ENGINEERING - Cambridge University Press India
ENGINEERING - Cambridge University Press India
ENGINEERING - Cambridge University Press India
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All the<br />
Mathematics You<br />
Missed<br />
But Need to Know for<br />
Graduate School<br />
Thomas A. Garrity<br />
Williams College,<br />
Massachusetts<br />
Algebraic Topology<br />
Allen Hatcher<br />
Cornell <strong>University</strong>,<br />
New York<br />
25. Lattices and Mobius inversion;<br />
26. Combinatorial designs and projective<br />
geometries; 27. Difference sets and<br />
automorphisms; 28. Difference sets and the group<br />
ring; 29. Codes and symmetric designs;<br />
30. Association schemes; 31. Algebraic graph<br />
theory: eigenvalue techniques; 32. Graphs:<br />
planarity and duality; 33. Graphs: colorings and<br />
embeddings; 34. Electrical networks and squared<br />
squares; 35. Polya theory of counting;<br />
36. Baranyai’s theorem; Appendices; Name index;<br />
Subject index.<br />
ISBN: 9780521718172 616pp ` 545.00<br />
This book will help students to see the broad<br />
outline of mathematics and to fill in the gaps in<br />
their knowledge. The author explains the basic<br />
points and a few key results of all the most<br />
important undergraduate topics in mathematics,<br />
emphasizing the intuitions behind the subject. The<br />
topics include linear algebra, vector calculus,<br />
differential geometry, real analysis, point-set<br />
topology, probability, complex analysis, abstract<br />
algebra, and more. An annotated bibliography<br />
then offers a guide to further reading and to more<br />
rigorous foundations. The book will be an essential<br />
resource for advanced undergraduate and<br />
beginning graduate students in mathematics, the<br />
physical sciences, engineering, computer science,<br />
statistics and economics who need to quickly learn<br />
some serious mathematics.<br />
Contents: 1. Linear algebra; 2. e and d real<br />
analysis; 3. Calculus for vector-valued functions;<br />
4. Point set topology; 5. Classical Stokes’<br />
theorems; 6. Differential forms and Stokes’<br />
theorem; 7. Curvature for curves and surfaces;<br />
8. Geometry; 9. Complex analysis;<br />
10. Countability and the axiom of choice;<br />
11. Algebra; 12. Lebesgue integration; 13. Fourier<br />
analysis; 14. Differential equations;<br />
15. Combinatorics and probability theory;<br />
16. Algorithms; A. Equivalence relations.<br />
ISBN: 9780521670340 374pp ` 445.00<br />
This introductory textbook is suitable for use in a<br />
first-year graduate course or for self-study,<br />
featuring broad coverage of the subject and a<br />
readable exposition, with many examples and<br />
exercises. Along with the basic material on<br />
fundamental group and covering spaces,<br />
homology and cohomology, higher homotopy<br />
groups, and homotopy theory, the book includes<br />
many optional topics for which elementary<br />
expositions are hard to find.<br />
Contents: Part I. Some Underlying Geometric<br />
Notions: 1. Homotopy and homotopy type;<br />
2. Deformation retractions; 3. Homotopy of maps;<br />
4. Homotopy equivalent spaces; 5. Contractible<br />
spaces; 6. Cell complexes definitions and<br />
examples; 7. Subcomplexes; 8. Some basic<br />
constructions; 9. Two criteria for homotopy<br />
equivalence; 10. The homotopy extension<br />
property; Part II. Fundamental Group and<br />
Covering Spaces: 11. The fundamental group,<br />
paths and homotopy;12. The fundamental group<br />
of the circle; 13. Induced homomorphisms;<br />
14. Van Kampen’s theorem of free products of<br />
groups; 15. The van Kampen theorem;<br />
16. Applications to cell complexes; 17. Covering<br />
spaces lifting properties; 18. The classification of<br />
covering spaces; 19. Deck transformations and<br />
group actions; 20. Additional topics: graphs and<br />
free groups; 21. K(G,1) spaces; 22. Graphs of<br />
groups; Part III. Homology: 23. Simplicial and<br />
singular homology delta-complexes; 24. Simplicial<br />
homology; 25. Singular homology; 26. Homotopy<br />
invariance; 27. Exact sequences and excision;<br />
28. The equivalence of simplicial and singular<br />
homology; 29. Computations and applications<br />
degree; 30. Cellular homology; 31. Euler<br />
characteristic; 32. Split exact sequences;<br />
33. Mayor-Vietoris sequences; 34. Homology with<br />
coefficients; 35. The formal viewpoint axioms for<br />
homology; 36. Categories and functors;<br />
37. Additional topics homology and fundamental<br />
group; 38. Classical applications; 39. Simplicial<br />
approximation and the Lefschetz fixed point<br />
theorem; Part IV. Cohomology: 40. Cohomology<br />
groups: the universal coefficient theorem;<br />
41. Cohomology of spaces; 42. Cup product the<br />
cohomology ring; 43. External cup product;<br />
44. Poincare duality orientations; 45. Cup product;<br />
46. Cup product and duality; 47. Other fortes of<br />
duality; 48. Additional topics the universal<br />
coefficient theorem for homology; 49. The Kunneth<br />
formula; 50. H-spaces and Hopf algebras; 51. The<br />
cohomology of SO(n); 52. Bockstein<br />
homomorphisms; 53. Limits; 54. More about ext;<br />
55. Transfer homomorphisms; 56. Local<br />
coefficients; Part V Homotopy Theory:<br />
57. Homotopy groups; 58. The long exact<br />
sequence; 59. Whitehead’s theorem; 60. The<br />
Hurewicz theorem; 61. Eilenberg-MacLane<br />
spaces; 62. Homotopy properties of CW<br />
complexes cellular approximation; 63. Cellular<br />
models; 64. Excision for homotopy groups;<br />
65. Stable homotopy groups; 66. Fibrations the<br />
homotopy lilting property; 67. Fiber bundles;<br />
68. Path fibrations and loopspaces; 69. Postnikov<br />
towers; 70. Obstruction theory; 71. Additional<br />
topics: basepoints and homotopy; 72. The Hopf<br />
invariant; 73. Minimal cell structures;<br />
74. Cohomology of fiber bundles; 75. Cohomology<br />
theories and omega-spectra; 76. Spectra and<br />
homology theories; 77. Eckmann-Hilton duality;<br />
78. Stable splittings of spaces; 79. The loopspace<br />
of a suspension; 80. Symmetric products and the<br />
Dold-Thorn theorem; 81. Steenrod squares and<br />
powers; Appendix: topology of cell complexes; The<br />
compact-open topology.<br />
ISBN: 9780521541862 500pp ` 495.00<br />
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