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...

We propose a novel SAS macro to utilize PROC NLMIXED in SAS to estimate the standard error for AUC when the AUC estimator is based on the trapezoidal rule. Our method accounts for the trapezoidal region-by-region correlation structure. An example is provided for the standard error of the AUC under the ROC curve.

In this paper, a numerical method for integration of fuzzy functions is considered. Fuzzy Newton-Cotes formula, such as fuzzy trapezoidal method and fuzzy Simpson method are calculated by integration of fuzzy functions on two and three equally space points. Also the composite fuzzy trapezoidal and composite fuzzy Simpson method are proposed for n equally space points. The proposed method are il...

Q : dimensionless heat loss from the fin, q/(klw f φ ) Abstract Rt: dimensionless thermal resistance A reversed trapezoidal fin with fluid in the inside wall is analyzed using a two-dimensional analytical method. Thermal resistance for the reversed trapezoidal fin is presented as a function of the fin length and fin base height. The relationship between the inside fluid convection characteristi...

In a framework with two parties, deterministic voter preferences and a type of geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts one party can win, given the distribution of individual votes (the " optimal gerrymandering rule "). As a corollary, we obtain that no districting rule can sa...

By using geometry, a fairly complete analysis of Kemeny’s rule (KR) is obtained. It is shown that the Borda Count (BC) always ranks the KR winner above the KR loser, and, conversely, KR always ranks the BC winner above the BC loser. Such KR relationships fail to hold for other positional methods. The geometric reasons why KR enjoys remarkably consistent election rankings as candidates are added...

In this paper we study convolution operators Tμ with measures μ in R of the form μ (E) = ∫ B χE (x, φ (x)) dx, where B is the unit ball of R, and φ is a homogeneous polynomial function. If infh∈S1 ∣∣det (d2xφ (h, .))∣∣ vanishes only on a finite union of lines, we prove, under suitable hypothesis, that Tμ is bounded from L into L if ( 1 p , 1 q ) belongs to a certain explicitly described trapezo...

The rules of many sports are not fair—they do not ensure that equally skilled 3 competitors have the same probability of winning. As an example, the penalty shootout in soccer, 4 wherein a coin toss determines which team kicks first on all five penalty kicks, gives a substantial 5 advantage to the first-kicking team, both in theory and practice. We show that a so-called Catch6 Up Rule for deter...

If In is the approximation of a deenite integral I = Z b a fxdx with step b , a 2 n using the trapezoidal rule respectively Simpson's rule, if C a;b denotes the number of signiicant digits common to a and b, w e show, in this paper, that CI n ;