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1942
Volume 06
1176+03
1176+04
Break surface exploring Organisation exploring organisation Summary: A field can be explored easily, but break-surfaces are destroyed by their discovery. This may involve curious philosophical
properties.
Break surface controlling substitution 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.
Adaptation defined Summary: "Adaptation" is more properly divided into: the adapted state after this has been reached, and the process of finding this
state.
Adaptation non-adaptation Survival failure of 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
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.
Adaptation and mutations Adaptation chains of Evolution as law and chance Holism in mutations Natural Selection [13]: In adaptation by heredity there is the recombination effect 1254. Quotation 2163. Summary: Notes on adaptation to "internal" environment; and an example of how a set of adaptations can collapse.
Adaptation growing Organisation must change only infinitesimally Summary: Huxley's book reviewed, and proof that a holistic set must be altered by infinitesimal steps.
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.
1291
1292
1293
1294
Summary: A view of Levy's book. He specifically notices that breaks are an essential feature of matter and not a trivial one.
1295
1296
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.
1297
1297+01
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)
1298
1298+01
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: The concept of "parameter" should be replaced, (except in simple cases), by the idea of a variable having some special properties,
These are given. The fundamental is [x-k=0]. (But see 1324)
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.
1315
1316
Summary: "Step-function" in practice is not usually so restricted as on 1279.
1317
1318
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.
1323
1324
Summary: A single permanent zero in [f] introduces a slight, permanent restriction in the field.
1325
1326
Summary: The non-zero elements in [f] correspond, in a sense, to dendrons.
1327
1328
1329
1330
1331
1332
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.
1333
1334
1335
1336
1337
1338
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.
1339
1340
1341
1342
Summary: The solutions of a complete system form a finite continuous group of order one.
1343
1344
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 1474!)
1349
1350
1351
1352
Summary: The relations of "complete sets which contain complete sets which ..." can be shown accurately by an isomorphic diagram.
1353
1354
1355
1356
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: An unsolved problem in organisation. (Now see 1420)
1359
1360
1361
1362
1363
1364
1365
1366
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)
1367
1368
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)
1369
1370
Summary: The 1st Jacobian Matrix (1) cannot be filled in arbitrarily (2) does not accurately specify a dynamic system.
1371
1372
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.
1373
1374
Summary: As first approximation, the "largest of a sample of n" tends to increase as log n.
1375
1376
Input control possible Parameter degrees of freedom 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!
Summary: The problem of several complete systems joining into an interacting system without losing (entirely) their completeness is
discussed and partially solved.
1389
1389+01
1389+02
1390
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).
1391
1392
1393
1394
Summary: A proof, with modern technique, of the old problem, showing that two stable machines can be joined to form an unstable one.
1395
1396
1397
1398
Summary: A test to see whether a neutral point is stable or unstable. (Test for neutral cycle, 1494)
1399
1400
1401
1402
1403
1404
1405
1406
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.
1407
1408
Oddments [6]: "Almost absolute" systems, defined 1409, some properties 1386. Summary: Contrary to p.____ [0599], the concept of equilibrium does not depend on a circuit.
Summary: Definition of an "almost" complete system.
Higher geometry of fields and matrix theory [5]: If some variables in a complete system are unobservable, a path fixed by n t,x combinations can be accertained empirically,
by using substitute variables, 1413, numerical example 1470.
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 1424)
1413
1414
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.
1415
1416
1417
1418
Hour glass system properties 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.
1419
1420
1421
1422
Summary: It is shown that the "hour-glass" type of organisation will differ little from others in its properties of adaptation.
1423
1424
Higher geometry of fields and matrix theory [4]: A path is fixed (in a given field) by any n combinations of t and x, 1425. Summary: If, in a system of n variables complete or not, we are given n coordinate-time pairs, the particular path is fixed.
Summary: Six definitions of a "complete" system are given and are all proved equivalent.
1431
1432
Summary: Some references to amoeboid activity in nerve cells.
1433
1434
Summary: The brain is an equilibrium-trap. And if the equilibrium can only occur on certain conditions then the brain will trap those
conditions too! 1487.
1435
1436
1437
1438
Summary: Stabilising some variables almost certainly stabilises those other variables connected with them.
