Narrow Results

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal ℙ²-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal ℙ²-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and ℙ²-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3-manifolds to be formed.

The following sections are included:

• INTRODUCTION

• DATA COLLECTION
  • Primary and Secondary Sources of Data

• DATA PRESENTATION: TABLES
  • Annual CPI, T -Bill Rate, and Prime Rate

• DATA PRESENTATION: CHARTS AND GRAPHS

• APPLICATIONS
  • Analysis of Stock Rates of Return and Market Rates of Return
  • Financial Health of the Banking Industry: 1981 Versus 1989
  • How Congress Voted on the Gulf Resolution
  • Bar Charts Reveal How Several Economic Indicators Are Related

• Summary

• Appendix 2A Using Microsoft Excel to Draw Graphs

• Appendix 2B Stock Rates of Return and Market Rates of Return

• Appendix 2C Financial Statements and Financial Ratio Analysis
  • Review of Balance Sheets and Income Statements
  • Financial Ratio Analysis

• Questions and Problems

The following sections are included:

• INTRODUCTION

• SAMPLING FROM A POPULATION
  • Sampling Error and Nonsampling Error
  • Selection of a Random Sample

• SAMPLING COST VERSUS SAMPLING ERROR
  • Sample Size and Accuracy
  • Time Constraints

• SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
  • All Possible Random Samples and Their Mean
  • Mean and Variance for a Sample Mean
  • Sample Without Replacement from a Finite Sample

• SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTION
  • Calculating the Probability of Defective Chips

• THE CENTRAL LIMIT THEOREM
  • Illustrating the Central Limit Theorem

• OTHER BUSINESS APPLICATIONS
  • Audit Sampling
  • Patient Waiting Time⁸

• Summary

• Appendix 8A Sampling Distribution from a Uniform Population Distribution

• Questions and Problems

Population study is a fundamental unit in social science. Demography concerns itself with the dynamics of population behavior, primarily with respect to the size and structure, as well as movement. China is the largest country in the world, with an immense population of more than 1.3 billion people today, which constitutes almost one-fifth of the world's total. China has a huge population, but a poor foundation with relatively inadequate and unevenly distributed natural resources. These are China's basic national conditions. Population changes may have profound consequences for a wide range of areas in sociology, demography, social policy and others. Insight into these changes is of great value for many purposes, and will benefit researchers and policy makers in different fields. This chapter deals mainly with questions of Chinese population, in particular, the size, characteristics, growth, and the factors affecting the changes in population.