الفهرس 
SummaryOnly 14 pages are availabe for public view
 |  | from | 290 |  |  |
| | | from | 290 | | |
Abstract
This thesis consists of four chapters. Chapter one (Fundamental Concepts and Auxiliary Results), contains a brief review of some fundamental concepts and auxiliary results which are used in the later chapters. For example, we give the definitions of floor, ceiling functions, Chebyshev polynomials of first, second, third and forth types, some auxiliary properties of general matrices. The linear difference equations of the first two orders are briefly reviewed. Some fundamentals of Markov chains are also introduced. In Chapter two, (Power of a Matrix and Some Methods of Matrix Inversion), we have established formulae for computing the -step transition probability matrix of finite Markov chain using algebra and stochastic processes . This chapter consists of three sections. In the first section, three methods for computing the th power of a matrix are introduced. In the first method has proved that any positive integer power of such a matrix can be expressed as a linear combination of such matrix with its eigenvalues and the identity matrix. In the second method, the th power of the matrix is expressed directly in terms of the trace and the determinant of the matrix. In the third one, the -th power of the matrix is expressed directly in terms of the entries of the matrix. Algorithms are introduced for each method in this section and applied in the end of this thesis. In the second section formulae for the high-order transition probability matrix of finite Markov chain are established for the following two cases: (1) the two state transition probability matrix and (2) the step transition probability matrix. Some mistakes were found in Li, Z. and Wang, W., ”Computer aided solving the higher-order transition probability matrix of the finite Markov chain”, Appl. Math. Comput. 172 (2006) 267--285, during studying the power of high order transition probability matrix. Here, we introduce their study with corrections. In the third section, six different methods for inverting square matrix and an algorithm in the tridiagonal case corresponding to each method are offered. In Chapter three, (Periodic Tridiagonal Matrix Inversion), analytical forms for the inversion of general periodic tridiagonal matrices are presented, and for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices, and some of them are given in terms of chebyshev polynomials, and also in case that the elements of the matrix are transition probabilities. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form are given. It is suited for implementation using Computer algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. Examples are given to illustrate the algorithm. Also Some special cases are introduced. In Chapter four, (Applications for Finite Markov Chains), we present examples of the applications of finite Markov chains which span applications from to planning. In the first example (Mathematical Problem Solving Process), we study the mathematical problem solving process, in this part the level of students’ mathematical problem-solving ability, the ’behavior’ of the group of solvers and also teachers ’variability’ are determined. The second example (Copolymerization), studies copolymerization in chemistry. We start with binary copolymerization and ternary copolymerization and then component copolymerization. In each case the mole fraction of the monomers in the copolymer, and the portion of monomer units in the macromolecules in a large number of steps in growth of the latter are studied. In the third example (A Human Resource Planning and Valuation Model), the periods that an employment will stay in the different positions in a firm are studied . In the fourth example (A Markovian Approach to Determining Optimum Process Target Levels for a Multi-Stage Serial Production System), the expected profit per item for both single-stage and the two-stage systems are studied. In the fifth example (Predicting Duration of Stay in a Pediatric Intensive Care Unit), we study the periods of staying in a pediatric intensive care unit (PICU) as the patient’s health demand