By J. F. Adams, G. C. Shepherd

This set of notes, for graduate scholars who're focusing on algebraic topology, adopts a singular method of the instructing of the topic. It starts with a survey of the main invaluable components for examine, with innovations concerning the most sensible written bills of every subject. simply because the various resources are particularly inaccessible to scholars, the second one a part of the publication includes a set of a few of those vintage expositions, from journals, lecture notes, theses and convention lawsuits. they're attached via brief explanatory passages written by means of Professor Adams, whose personal contributions to this department of arithmetic are represented within the reprinted articles.

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4. Conclusion In this paper we have studied long-tailed distribution of biological systems. We have found that the lognormal distribution is characteristic of the statistics of the duration distribution of life for disability. We believe that the main lesson from the argument of the duration distribution for disability in aged people is that any hospital for the aged must be a polyclinic which can cope with various diseases simultaneously in a coordinated fashion, because diseases in aged people usually proceed multiplicatively.

2. Long-tailed Distribution of Duration of Disability for Aged People Disability and the resulting lowered quality of life are serious problems accompanying increased longevity. Curiously, despite its potential contribution to aging theory,12 complete statistical and etiological structures of this common and unwelcome aging phenotype before death have not been well studied. The consequences of aged people’s diseases are not only very December 26, 2006 16:0 WSPC - Proceedings Trim Size: 9in x 6in 060421DRAFT4 42 important to the aged people themselves but also young people as well because of the ﬁnancial and social problems such as payment of various kinds of pensions and nursing of the aged.

If r 1−r λ0 < 1, the probability that the disease propagates from node i to j through a path whose length is l decreases exponentially. This discussion r λ0 < 1 the disease will be eradicated. Therefore we suggests that if 1−r obtain the result that Kc > ln 1 + λ10 . 5 Another important application of our idea is the estimation of the “importance” of each node. In the SIS model on inhomogeneous networks, we can prevent the spread of disease by immunizing “important” nodes. 6 Such a targeted immunization works well for some network models, however, it is not versatile.