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...

In some multi-attribute decision making (MADM) problems, various relationships among the decision attributes should be considered. This paper investigates the prioritization relationship of attributes in MADM with trapezoidal intuitionistic fuzzy numbers (TrIFNs). TrIFNs are a special intuitionistic fuzzy set on a real number set and have the better capability to model ill-known quantities. Fir...

The variance of fuzzy random variables often appears in fuzzy random programming problems. Based on the definition of the variance of a fuzzy random variable, this paper attempts to deduce several formulas for the variances of trapezoidal fuzzy random variables, in which the randomness is characterized by uniform distribution. Firstly, we give the moment formulas for trapezoidal fuzzy variables...

In this paper, we present a new prioritization method for information fusion under fuzzy environment. We propose a new measure to measure the similarity between generalized trapezoidal fuzzy numbers. Based on the proposed similarity measure, we develop an utility function of generalized trapezoidal fuzzy numbers. With the utility function of generalized trapezoidal fuzzy numbers, we propose a n...

This paper proposes the concepts of a neutrosophic number and a trapezoidal neutrosophic number (TNN), the basic operational relations of TNNs, and the score function of TNN. Then, we develop a trapezoidal neutrosophic weighted arithmetic averaging (TNWAA) operator and a trapezoidal neutrosophic weighted geometric averaging (TNWGA) operator to aggregate TNN information and investigate their pro...

Standard rank-revealing factorizations such as the singular value decomposition (SVD) and column pivoted QR factorization are challenging to implement efficiently on a GPU. A major difficulty in this regard is inability of standard algorithms cast most operations terms Level-3 BLAS. This article presents two alternative for computing form = U T V ∗ $$ \mathbf{\mathsf{A}}=\mathbf{\mathsf{UT}}{\m...

We show that the E +-class can be decided by a liftable order, when it is applied a posteriori. This is a surprising result, because the E +-class was one of the motivations for the study of non-liftable orders. Also surprising is the behaviour of the resolution process. Initially the variable depth of the clauses may grow, but it will not grow deeper than a certain bound. We do not make use of...