Applied Mathematical Sciences

I Application of Classical Topology to Graphs and Networks.- I. A Resume of Linear Algebra.- 1. Matrices.- 2. Vector and Vector Spaces.- 3. Column Vectors and Row Vectors.- 4. Application to Linear Equations.- II. Duality in Vector Spaces.- 1. General Remarks on Duality.- 2. Questions of Nomenclature.- 3. Linear Functions on Vector Spaces. Multiplication.- 4. Linear Transformations. Duality.- 5. Vector Space Sequence of Walter Mayer.- III. Topological Preliminaries.- 1. First Intuitive Notions of Topology.- 2. Affine and Euclidean Spaces.- 3. Continuity, Mapping, Homeomorphism.- 4. General Sets and Their Combinations.- 5. Some Important Subsets of a Space.- 6. Connectedness.- 7. Theorem of Jordan-Schoenflies.- IV. Graphs. Geometric Structure.- 1. Structure of Graphs.- 2. Subdivision. Characteristic Betti Number.- V. Graph Algebra.- 1. Preliminaries.- 2. Dimensional Calculations.- 3. Space Duality. Co-theory.- VI. Electrical Networks.- 1. Kirchoff's Laws.- 2. Different Types of Elements in the Branches.- 3. A Structural Property.- 4. Differential Equations of an Electrical Network.- VII. Complexes.- 1. Complexes.- 2. Subdivision.- 3. Complex Algebra.- 4. Subdivision Invariance.- VIII. Surfaces.- 1. Definition of Surfaces.- 2. Orientable and Nonorientable Surfaces.- 3. Cuts.- 4. A Property of the Sphere.- 5. Reduction of Orientable Surfaces to a Normal Form.- 6. Reduction of Nonorientable Surfaces to a Normal Form.- 7. Duality in Surfaces.- IX. Planar Graphs.- 1. Preliminaries.- 2. Statement and Solution of the Spherical Graph Problem.- 3. Generalization.- 4. Direct Characterization of Planar Graphs by Kuratowski.- 5. Reciprocal Networks.- 6. Duality of Electrical Networks.- II The Picard-Lefschetz Theory and Feynman Integrals.- I. Topological and Algebraic Considerations.- 1. Complex Analytic and Projective Spaces.- 2. Application to Complex Projective n-space Pn.- 3. Algebraic Varieties.- 4. A Resume of Standard Notions of Algebraic Topology.- 5. Homotopy. Simplicial Mappings.- 6. Singular Theory.- 7. The Poincare Group of Paths.- 8. Intersection Properties for Orientable M2n Complex.- 9. Real Manifolds.- II. The Picard-Lefschetz Theory.- 1. Genesis of the Problem.- 2. Method.- 3. Construction of the Lacets of Surface ?z.- 4. Cycles of ?z. Variations of Integrals Taken On ?z.- 5. An Alternate Proof of the Picard-Lefschetz Theorem.- 6. The ?1-manifold M. Its Cycles and Their Relation to Variations.- III. Extension to Higher Varieties.- 1. . Preliminary Remarks.- 2. First Application.- 3. Extension to Multiple Integrals.- 4. The 2-Cycles of an Algebraic Surface.- IV. Feynman Integrals.- 1. On Graphs.- 2. Algebraic Properties.- 3. Feynman Graphs.- 4. Feynman Integrals.- 5. Singularities.- 6. Polar Loci.- 7. More General Singularities.- 8. On the Loop-Complex.- 9. Some Complements.- 10. Examples.- 11. Calculation of an Integral.- 12. A Final Observation.- V. Feynman Integrals. B.- 1. Introduction.- 2. General Theory.- 3. Relative Theory.- 4. Application to Graphs.- 5. On Certain Transformations.- Subject Index Part I.- Subject Index Part II.