RANDOM GRAPHS APPLIED TO THE IMMUNE NETWORK.
Item
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Title
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RANDOM GRAPHS APPLIED TO THE IMMUNE NETWORK.
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Identifier
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AAI8302558
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identifier
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8302558
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Creator
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PERLIN, MARK WILLIAM.
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Contributor
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Stanley Kaplan
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Date
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1982
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Chapter I treats the classical pre-1970 clonal selection theory of immune memory. An explanation of the genetic restrictions on the immune response is given. We develop some estimates of clonal connectivity which are used in later chapters.;In Chapter II we discuss Jerne's network theory of the immune system. We look at the Boolean matrix products of large 0-1 matrices. A variance estimate shows us how changes in the probability p of a matrix entry equalling 1 produce changes in the immune network's memory. We also apply some results of Erdos and Renyi to the immune network.;In Chapter III, we look at the specificity labelling, or rank, problem as it originally arose in immunogenetics. We develop some probabilistic formulas for this lattice embedding problem which we use in Chapter IV. In the final section we extend some of these ideas to matrices with entries in an arbitrary lattice, not just {lcub}0,1{rcub}. We show that such an extension is possible if and only if the lattice is a chain lattice.;We introduce a probabilistic version of the rank lattice embedding problem in Chapter IV. In particular, we conjecture that the probabilistic rank function rises from a value of 0 at p = 0 to a maximum of rank n at p = log n/n, and then decreases linearly to the value 1 at p = 1. This is applied to immunology and discrete mathematics.;In Chapter V we discuss the population dynamics and stability of the interacting clones in a suppressive immune network. We use the May-Wigner stability theorem, a result about the eigenvalues of random matrices. This stability theorem is used to describe the aging of the immune system in an individual and to explain the evolution of the antigen receptor.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
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Program
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Mathematics
