Triangulations and Simplicial Methods

1 Introduction.- 2 Preliminaries.- 2.1 Notations.- 2.2 Fixed Point Theorems.- 2.3 Applications.- 2.3.1 A Pure Exchange Economy.- 2.3.2 A Finite n-Person Normal Form Game.- 2.3.3 An Exchange Economy with Linear Production Technologies.- 2.3.4 Convex Programming.- 2.3.5 A Balanced Game without Side Payments.- 3 Existing Triangulations.- 3.1 Existing Triangulations of Sn.- 3.2 Existing Triangulations of Rn.- 3.3 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Sn.- 3.4 Existing Triangulations of Continuous Refinement of Grid Sizes of (0,1) × Rn.- 4 The D1-Triangulation of Rn.- 4.1 The D1-Triangulation of Rn.- 4.2 Pivot Rules of the D1-Triangulation.- 4.3 The Number of Simplices of the D1-Triangulation in a Unit Cube.- 4.4 The Diameter of the D1-Triangulation.- 4.5 The Average Directional Density of the D1-Triangulation.- 5 The T1-Triangulation of the Unit Simplex.- 5.1 The T1-Triangulation.- 5.2 Pivot Rules of the T1-Triangulation.- 5.3 Comparison of the Triangulations of the Unit Simplex.- 6 The D1-Triangulation in Variable Dimension Algorithms on the Unit Simplex.- 6.1 The Dv1-Triangulation.- 6.2 Pivot Rules of the Dv1-Triangulation.- 6.3 The (n + 1)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 6.4 The (2n+1 - 2)-Ray Variable Dimension Method Based on the Dv1-Triangulation.- 7 The D1-Triangulation in Variable Dimension Algorithms on the Euclidean Space.- 7.1 The Dv2-Triangulation.- 7.2 Pivot Rules of the Dv2-Triangulation.- 7.3 The 2n-Ray Variable Dimension Method Based on the D1-Triangulation.- 7.4 The 2n-Ray Variable Dimension Algorithm Based on the Dv2-Triangulation.- 8 The D3-Triangulation for Simplicial Homotopy Algorithms.- 8.1 Definition of the D3-Triangulation.- 8.2 Construction of the D3-triangulation.- 8.3 Pivot Rules of the D3-Triangulation.- 8.4 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 9 The D2-Triangulation for Simplicial Homotopy Algorithms.- 9.1 Construction of the D2-Triangulation.- 9.2 Description of the D2-Triangulation.- 9.3 Pivot Rules of the D2-Triangulation.- 9.4 Description of the D2*-Triangulation.- 9.5 Pivot Rules of the D2*-Triangulation.- 9.6 Comparison of Several Triangulations for Simplicial Homotopy Algorithms.- 10 Conclusions.