Summary: A field can be explored easily, but break-surfaces are destroyed by their discovery.
This may involve curious philosophical properties. Break surface exploring Organisation exploring organisation
Summary: It has been shown that a representative point, staying within a region bounded by
a layer of break-surfaces, can act as a "variable" in a substitution composed of n such points provided the representative points move with a velocity of a higher "order"
than that of the substitution. "Order" is defined and explained. The ordinary substitution
can be considered as the limit of this type. Break surface controlling substitution
Summary: A method is described by which a machine can show increasing adaptation, by one part after another getting into equilibrium. A clear explanation of "threshold" and "summation"
in the Central Nervous System follows. It is concluded that between a sense organ
and the adaptive part a "distributor" must occur. 5345 Adaptation non-adaptation Survival failure of
Summary: An attempt is made to classify and exhaust the causes of non-adaptation; but it seems
that non- adaptation must be taken as fundamental, adaptation occuring only if there
is some special reason for it.
Summary: A description is given of relations between differential equations and solutions when
certain variables are not present in some of the equations. Two matrices |f| and |F| are defined. Particularly it is shown that the "independence" test of p applies to either.
Summary: A view of Levy's book. He specifically notices that breaks are an essential feature
of matter and not a trivial one.
Summary: The concept of "dominance" involves an inverted way of looking at things, and is better
replaced by the same variables being "independent of the others" in a system.
Summary: We may not write arbitrary functions in the solutions xi=Fi(xo;t), for the f's are to be free from t. This means that there are restrictions on the F's, and it is shown that suitable F's will satisfy certain equations. (Cf. 1315)(and 1341)
Summary: Definition of the First and Second Jacobian matrices of a dynamic system, with a note
that "completion" applies to the Second and not the First.
Summary: Exploring the interaction of a given set of variables means finding the F's in xi=Fi(xo;t). (Assembling a machine gives us the [xoi=fi(x)] equations). By the independence test on the Second Jacobian Matrix applied in one
stroke we eliminate what is not wanted. That its behaviour is reproducible is equivalent
to the requirement that t is explicitly absent from the f's. This restricts possible F's. An equation is given which they must satisfy. It is proved that under these conditions
the F's are always completed.
Summary: "Step-function" in practice is not usually so restricted as on 1279.
Summary: At last an exact meaning can be given to the idea of whether one variable does, or
does not, affect another. It can only be tested when the complete system containing
the affected one is obtained. A set, independant of the others, contained in a complete
set, must itself be complete.
Summary: A definition of a complete system, and some elementary properties.
Summary: Parameters which are regarded as constant "variables" thereby lose some freedom, perhaps
too much sometimes.
Summary: A single permanent zero in [f] introduces a slight, permanent restriction in the field.
Summary: The non-zero elements in [f] correspond, in a sense, to dendrons.
Summary: The chance that n variables should all independently be in equilibrium is discussed and this gives
an estimate of the time required to reach equilibrium. The fastest method of getting
equilibrium will be the one found in practice, for the system selects the fastest.
And this suggests that the brain will automatically manifest an "analysing" tendency.
Summary: The environment (probably) consists of many small complete systems contained in larger
complete systems, etc slow time changes upsetting all. Two more ways of graduating
adaptation are noted. The dynamic form of "whole" and "part" is clarified.
Summary: The solutions of a complete system form a finite continuous group of order one.
Summary: Notes from Bieberbach on finite continuous groups.
Summary: Variables changing at different orders of velocity hardly interact. A study of interaction
must therefore assume the variables are of the same order of velocity (Now turn to
Summary: The relations of "complete sets which contain complete sets which ..." can be shown
accurately by an isomorphic diagram.
Summary: Assuming each variable has a fixed chance of getting equilibrium, it is shown that
a system of n1, variables dominating n2 will in 1-pn2 cases get equilibrium by getting it in the n1 and then in the n2, while in pn2 cases it will get the whole simultaneously, the latter proportion being vanishingly
small. Experiment will therefore demonstrate the equilibrium appearing in stages.
Summary: If a complete system has n variables and r parameters [x-i=fi(x;λ)], then the λ's can, from given starting point, control the movement of the x-point within an r-dimensional space which moves with time through the n-space, but the λ's cannot control the movement of the r-space. (Now see 1376)
Summary: A Permanent zero in the 1st. Jacobian Matrix, i.e. incomplete joining, means that
a sudden change of the variables does not immediately alter the path as projected
on to the other variable's axis. (Continued 1372)
Summary: The 1st Jacobian Matrix (1) cannot be filled in arbitrarily (2) does not accurately
specify a dynamic system.
Summary: If each break (a) depends only on one variable, (b) affects, or appears in only that
variables' f, then each variable will become stabilised almost independently of the others. Under
these conditions the time taken by n is of the order of log n.
Summary: As first approximation, the "largest of a sample of n" tends to increase as log n.
Summary: If r parameters controlling a complete system are arbitrarily under our control, then
we can, by controlling the parameters, force an arbitrarily selected set of r variables to behave as we chose. The detailed control can, so to speak, be transmitted
through the many other variables without any loss of control! Input control possible Parameter degrees of freedom
Summary: The problem of several complete systems joining into an interacting system without
losing (entirely) their completeness is discussed and partially solved.
Summary: The solutions are given of the problems of: Given the f's (or the F's), to find the F's (or the f's).
Summary: A proof, with modern technique, of the old problem, showing that two stable machines
can be joined to form an unstable one.
Summary: A test to see whether a neutral point is stable or unstable. (Test for neutral cycle,
Summary: The old case of several variables affecting one another chain-fashion is re-examined.
It is shown that if an "increase" leads back to a "decrease" the system will be stable,
though probably with oscillations (of decreasing amplitude). If it leads to an "increase"
the system may still be stable.
Summary: If the study of a complete system of n variables is restricted to some of the variables only, the others being hidden, the
behaviour of the visible variables can be predicted correctly when we know any n coordinate-time combinations. A machine may appear to show imagination. (Restated
Summary: The (real) environment may be absolutely anything. But we can devise theoretical systems
to which a given brain could and would adapt, and we then examine the real world to see if such sorts exist.
Summary: The idea of a "constraint" added to a dynamic system may have meaning with Newtonian
dynamics but it has no general meaning. And the idea of thereby losing a "degree of
freedom" is also of restricted applicability. Hour glass system properties
Summary: It is shown that the "hour-glass" type of organisation will differ little from others
in its properties of adaptation.