate the performance of the methods.

Multiplicative operations for fuzzy numbers raise several problems both from the theoretical and practical point of view in fuzzy arithmetic. The multiplication based on Zadeh's extension principle and its triangular and trapezoidal approximation is used in several recent works in applications in geology. Recently, new product-type operation are introduced and studied, as e.g. the cross product...