The ever-growing applications and richness of approaches to the Riordan group is captured in this comprehensive monograph, authored by those who are among the founders and foremost world experts in this field. The concept of a Riordan array has played a unifying role in enumerative combinatorics over the last three decades. The Riordan arrays and Riordan group is a new growth point in mathematics that is both being influenced by, and continuing its contributions to, other fields such as Lie groups, elliptic curves, orthogonal polynomials, spline functions, networks, sequences and series, Beal conjecture, Riemann hypothesis, to name several. In recent years the Riordan group has made links to quantum field theory and has become a useful tool for computer science and computational chemistry. We can look forward to discovering further applications to unexpected areas of research. Providing a baseline and springboard to further developments and study, this book may also serve as a text for anyone interested in discrete mathematics, including combinatorics, number theory, matrix theory, graph theory, and algebra.

The ever-growing applications and richness of approaches to the Riordan group is captured in this comprehensive monograph, authored by those who are among the founders and foremost world experts in this field.  The concept of a Riordan array has played a unifying role in enumerative combinatorics over the last three decades. The Riordan arrays and Riordan group is a new growth point in mathematics that is both being influenced by, and continuing its contributions to, other fields such as Lie groups, elliptic curves, orthogonal polynomials, spline functions, networks, sequences and series, Beal conjecture, Riemann hypothesis, to name several. In recent years the Riordan group has made links to quantum field theory and has become a useful tool for computer science and computational chemistry. We can look forward to discovering further applications to unexpected areas of research. Providing a baseline and springboard to further developments and study, this book may also serve as a text for anyone interested in discrete mathematics, including combinatorics, number theory, matrix theory, graph theory, and algebra.

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This book is adapted from Dr. Wang's dissertation research, part of which has been accepted by peer-reviewed journal "teacher and teaching education." The whole picture of the research was depicted in this book. This book is divided into five chapters. The background of the research is described in chapter I, in which, the readers will learn about the educationreform in China, particularly in Shanxi Province, and the concept andcontext of 2+2 program. The author offers the reader the opportunity to learn what have been studies previously in terms of teacher education and Teacher collaboration in Chapter II. The gaps in the present knowledge are discussed in this chapter. Chapter III is the core of the study, in which the reader will learn how the authors created the research questions and how the study was conducted. Chapter IV lays out all the findings followed by Chapter V with the detailed discussion on the meanings and potential implication of the findings.