نشریه پژوهشهای ریاضی
سال چهارم شماره 1 (پیاپی 8، بهار و تابستان 1397)

Longitudinal study designs are common in a lot of scientific researches, especially in medical, social and economic sciences. The reason is that longitudinal studies allow researchers to measure changes of each individual over time and often have higher statistical power than cross-sectional studies. Choosing an appropriate sample size is a crucial step in a successful study. A study with insufficient sample size may have a small statistical power to detect significant effects and may lead to incorrect answers to many important research questions. On the other hand, a study with more than what a sample size should be wastes the resources and budget. In longitudinal studies, because of the complexity of the design of experiment, including the selection of the number of individuals and the number of repeated measurements, the sample size determination is less studied. This paper uses a simulation-based method to determine the sample size from a Bayesian perspective. For this purpose several Bayesian criteria of sample size determination for a longitudinal study using marginal model are used. Most of the methods of determining the sample size are based on creation of one hypothesis. In this paper, in addition to using this method, we also present a method to determine the sample size for multiple hypothesis testing. Using several examples the proposed Bayesian methods are discussed. Material and method: Based on the Bayesian sample size determination approach proposed by Wang and Gelfand (2002), which is a simulation-based approach, two sets of priors are considered. The first set is called the “fitting” or “analysis” priors which are used for analyzing data. The other set of priors is called “design” or “sampling” sets which drawing upon expertise, we may speculate upon a variety of informative scenarios regarding the unknown parameters and capture each with a suitable sampling prior. In the first step of the simulation-based approach, one generates the parameters from the design priors then a sample from the generated parameters is obtained. Note that the frequentist/classical viewpoint of the sample size determination approach require a point estimate of the variance and the smallest meaningful difference (Diggle et al., 2002). This information is elicited based on pilot data or the opinion of experts. The use of design priors in the proposed approach of Wang and Gelfand (2002) replaces this part of the classical sample size determination approach. After generating data using the design priors in this stage, they are analyzed by the fitting priors. In this paper, non-informative priors are used as fitting priors, as we assume there is no prior information or any expert opinion to be used to construct a prior distribution. If there is any prior information to be used to elicit an appropriate prior distribution, one may use it in the step of fitting prior and for sure, if the specification of the prior is correct, will achieve a better result. Also, in this paper, four criteria are used for determining sample size in longitudinal studies: Bayesian power criterion (BPC), average length criterion (ALC), average posterior variance criterion (APVC) and average converge criterion (ACC).
The simulation-based approach of Wang and Gelfand (2002) can be summarized as follows: Specify the value of the effect of an important regression coefficient which is the interest parameter and specify the design and fitting priors.
For each sample size, the following steps are repeated M times: Generate values of the unknown parameters from their design priors.
Simulate the values of covariates from continuous or discrete distributions and the response variable from its distribution.
Analyze the generated data set of step (ii) using the fitting priors.
Calculate BPC.
Fit a curve or surface through Bayesian power values and find an adequate sample size for a desired power using interpolation. In this paper, the curve is fitted using a polynomial regression. In step 2(iv), one can calculate ACC, APVC, or ALC and determine the sample size based on these criteria.

In industrial designing and manufacturing, it is often required to generate a smooth function approximating a given set of data which preserves certain shape properties of the data such as positivity, monotonicity, or convexity, that is, a smooth shape preserving approximation.
It is assumed here that the data is sufficiently accurate to warrant interpolation, rather than least squares or other approximation methods. The shape preserving interpolation problem seeks a smooth curve/surface passing through a given set of data, in which we priorly know that there is a shape feature in it and one wishes the interpolant to inherit these features. One of the hidden features in a data set may be its boundedness. Therefore, we have a data set, which is bounded, and we already know that. This happens, for example, when the data comes from a sampling of a bounded function or they reflect the probability or efficiency of a process.
Scientists have proposed various shape-preserving interpolation methods and every approach has its own advantages and drawbacks. However, anyone confesses that splines play a crucial role in any shape-preserving technique and every approach to shape-preserving interpolation, more or less, uses splines as a cornerstone.
This study concerns an interpolation problem, which must preserve boundedness and needs a smooth representation of the data so the cubic Hermite splines are employed.

Fractional differential equations (FDEs) have attracted in the recent years a considerable interest due to their frequent appearance in various fields and their more accurate models of systems under consideration provided by fractional derivatives. For example, fractional derivatives have been used successfully to model frequency dependent damping behavior of many viscoelastic materials. They are also used in modeling of many chemical processed, mathematical biology and many other problems in engineering. The history and a comprehensive treatment of FDEs are provided by Podlubny and a review of some applications of FDEs are given by Mainardi.

In classical methods of statistics, the parameter of interest is estimated based on a random sample using natural estimators such as maximum likelihood or unbiased estimators (sample information). In practice, the researcher has a prior information about the parameter in the form of a point guess value. Information in the guess value is called as nonsample information. Thompson (1968) proposed a shrinkage estimator by combining sample and nonsample information which is a linear combination of a natural estimator and the guess value.
The best linear unbiased estimator and it's risk is computed under the considered loss function. Shrinkage testimators are proposed based on the acceptance or rejection of a null hypothesis of the form equality or guess value and the true value of the parameter and their risks are computed. The relative efficiency of shrinkage testimators and the best linear unbiased estimator is calculated for the comparison of them. In Bayesian approaches, a Bayes estimator is derived by employing a flexible prior distribution for the parameter of interest. A Bayesian shrinkage estimator is provided using a Bayesian shrinkage approach and its performance is compared with the best linear unbiased estimator via the relative efficiency of them. Finally, a numerical example is used for illustrating the results. Results and discusion: The results show that the shrinkage testimators have higher efficiency than the best linear estimator when the guess value is close to the true value of the parameter. Our findings show that the Bayes shrinkage estimator outperforms the best linear estimator if the prior point information about the value of the parameter is not too far from its true value. Moreover, the Bayes shrinkage estimator performs well with respect to the best linear estimator for guess value in the vicinity of true value. In this case, the Bayes shrinkage estimators with larger values of hyperparameters outperforms other estimators in neighborhood guess value, when other parameters held fixed. In this paper, some shrinkage testimators and a Bayes shrinkage estimator are proposed for the scale parameter of Rayleigh lifetime distribution based on censored samples under a scale invariant squared error loss function. The following conclusions were drawn from this research: - The shrinkage testimators are better than the best linear estimator for guess value in neighborhood of true value of parameter. Moreover, the first proposed shrinkage testimator have higher efficiency than other testimators for small sample size when the guess value is close to the true value of the parameter.
- The proposed Bayes shrinkage estimator performs well with respect to the best linear estimator for guess value in the vicinity of true value. Also, the Bayes shrinkage estimators with larger values of hyperparameters outperforms other estimators in neighborhood guess value, when other parameters held fixed.

This paper aims to introduce the original ideas of normal form theory and bifurcation analysis and control of small amplitude limit cycles in a non-technical terms so that it would be comprehensible to wide ranges of Persian speaking engineers and physicists. The history of normal form goes back to more than one hundreds ago, that is to the original ideas coming from Henry Poincare. This tool plays an important role in the bifurcation analysis of dynamical systems. Many phenomena in chemistry, physics, and engineering are modeled by parametric nonlinear differential systems .These systems demonstrate a complicated dynamics, when the parameters reach singular values. Normal form theory is an efficient tool for the local bifurcation analysis of singular nonlinear systems. The main idea relies in using a nonlinear change of coordinates to convert a given vector field into a simple form, which is named normal form.
In most nonlinear systems, behavior and dynamics is involved with local appearance or disappearance of small amplitude limit cycles. These are called bifurcations of limit cycles and they mostly occur through Hopf or degenerate Hopf bifurcations. Center manifolds and normal form theory are considered as among the most powerful tools for the bifurcations and stability analysis of limit cycles. Indeed, the analysis requires the computation of Lyapunov coefficients. Here we present the first (well-known) and the second Lyapunov coefficients for Hopf and degenerate bifurcations. Lyapunov coefficients: In this paper, we discuss normal form of systems whose linear part has a pair of imaginary eigenvalues (usually referred by Hopf singularity) and compute the first and second Lyapunov coefficients.
In order to use the bifurcation analysis of a Hopf singularity for bifurcation control purposes, one needs to be able to derive the parametric normal forms of the original system so that the bifurcation analysis for the parametric normal form system would give rise to a bifurcation analysis in terms of the parameters of the original parametric system. We have designed a Maple program for this purpose. Here we merely present the Lyapunov coefficients. The program computes Lyapunov coefficients for systems with parameters and symbolic coefficients.

Application: Computation of Lyapunov coefficients is not important only in theatrical aspects but also in real world applications. So in last section of the Persian language paper, we apply the method on a well-known system, called Linard system, and a non-linear electrical system in order to illustrate how these tools can be applied in applications.
We detect Hopf singularity for these two application models and then compute their normal form up to degree 3.

Many problems which appear in different sciences such as physics, engineering, biology, applied mathematics and different branches can be modeled by using deterministic integral equations. Weakly singular integral equation is one of the principle type of integral equations which was introduced by Abel for the first time. These problems are often dependent on a noise source which are neglected due to poor computational tools. So, it is satisfactory to use stochastic models to describe the behaviour of them. Thus, researchers added uncertainty term in the deterministic models and this leads to the stochastic models such as stochastic partial differential equations or stochastic integral equations. Since 1960, by increasing computational power, some random factors are inserted to deterministic integral equations and are created various kinds of stochastic integral equations such as Ito-Volterra integral equations, Ito-Fredholm integral equations, or weakly singular Ito-Volterra integral equations. In more cases, the analytical solution of these equations do not exist or finding their analytic solution is very difficult. Thus, presenting an accurate numerical method is an essential requirement in numerical analysis. Numerical solution of stochastic integral equations because the randomness has its own difficulties. In recent years, some different basis functions have been used to estimate the solution of stochastic integral equations. In this paper, we develop operational matrix method based on Euler polynomials to solve weakly singular Ito-Volterra integral equations. Euler polynomials have received considerable attention in dealing with various problems and equations. Material and methods: In this scheme, we first calculate operational matrix of integration and stochastic operational matrix of integration based on Euler polynomials and then by using these matrices, weakly singular Ito-Volterra integral equation is transformed to a system of algebraic equations which can be solved via a suitable numerical method. We solve some test examples by using present technique to demonstrate the efficiency, high accuracy and the simplicity of the present method, then compare the proposed method with block-pulse method. The reported results demonstrate that there is a good agreement between approximate solution and exact solution. Also, the numerical results reported in the tables indicate that the accuracy improve by increasing the N. Therefore, to get more accurate results, using the larger N is recommended. Note that, obtained results confirm that proposed method enables us to find some more reasonable approximate solutions than block-pulse method. The following conclusions were drawn from this research. Coefficients of the approximate function via Euler polynomials are found very easily and therefore many calculations are reduced.
Euler polynomials are simple basis functions, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems with little additional works.
The main characteristic of this method is that it reduces tht considered problem to a system of algebraic equations which can be easily solved by using direct method or iterative method.

According to the classic sampling theory, errors that are mainly considered in the estimations are sampling errors. However, most non-sampling errors are more effective than sampling errors in properties of estimators. This has been confirmed by researchers over the past two decades, especially in relation to non-response errors that are one of the most fundamental non-immolation errors. More done researches, had studied non-sampling errors individually, however, in practice, two or more non-sampling errors occur together. In this paper, are considered two types of non-sampling errors in the stage of estimation of population mean: measurement error and non-response error. Has been proposed a new estimator for estimating the mean population in the presence of measurement and non-response errors. It has been shown theoretically and experimentally that the proposed estimator is more efficient than the Hansen and Hurwitz's unbiased estimator and other existing estimators. Material and methods: Since we can have populations for which the correlation between the studied variable and the auxiliary variable can be positive or negative simultaneously, for the positive correlation, the ratio estimator, and for the negative correlation, we need an product estimator. Therefore, we need an estimator that can express both conditions. For such populations, it is suggested that the estimator combines the ratio exponentiol estimator and the product exponentiol estimator using the probability balance method. The effect of measurement and non-response errors on the population mean is simultaneously examined and has been proposed a class of new estimators for the estimation of the mean of the variable population studied, when the measurement and non-response errors occur in both variable studied, and the proposal . The proposed estimator is compared with some of the existing estimators both theoretically and empirically, and it has been shown that in either case the proposed estimator is more efficient than the other estimators. The following conclusions were drawn from this research. In each of the six communities, the proposed estimator is better than the other estimators.
proposed method is easy to implement and it is a powerful mathematical tool to obtain the new population mean estimators with little additional works.