The target of this paper is to examine the average–per–observation information matrix for the truncated cosinor model. We state and prove an orthogonal decomposition of this matrix so that the total information can be obtained as a result of three particular parts. The total information is presented piece–wise in three components. Each component is easily represented. Therefore the total information can be checked through three different points on the collection of the information.

The convolution sum m<n=8 _3(m)_3(n - 8m) is evaluated for all n 2 N. This evaluation is used to determine the formulae of the convolution sums etc., and the number of representations of n through the sum of eight triangular numbers.

Aims: This paper presents an analysis and performance evaluation of the position control of an AC motor. Study Design: Mathematical model which represents the position control of an AC motor is used in this Paper, where is obtained from experimental data of the AC motor. Place and Duration of Study: Computer and software engineering department, collage of engineering, Diyala University-Iraq between October 2011 and June 2012. Methodology: Different types of controllers are used to analyze and evaluate the performance of the position control of AC motor. Classical Proportional- Integral-Derivative CPID, Fuzzy-like Proportional-Derivative (FPD), Fuzzy-like Proportional-Integral (FPI) and Fuzzy-like Proportional- Integral-Derivative (FPID) are used for the purpose of this paper. Results: For classical controllers, the rising time was decreased with most types, classical Ziegler-Nichols is used to tune the CPID parameters. The effective values of Kp are from 4.2 to 14.4, where the overshoot is increased from 0.2 to 0.5 and the steady state error is decreased from 2.5 to 2. The response performance of the system is improved with the fuzzy controllers’ types. FPD eliminates the overshoot with small change on the error where the FPD with three Membership Function (3FPD) versions I is 0.005 and with 3FPD version II is zero overshoot, while the steady state error is zero in both versions of 3FPD. Fuzzy PD with 5 MSF (5FPD) has zero overshoot with steady state error equal to 0.005 and faster system response. Conclusion: Fuzzy PD with 8 MSF (8FPD) provides system response with zero overshoot versus big error bigger than 0.5 and slower system response due to the huge number of rules where was 64 fuzzy rules. Fuzzy PI with 5 MSF (5FPI) eliminates the error but with big overshoot equal to 0.2. FPID got the better response performance of the control system but slower than FPD.

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph(digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we consider the eccentric digraphs of different products of graphs, viz., cartesian, normal, lexicographic, prism, etc.

In this paper we define a new multiplication on Banach algebra A using commute idempotent endomorphisms of A. Then we consider types of amenability and contractibility of A with this new multiplication. We will show that this new Banach algebra has better amenable properties than Banach algebra A.

In this paper, by using the probability density function we introduce the mild solution of fractional differential equations with impulsive conditions and obtain various criteria on the existence of mild solutions by using fixed point theorem.

Although not as efficient as simple random sampling, cluster sampling has been regarded as a valid sampling technique when the researcher is attempting to save cost. In order to do so, it is necessary that random selection occurs in all stages of sampling. This simulation study examines purposeful selection of cluster sampling in the second stage of a two stage cluster design.

The Iterative Weighted Least Squares (IWLS) method is one of the estimation procedures in logistic regression modeling. In consideration of the strategic role played by this model, especially in biometrics, the need to source for alternative logistic regression estimators has continued to resurface in the literature. In this paper, a modified IWLS method is developed by exponentiating the response probability. As a consequence, both the weight function and the adjusted dependent variate are modified. The resulting estimator is compared with the existing IWLS estimator using variances of parameter estimates and confidence intervals of the estimates. Four subpopulations were used in three illustrative examples with gender, Fasting Blood Sugar and Body Mass Index as explanatory variables. It is shown in this paper that the new update is superior to the traditional IWLS update in terms of variance reduction of parameter estimates and for the fact that it provides a more strict confidence interval for the test of hypotheses. In the first example, the confidence intervals for the parameter estimates of the existing IWLS scheme are (0.2183, 1.7197), (-1.9993, -0.1687), (1.7993, 0.0313), while those of the proposed method are (-2.6588, -2.2240), (1.1228, 1.6826), (0.8552, 1.4150). The proposed estimator allows for improved goodness-of-fit. By a careful formulation of the model, the proposed estimator is made to behave like a survival function in the sense that it can be used to model the probability of extra survival time in survival analysis.

We calculate some special convolution sums related to the odd divisors multiplied by the even divisors. Also according to the Ramanujan’s functions implying σ1(n), σ3(n), and σ5(n), we obtain various convolution formulas.

This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compare favorably with the existing methods.