MATHEMATICAL RESEARCHES
Year:2018 | Volume:4 | Issue:1 |Papers Count:8

Introduction 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. ...

In this paper, we consider a membrane with shape Ω which is fixed at its boundary. Assume the membrane is made using two different materials and its mass is prescribed. We need that certain regions (called patches) in Ω be made out of both materials in a way that the basic frequency of the resulting membrane is either minimal or maximal. ....

Introduction 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. ...

Introduction 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. ...

Introduction 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. ...

Introduction 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. ...

Introduction 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. ...

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. ...