By John B. Conway

This textbook in aspect set topology is aimed toward an upper-undergraduate viewers. Its mild speed should be worthwhile to scholars who're nonetheless studying to put in writing proofs. must haves contain calculus and at the very least one semester of study, the place the scholar has been correctly uncovered to the information of simple set conception comparable to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the actual line. Appendices are incorporated to bridge the distance among this new fabric and fabric present in an research path. Metric areas are one of many extra customary topological areas utilized in different parts and are as a result brought within the first bankruptcy and emphasised in the course of the textual content. This additionally conforms to the procedure of the publication to begin with the actual and paintings towards the extra basic. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of notion, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued capabilities, culminating in a improvement of paracompact areas.

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Grimmett, R. B. Schinazi. Dependent random graphs and spatial epidemics. Ann. Appl. Prob. 8(2), 317–336 (1998). December 26, 2006 15:57 WSPC - Proceedings Trim Size: 9in x 6in hr 18 HOMOTOPY REDUCTION OF COMPLEX NETWORKS Y. jp T. jp In this paper, the notion of homotopy reductions is presented to derive a skeleton structure focusing on loops in complex networks. The main result is to show scale-free networks as skeleton structures of random networks under some rules. In addition, homotopy reductions are applied to scale-free real networks(WWW and protein interaction networks1 ).

Watts and S. Strogatz, Nature 393, 440 (1998). 6. A. Barabasi and R. Albert, Science 286, 509 (1999). 7. J. Bessen and E. htm#ip2. 8. J. Lahtinen, J. Kert´esz and K. Kaski, Physica A 349, 535 (2004). December 26, 2006 16:0 WSPC - Proceedings Trim Size: 9in x 6in 060421DRAFT4 40 LONG-TAILED DISTRIBUTIONS IN BIOLOGICAL SYSTEMS: REVISIT TO LOGNORMALS∗ N. KOBAYASHI∗ and K. jp O. MORIYAMA, Y. SASAKI and M. jp S. MATSUSHITA Tokyo Metropolitan Geriatric Hospital, Sakae-cho, Itabashi-ku, Tokyo, 173-0015, Japan Long-tailed distributions in biological systems have been studied.

E. J. Newman. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002). 17. M. E. J. Newman. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003). 18. R. Pastor-Satorras, A. Vespignani. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001). 19. R. Pastor-Satorras, A. Vespignani. Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63, 066117 (2001). 20. R. B. Schinazi, J. Stat. Phys. 97(1/2), 409–417 (1999). December 26, 2006 16:5 WSPC - Proceedings Trim Size: 9in x 6in ws˙cps 17 7 21.