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- Author or Editor: Gábor Tusnády x

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Summary DCT Given a finite set of points in an Euclidean space the \emph{spanning tree} is a tree of minimal length having the given points as vertices. The length of the tree is the sum of the distances of all connected point pairs of the tree. The *clustering tree with a given length* of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be *column monotone orderable* if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. *Decomposition clusters of monotone orderings* of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be *column partitionable* if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the *partition sum of squares of the matrix* is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. *Decomposition of the rows of a matrix into clusters of partitions* is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.

Current wisdom describes the immune system as a defense against microbial pathogens. It is claimed that the virgin immune system has a capacity to produce antibodies against the entire antigenic universe. We assume, by contrast, that the responding capacity of the immune system is limited. Thus it cannot stand in readiness to deal with a practi- cally endless diversity and abundance of microbes. Axioms and theorems are suggested for a mathematician audience delineating how the immune system could use its limited resources economically. It is suggested that the task of the immune system is twofold: (i) It sustains homeostasis to preserve the genome by constant surveillance of the intracellular antigenic milieu. This is achieved by standardization of the T cell repertoire through a positive selection. The driving force of positive selection is immune cell survival. T cells must constantly seek contact with complementary MHC structures to survive. Such contact is based on molecular complementarity between immune cell receptors and MHC/self-peptide complexes. At the highest level of complementarity a local free energy minimum is achieved, thus a homeostatic system is created. Homeostatic interactions happen at intermediate afinity and are reversible. Alteration in the presented peptides typically decreases complementarity. That pushes the system away from the free energy minimum, which activates T cells. Complementarity is restored when cytotoxic T cells destroy altered (mutated/infected) host cells. (ii) B cells carry out an immune response to foreign proteins what requires a change in the genome. B cells raised under the antigenic in uence of the normal intestinal micro o- ra, self-proteins and alimentary antigens must go through a hypermutation process to be able to produce specific antibodies. It has a certain probability that hypermutation will successfully change the genome in some clones to switch from low afinity IgM antibody production to high afinity IgG production. Interactions (typically antibody antigen reac- tions) in an immune response happen at high afinity and are irreversible. High afinity clones will be selected, stimulated and enriched by the invading microbes. A complete account of the course of an infectious disease must also include a descrip- tion of the ecology of the immune response. It is therefore suggested that during prolonged interaction between host and infectious organism, carried on across many generations, the adaptive antibody population may facilitate the evolution of the natural antibody reper- toire, in accordance with the Baldwin effect in the evolution of instinct (see Appendix 6).

## Abstract

There is a uniquely defined random graph model with independent adjacencies in which the degree sequence is a sufficient statistic. The model was recently discovered independently by several authors. Here we join to the statistical investigation of the model, proving that if the degree sequence is in the interior of the polytope defined by the Erdős–Gallai conditions, then a unique maximum likelihood estimate exists.