On relations between sets

With a view to future applications in relational biology, the notion of relations between sets is introduced and several theorems are demonstrated.     

On relations between sets: II

The notion of relations between sets, defined in a previous publication (Bull. Math. Biophysics,23, 233–235, 1961) is generalized and some biological examples are given. A generalization ton-ary relation is suggested.     

A note on relations between sets

From the definition of a strong and weakn-ary relation betweenn sets, given in a previous paper (Bulletin of Mathematical Biophysics,27, 477–492), it follows that for a given set ofn sets and givenn-ary relationR between them there can exist only one strong relation, but a large number of weak ones. An expression for the total number of possible weak relations is derived and the notion of the degree of weakness of a relation is introduced and discussed.     

Elementary propositions on relations between sets

Some elementary propositions dealing with set relations (as introduced in an earlier work by Rashevsky) are presented.     

On dominance relations and the structure of animal societies: III The condition for a score structure

The necessary and sufficient condition is given forn integers to be the score structure of a society with a dominance relation. A proof is also given for a theorem showing that there are members who dominate every other member either directly or indirectly through a single intermediate member.     

On mutant sets

This paper sketches the outline of a new, general mathematical theory concerning the nature of a relative anti-closure property for subsets of general algebraic systems. It thus quite naturally fits into any exhaustive theory of relations and, in particular, into a theory of relations for abstract mathematical molecular biology. In addition, the theory possesses certain intuitive, but naive relationships to basic analytical studies of biological mating and mutation. At the purely mathematical level one can produce an abundance of theorems from the theory with interpretations in the context of the frequently appearing mathematical structures of groups and rings, among other algebraic structures.     

On evolutionarily stable sets

As an extension of the concept of an evolutionarily stable strategy (ESS) evolutionarily stable sets are introduced, i.e. sets of equilibrium strategies (EQS) which have much of the properties of an ESS. They are primarily used with evolutionary game models that allow a continuum of EQSs, none of which can be an ESS, but also include common ESSs as a special case. For a large class even of nonlinear models it can be shown that the standard dynamics converge towards some equilibrium point in an ES set if started within a neighbourhood of the set. Important applications of ES sets include e.g. mixed-strategist models and evolutionary game models in sexual populations.     

Calibration of multivariate scatter plots for exploratory analysis of relations within and between sets of variables in genomic research

The scatter plot is a well known and easily applicable graphical tool to explore relationships between two quantitative variables. For the exploration of relations between multiple variables, generalisations of the scatter plot are useful. We present an overview of multivariate scatter plots focussing on the following situations. Firstly, we look at a scatter plot for portraying relations between quantitative variables within one data matrix. Secondly, we discuss a similar plot for the case of qualitative variables. Thirdly, we describe scatter plots for the relationships between two sets of variables where we focus on correlations. Finally, we treat plots of the relationships between multiple response and predictor variables, focussing on the matrix of regression coefficients. We will present both known and new results, where an important original contribution concerns a procedure for the inclusion of scales for the variables in multivariate scatter plots. We provide software for drawing such scales. We illustrate the construction and interpretation of the plots by means of examples on data collected in a genomic research program on taste in tomato.     

Molecular set theory. III: The wide sense kinetics of molecular sets

An effort is made to begin widening the scope of kinetics by merging the concepts and point of view of molecular set theory with the stochastic approach to kinetics, beginning with the simplest unimolecular molecular set transformation. In this spirit the new concept ofmolecular set variable is introduced as the basic unit of kinetics as opposed to simply the traditionalconcentration (or cardinality) unit, connoting that the composition as well as the size of a molecular set are significant dynamic features of a system. The changes in state (or “value”) of the molecular set variable are characterized by a Markovian stochastic process and the relationship between this process and the corresponding unimolecular process for the concentration variable introduced earlier is discussed. The possible role of molecular set theory in terms of the underlying biomathematical structure of relational biology is also considered.     

Contribution to the theory of organismic sets,III

In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern>1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.     

On chains of related sets

IfK is a partition of a setK which is partially ordered by the relationR andR is a collection of pairs of sets ofK such that the sets of each pair are related byR in the sense of Rashevsky, thenR is a relation which partially ordersK. Necessary and sufficient conditions thatK be a chain are obtained, and ifK is a chain under these conditions, it is shown thatK is unique.     

Topological relations between olfactory neurons

Summary The olfactory mucosa of several vertebrates has been studied with the aid of light and electron microscopes. It has been found that the receptor neurons vary in density per square area from one region to the other of the olfactory mucosa in each animal. The neurons, moreover, can be arranged in interconnecting rings forming a complex pattern where each element directly contacts the others along different portions of the cell body. These contacts and their different ultrastructural details are illustrated and discussed. The data are elaborated in view of their functional importance in regard to the reception of stimuli.With the technical assistance of Mrs. H.T. Gagne and Mrs. L.K. Stuart. 31 Z. Zellforsch., Bd. 118     

On the establishing of permissible relations between MACC- and MACT-values of harmful substances

The establishment of MACC (Ceiling Values) and MACT (Time Weighted Average Values) for chemical substances permits a substantial more precise evaluation of the exposure to harmful substances at the work place as if only one limit value is given without reference to a Ceiling Value or a Time Weighted Average Value, respectively. The derivation of MACC- and MACT-Values is proposed from toxicometric data that are gained during animal experiments with chemical substances. For acutely toxic substances (e.g. irritants) the quotient F = MACC/MACT ought to be = 1. For cumulative acting substances the permissible relation of both values is determined from the magnitude of the zone of chronic effect (Zch) as F = 0.4. Zch. The condition MACC less than less than Limac takes into consideration at the same time. The frequency and the duration of MACT exceedings need not to be regulated with this procedure. A method is proposed for the derivation of the length of sampling periods necessary from the toxicological point of view starting from the Zch.     

On the relations between the hypothalamus and the anterior pituitary in Rana temporaria

Summary Dawson (1952) showed that, in Rana, many nerve terminals end round the capillaries of the venous portal system of the median eminence. Our observations not only confirm these facts but show as well that, besides the posterior lobe, the outer zone of the median eminence is also a centre of accumulation of neurosecretory material derived from the pre-optic nucleus. The amount of this neurosecretory material is moreover subject to a certain variation in connection with the seasons and with experimental conditions. The idea is put forward that the outer zone of the median eminence of Rana has to be considered as an analogical organ of the posterior lobe. Indeed, in the median eminence, the same morphological facts can be found, which are advanced for the conception that the posterior lobe is a centre of accumulation and a place of release of neurosecretory material originating from the pre-optic nucleus. On the basis of our observations we presume that the hypothalamo-hypophyseal neurosecretory system of Rana can be divided in two sub-units. System I ends in the posterior lobe; system II in the outer zone of the median eminence. The data of the literature indicate that the two systems also probably exist in the higher vertebrates. From these facts it is obvious that the neurosecretory material is transported from the pre-optic nucleus, via the tract, towards the outer zone of the median eminence.