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IntellectualArchive

Volume 1, Number 4

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Intellectual

ArchiveVolume 1 Number 4 August 2012

Table of Contents

Mathematics

Josimar da SilvaRocha,Said Najati Sidki

The n-ary adding machine and solvable groups ................................... 1

Alexander A.Ermolitski

Three–dimensional compact manifolds and the Poincare conjecture .. 51

Alexander Krasulin Five-Dimensional Tangent Vectors in Space-Time: III. SomeApplications .......................................................................................... 63

Physics

V.Volov Fractal-Cluster Theory and Thermodynamic Principles of BiologicalSystems Control and Analysis .............................................................. 75

Philosophy

J.C. Hodge Survival is the only moral goal of life .................................................... 80

Education

Baisheva MI Teacher security and counter-terrorism in DOW as the direction ofscientific research department (English / Russian) ................................. 119

Natalia Filatova Polycultural education of students based on the pedagogicalprojection............................................................................................... 131

Lena Maximova Professional training of teachers-bachelors in the conditions ofmodernization of research activity of higher education ........................ 138

Toronto, August 2012

THE n-ARY ADDING MACHINE AND SOLVABLE GROUPS

JOSIMAR DA SILVA ROCHA AND SAID NAJATI SIDKI

Abstract. We describe under a various conditions abelian subgroups ofthe automorphism group Aut(Tn) of the regular n-ary tree Tn, which arenormalized by the n-ary adding machine τ = (e, ..., e, τ)στ where στ is then-cycle (0, 1, ..., n− 1). As an application, for n = p a prime number, andfor n = p2 when p = 2, we prove that every finitely generated soluble sub-group of Aut(Tn), containing τ is an extension of a torsion-free metabeliangroup by a finite group.

Contents

1. Introduction 22. Preliminaries 33. The holomorph of the n-adic integers 63.1. Powers of τ . 63.2. Centralizer of τ . 73.3. Normalizer of the topological closure 〈τ〉 84. Abelian groups B normalized by τ 115. The case β ∈ B with σβ ∈ 〈στ 〉 146. Solvable groups for n = p, a prime number. 227. Two cases for n even 247.1. The case σβ = (στ )

n2 24

7.2. The case σβ transposition 278. Solvable groups for n = 4. 348.1. Cases σβ ∈ (0, 3)(1, 2), (0, 1)(2, 3) 358.2. Cases σβ ∈ (0, 2), (1, 3) 35

8.3. The case σβ = (στ )2 = (0, 2) (1, 3) 37

8.4. Cases σβ ∈ e, στ , σ−1τ 41

8.5. Final Step 46References 49

Date: August 2011.Key words and phrases. Adding machine, Tree automorphisms, Automata, Solvable

Groups.1

1IntellectualArchive Vol. 1, No. 4

2 JOSIMAR DA SILVA ROCHA AND SAID NAJATI SIDKI

1. Introduction

Adding machines have played an important role in dynamical systems, andin the theory of groups acting on trees : see [1, 2, 5, 4, 10].

An element α in the automorphism groupAn = Aut(Tn) of the n-ary tree Tn,is represented as α = α|φ = (α|0, ..., α|n−1)σα where φ is the empty sequencefrom the free monoid M generated by Y = 0, 1, .., n− 1, where α|i ∈ An(i ∈ Y )-called 1st level states of α- and where σα (the activity of α) is apermutation in the symmetric group Σn on Y extended ’rigidly’ to act on thetree; if = e, we say α inactive. In applying the same representation to α|0 weproduce α|0i where i ∈ Y and in general we produce α|u | u ∈M the set ofstates of α. Following this notation, the n-ary adding machine is representedas τ = (e, ..., e.τ)στ where e is the identity automorphism an στ is the regularpermutation σ = (0, 1, ..., n− 1). In this sense the adding machine may beviewed as an infinite variant of the regular permutation which often appearsin geometric and combinatorial contexts.

A characteristic feature of τ is that its n-th power τn is the diagonal au-tomorphism of the tree (τ, ..., τ). This fact implies that the centralizer of thecyclic group 〈τ〉 in An is equal to its topological closure < τ > in An seen asa topological group with respect to the the natural topology induced by thetree.

A large variety of subgroups of An which contain τ have been constructed,including finitely generated groups which are torsion-free and just non-solvable,yet without free subgroups of rank 2 [3, 6], and generalizations thereof [9], aswell as constructions of free groups of rank 2 [11]. Yet solvable groups whichcontain τ are expected to have restricted structure [2]. For nilpotent groupswe show

Proposition. Let G be a nilpotent subgroup of An which contains then-adic adding machine τ . Then G is a subgroup of < τ > .

Let Zn be the ring of n-adic integers and U (Zn) its subgroup of units. Thenormalizer of < τ > in An is isomorphic to the holomorph of Zn, the semi-direct product Zn o U (Zn), and is therefore metabelian.

The most visible examples of finitely generated solvable groups containingτ are conjugate to subgroups of those belonging to the sequence of groups

Γ0 = NAn< τ >,Γ1 = (×nΓ0) oG1, ...,Γi+1 = (×nΓi) oGi+1, ...

where ×nΓi is a direct product of n copies of Γi (seen as a subgroup of the 1stlevel stabilizer of the tree) and where Gi is a solvable subgroup of Σn in itscanonical action on the tree, containing the cycle στ . We note that for all i,the groups Γi are metabelian by ’finite solvable subgroups of Σn’. It was shownby the second author that for n = 2, that finitely generated solvable groupswhich contain the binary adding machine are conjugate to some subgroups ofΓi acting on the binary tree [7].


THE n-ARY ADDING MACHINE AND SOLVABLE GROUPS 3

The description for degrees n > 2 requires a classification of solvable sub-groups of Σn which contain the cycle σ = (0, 1, ..., n− 1)[8]. This is an openproblem, even for metabelian groups. On the other hand, the answer for prim-itive solvable subgroups of Σn is simple and classical. For then, n is a primenumber p or n = 4. In case n = p, the solvable subgroups Gi can all be takento be the normalizer F = NΣn (〈σ〉) of order p (p− 1) and in case n = 4, theGi’s can all be taken to be the symmetric group Σ4.

Given this background, the main theorem of this paper isTheorem A. Let n = p, a prime number, or n = 4. Then any finitely

generated solvable subgroup of An,which contains the n-ary machine τ is con-jugate to a subgroup of Γi for some i.

The result follows first from general analysis of the conditions [β, βτx] = e

(for some β ∈ An and all x ∈ Z), their impact on the 1st level states of thesubgroup 〈β, τ〉 and then how these in turn translate successively to conditionson states at lower levels. It is somewhat surprising that the process convergesto a clear global description for trees of degrees p and 4.

If σβ is either a power of στ or a transposition, we describe abelian subgroupsnormalized by τ .

Theorem B. Let B be an abelian subgroup of An normalized by τ , letβ = (β|0, β|1, · · · , β|n−1)σβ ∈ B and define the subgroup H = 〈β|i (i ∈ Y ) , τ〉generated by the states of β and τ .(I) Suppose σβ = (στ )

s for some integer s and set m = ngcd(n,s)

. Then, H is

metabelian-by-finite. Indeed,on defining the subgroup

K =⟨

[β|i, τ k], β|iβ|i+sβ|i+2s · · · β|i+(m−1)s | k ∈ Z, i ∈ Y⟩

(the bar notation means ’modulo m’) then K is a normal subgroup of H andO = K 〈τ〉 is a metabelian normal subgroup of H where H

Ois a homomorphic

image of a subgroup of the wreath product Cm oCn of the cyclic groups Cm, Cn.(II) Let n be an even number. Then H is a metabelian group if s = n

2or σβ

is a transposition.Let P be a subgroup of Σn. The layer closure of P in An is the group L (P )

formed by elements of An all of whose states lie in P . The following result isyet another characterization of the adding machine.

Theorem C. Let n be an odd number, σ = (0, · · · , n − 1) ∈ Σn and letL = L (〈σ〉), the layer closure of 〈σ〉 in An. Let s be an integer relativelyprime to n and let β = (β|0, β|1, · · · , β|n−1)σs ∈ L be such that [β, βτ

x] = e

for all x ∈ Z. Then β is a conjugate of τ in L.

2. Preliminaries

We start by introducing definitions and notation. The n-ary tree Tn can beidentified with the free monoidM =< 0, 1, .., n−1 >∗ of finite sequences fromY = 0, 1, ..., n− 1, ordered by v ≤ u provided u is an initial subword of v.



The identity element of M is the empty sequence φ. The level function forTn, denoted by |m| is the length of m ∈M; the root vertex φ has level 0.

∅

0

88qqqqqqqq1

ffMMMMMMMM

00

AA01

]]<<<<

10

AA11

]]<<<<

BB `` >> `` >> `` >> \\

Figure 1. The Binary Tree

The action ρ : i → j of a permutation ρ ∈ Σn will be from the right andwritten as (i) ρ = j or as iρ = j. If i, j are integers then the action of ρ oni is to be identified with its action on its representatives i in Y , modulo n .Permutations σ in Σn are extended ’rigidly’ to automorphisms of An by

(y.u)ρ = (y)ρ.u, ∀ y ∈ Y, ∀ u ∈M.

An automorphism α ∈ An induces a permutation σα on the set Y . Conse-quently, α affords the representation α = α′σα where α′ fixes Y point-wise andfor each i ∈ Y , α′ induces α|i on the subtree whose vertices form the set i ·M.If j is an integer the α|j will be understood as α|j where j is the representativeof j in Y modulo n.

Given i in Y , we use the canonical isomorphism i · u 7→ u between i · Mand the tree Tn, and thus identify α|i with an automorphism of Tn; therefore,α′ ∈ F(Y,An), the set for functions from Y into An, or what is the same, the1st level stabilizer Stab(1) of the tree. This provides us with the factorizationAn = F(Y,An) · Σn.

Let α, β, γ ∈ An. Then following formulas hold

(1) σα−1 = (σα)−1 , σασβ = σαβ,

(2) (α−1)|u = α|(u)α

−1 ,

(3) (αβ)|u = (α|u) (γ|u) where γ|u = β|(u)α

(4) γ = α−1βα⇔ σγ = σ−1α σβσα,

(5) γ|(i)σα = α|−1i β|iα|(i)σβ , ∀i ∈ Y .

(6) θ = [β, α] = β−1βα ⇒ σθ = [σβ, σα],

(7) θ|(i)σαβ =(β|(i)σα

)−1(α|i)−1 (β|i)

(α|(i)σβ

),∀i ∈ Y .



(8) (αm) |i = (α|i)(α|(i)σα

) (α|(i)σ2

α

)· · ·(α|(i)σαm−1

)(9) (βα) |u =

(β|(u)α−1

)α|(u)α−1 ,where β ∈ Stab(k) and |u| ≤ k.

An automorphism α ∈ An corresponds to an input-output automaton withalphabet Y and with set of states Q(α) = α|u | u ∈ M. The automaton αtransforms the letters as follows: if the automaton is in state α|u and readsa letter i ∈ Y then it outputs the letter j = (i)α|u and its state changes to

α|ui; these operations can be best described by the labeled edge α|ui|j−→ α|ui.

Following terminology of automata theory, every automorphism α|u is calledthe state of α at u.

The tree Tn is a topological space which is the direct limit of its truncationsat the n-th levels. Thus the group An is the inverse limit of the permutationgroups it induces on the n-th level vertices. This transforms An into a topo-logical group. An infinite product of elements An is a well-defined element ofAn provided for any given level l, only finitely many of the elements in theproduct have non-trivial action on vertices at level l. Thus, if α ∈ An and ξ=∑

i≥0 aini ∈ Zn then αξ = αa0 .αna1 ..αn

iai ... is a well define element of An.

The topological closure of a subgroup H in An will be indicated by H. Wenote that if H is abelian then

H = hξ|h ∈ H, ξ ∈ Zn .One of the characterizing aspects of the n-ary adding machine is that thecentralizer of τ is a pro-cyclic group; namely,

CAn(τ) = 〈τ〉 = τ ξ | ξ ∈ Zn.Let v = yu where y ∈ Y, u ∈M. The image of v under the action of α is

(v)α = (yu)α = (y)σα.(u)α|y.The action extends to infinite sequences (or boundary points of the tree) inthe same manner. A boundary point of the tree c = c0c1c2 . . .where ci ∈ Y forall i, corresponds also to the n-adic integer ξ =

∑cini|i ≥ 0 ∈ Zn. Thus the

action of the tree automorphism α can thus be translated to an action on thering of n-adic integers. We will indicate c0 by ξ which is ξ modulo n. In thecase of the automorphism τ = (e, e, ..., e, τ)σ, the action of τ on c is

(c) τ =

(c0 + 1) c1c2 . . . if 0 ≤ c0 ≤ n− 2,0(c1c2, . . .)

τ , if c0 = n− 1,

which translates to the n-ary addition

ξτ = 1 + ξ.



?>=<89:;τ1/0

II

0/1// ?>=<89:;e

0/0, 1/1

Figure 2. The binary adding machine

3. The holomorph of the n-adic integers

The holomorph of Zn is the extension Zn by the its group of units U(Zn) inits natural action on Zn. An element ξ is a unit in Zn if and only if ξ is a unit inZ modulo n. The subgroup of U(Zn) consisting of elements ξ with ξ = 1 is de-noted by by Z1

n. This subgroup has the transversal j | 1 ≤ j ≤ n− 1, gcd (j, n) = 1in Zn and therefore has index [U(Zn) : Z1

n] = ϕ (n) where ϕ is the Euler func-

tion. The normalizer of 〈τ〉 in the group of automorphisms of the tree is theholomorph of Zn.

Given α ∈ An we denote the diagonal automorphism (α, ..., α) by α(1) and

define inductively α(i+1) =(α(i))(1)

for all i ≥ 1.

3.1. Powers of τ . Let ξ =∑

i≥0 aini ∈ Zn. Then a0 = ξ and

∑i≥1 ain

i−1 =ξ−ξn

.

Lemma 1. Let ξ ∈ Zn. Then

τ ξ = (τξ−a0n , · · · , τ

ξ−a0n , τ

ξ−a0n

+1, · · · , τξ−a0n

+1︸︷︷︸a0 terms

)σa0τ .

Proof. For j an integer with 1 ≤ j ≤ n− 1, we have

τ j =

e, ..., e, τ, · · · , τ︸︷︷︸j terms

σjτ

and τn = (τ, ..., τ) = τ (1).Given ξ =

∑i≥0 ain

i, then



τa0 = (e, · · · , e, τ, · · · , τ︸︷︷︸a0 terms

)σa0τ ,(10)

τajnj

= τ (ajnj−1)n =

(τajn

j−1)(1)

,(11)

τ ξ = (τξ−a0n , · · · , τ

ξ−a0n , τ

ξ−a0n

+1, · · · , τξ−a0n

+1︸︷︷︸a0 terms

)σa0τ(12)

= (τξ−ξn , · · · , τ

ξ−ξn , τ

ξ−ξn

+1, · · · , τξ−ξn

+1︸︷︷︸ξ terms

)σξτ .(13)

As we have seen, the description of τ ξ involves the partition of the interval[0, ..., n − 1] into two subintervals. Therefore we introduce the step functionδ : Z

nZ ×ZnZ → 0, 1 given by

δ(i, j) =i+ j − i+ j

n=

0, if 0 ≤ i ≤ n− j1, otherwise

.

which we will call the Polarizer Function. With this,

τ ξ =(τξ−ξn

+δ(i,ξ))

0≤i≤n−1σξτ .

The function δ extends to Zn×Zn, simply by δ(η, κ) = δ(i, k) where i = η, k =κ. Note that

n−1∑i=0

δ(i, j) = j.

//

OO

0 1 2 3 iii

1

2

3

jjj

•__

•__

•__

____ • __ •

______ •

Figure 3. Polarizer Function for n = 4.



3.2. Centralizer of τ .

Lemma 2. CAn (τ) = 〈τ〉.

Proof. Let α ∈ An commute with τ . Then, [σα, στ ] = e and therefore σα =(στ )

s0 for some integer 0 ≤ s0 ≤ n− 1. Therefore, β = ατ−s0 = (β|0, ..., β|n−1)commutes with τ and σβ = e. Now,

βτ = ((β|n−1)τ , β|0, ..., β|n−1) = β

implies β|i = β|0 for all 0 ≤ s0 ≤ n− 1 and β|0 commutes with τ . Therefore

β = (β|0)(1) and β|0 replaces α in previous argument. Hence,

there exists an integer 0 ≤ s1 ≤ n − 1 such that γ = β|0τ−s1 = (γ|0)(1).From which we conclude

α = βτ s0 = (β|0)(1) τ s0

=(

(γ|0)(1) τ s1 , .., (γ|0)(1) τ s1)τ s0

= (γ|0)(2) τns1τ s0 = (γ|0)(2) τns1+s0 .

Inductively then, we obtain the desired form α = τ ξ where ξ = s0+ns1+....

A characterization of nilpotent groups which contain τ follows.

Proposition 1. Let G be a nilpotent subgroup of An which contains the n-adicadding machine. Then G is a subgroup of < τ > .

Proof. Suppose G is a nilpotent group of class k > 1 which contains τ . Then,the center Z (G) is contained in 〈τ〉. Let j be the maximum index such that

Zj (G) ≤ 〈τ〉; therefore j < k. Let α ∈ Zj+1 (G) \Zj (G); then [τ, α] = τ ξ and

ξ 6= 0. Now, [τ, α, α] =[τ ξ, α

]= e. Yet

[τ ξ, α

]= [τ, α]ξ = τ ξ

2= e and so,

ξ = 0 and [τ, α] = e; a contradiction.

3.3. Normalizer of the topological closure 〈τ〉.

Lemma 3. The group Γ0 = NAn

(〈τ〉)

is metabelian. Indeed, the derived

subgroup Γ′0 is contained in 〈τ〉.

Proof. Let α, β ∈ Γ0, then τα = τ ξ and τβ = τ η for some η, ξ ∈ U(Zn).Therefore,

τα = τ ξ, τ = (τ ξ)α−1

= (τα−1

)ξ,

τα−1

= τ ξ−1

.

Likewise, τβ−1

= τ η−1

. Thus, τ [α,β] = τ and Γ′0 ≤ CAn(τ) = 〈τ〉 follows.

We present a property of the polarizer function δ which we will use in thesequel.



Lemma 4. For all i, j ∈ Z, ξ ∈ Zn we have

jξ − jξn

− j(ξ − ξn

)+ δ(i, jξ) =

j−1∑k=0

δ(i+ kξ, ξ).

Proof. Since

(τ ξ)j|i = (τ ξ)|i.(τ ξ)|i+ξ · · · (τ ξ)|i+(j−1)ξ,

(τ ξ)|i = τξ−ξn

+δ(i,ξ)

the assertion follows from

τjξ−jξn

+δ(i,jξ) = τj(ξ−ξn

)+∑j−1k=0 δ(i+kξ,ξ).

Proposition 2. Suppose α ∈ An satisfies τα = τ ξ for some ξ ∈ U(Zn). Then:

(i)

α|i = α|0τµi , (1 ≤ i ≤ n− 1);where

µi = i(ξ − ξ)n

+i−1∑k=0

δ((v(α) + k)ξ, ξ)

and 0 ≤ v(α) ≤ n− 1 is such that

(0)σα = v(α)ξ;

(ii) (recursion) τα|0 = τ ξ;(iii)

(j)σα = (v(α) + j)ξ, (0 ≤ j ≤ n− 1).If ξ ∈ Z1

n then v(α) = 0, (j)σα = jξ = j, µi = i ξ−1n

.

Proof. Since σσατ = σξτ , we have

((0)σα, (1)σα, · · · , (n− 1)σα) = (0, ξ, 2ξ, · · · , (n− 1)ξ).

Therefore, there exists v(α) ∈ Y such that (0) σα = v(α)ξ and so,

(j)σα = (v(α) + j)ξ, ∀j ∈ Y .

Now, τα = τ ξ is equivalent to(σσατ = σξτ and α|(i)σsτ = ((τ s)|i)−1 α|i(τ ξs)|(i)σα ,∀i ∈ Y, ∀s ∈ Z, by...

).

The latter conditions are equivalent to(α|0 = α|(0)σnτ = ((τn)|0)−1 α|0(τ ξn)|(0)σα

and α|i = α|(0)σiτ= ((τ i)|0)

−1α|0(τ ξi)|(0)σα ∀i ∈ Y − 0

)and these in turn are equivalent to


10 JOSIMAR DA SILVA ROCHA AND SAID NAJATI SIDKI(α|i = α|0τ

ξi−ξin

+δ(v(α)ξ,ξi) = α|0τµi

where µi = i(ξ−ξn

)+∑i−1

k=0 δ((v(α) + k)ξ, ξ) ∀i ∈ Y − 0

).

Substitute i = 0 in

jξ − jξn

+ δ(i, jξ) = j

(ξ − ξn

)+

j−1∑k=0

δ(i+ kξ, ξ),∀i, ξ ∈ Z.

to get∑i−1

k=0 δ(kξ, ξ) = 0. The rest of the assertion follows directly.

Corollary 1. Let ξ ∈ U (Zn) and µi be as above. Then α = (α)(1) (e, τµ1 , ..., τµn−1)conjugates τ to τ ξ. In particular, if ξ ∈ Z1

n, then

α = (α)(1) (e, τξ−1n , τ 2 ξ−1

n , · · · , τ (n−1) ξ−1n )

denoted by λξ conjugates α to τ ξ.

Although we have computed above an automorphism which inverts τ , wegive another with a simpler description. Define the permutation

ε = (0, n− 1) (1, n− 2) ...

([n− 2

2

],

[n+ 1

2

]).

Then ε inverts στ = (0, 1, ..., n− 1) and

ι = ι(1)ε

inverts τ .Define

Λ = λξ | ξ ∈ Z1n,

Ψ = λξτ t | ξ ∈ Z1n, t ∈ Zn

and call Λ the monic normalizer of 〈τ〉.

Proposition 3. (i) Λ is an abelian group isomorphic to Z1n;

(ii) Ψ = Λ n 〈τ〉 ∼= Z1n n Zn;

(iii) the derived subgroup Ψ′ = 〈τn〉.

Proof. (i) Let ξ, θ ∈ Z1n. Then, as λξ, λθ and λξθ are inactive, its follows that

(λξλθλ−1ξθ )|i = (λξ)|i(λθ)|i ((λξθ)|i)−1

= λξτi ξ−1n λθτ

i θ−1n

(λξθτ

i ξθ−1n

)−1

= λξλθλ−1θ τ i

ξ−1n λθτ

i θ−1n τ−i

ξθ−1n λ−1

ξθ

= λξλθ

(τ iθ

ξ−1n τ i

θ−1n τ−i

ξθ−1n

)λ−1ξθ = λξλθλ

−1ξθ ,∀i ∈ 0, · · · , n− 1.

Therefore, λξλθ = λξθ. In addition, λξ = e if and only if ξ = 1.(ii) This factorization is clear.



(iii) Let θ = 1 + nθ′, η ∈ Zn. Then

[τ η, λθ] = τ−ηλθ−1τ ηλθ =

τ−ητ ηθ = τ η(θ−1) = (τn)ηθ′.

We prove below the existence of conjugates τα of τ in NAn

(〈τ〉)

, which lie

outside 〈τ〉. This fact provides us with the first important type of metabelian

groups 〈τ〉〈τα〉 containing τ .

Proposition 4. Suppose α = (α|0, α|1, · · · , α|n−1) ∈ An satisfies τα = λξτρ

for some ξ ∈ Z1n, and ρ = 1 + κn ∈ Z1

n. Then α|i+1 = (α|0)λξi+1τ1n

[ρ ξi+1−1ξ−1

−(i+1)

](0 ≤ i ≤ n− 2) ,

τα|0 = λξnτ1n

[ρ ξn−1ξ−1

].

.

The converse is true for n ≥ 3 and for n = 2 provided 4|ξ − 1.

Proof. From τα = λξτ1+κn, we obtain using (4) and (5),

λξτi ξ−1n

+κ = α|−1i αi+1, if i ∈ Y − n− 1

λξτ(n−1) ξ−1

n+κ+1 = α|−1

n−1τα|0.

Therefore,

α|i+1 = α|0λξτκλξτξ−1n

+κ · · ·λξτ iξ−1n

+κ, for i = 0, 1, · · · , n− 2,

α|0 = τ−1α|n−1λξτ(n−1) ξ−1

n+κ+1.

The first equations can be expresses as

α|i+1 = α|0λξi+1τκ∑ij=0 ξ

j+ ξ−1nξi∑ij=1 j(ξ

−1)j

= α|0λξi+1τ1n

[(1+κn) ξ

i+1−1ξ−1

−(i+1)

]

and the last as

α|0 = τ−1α|0λξnτξn

[(1+κn) ξ

n−1−1ξ−1

−(n−1)

]τ (n−1) ξ−1

n+κ+1

= λξnτ1n

[(1+κn) ξ

n−1ξ−1

].

If n ≥ 3 then τα|0 = λξnτ1n

[(1+κn) ξ

n−1ξ−1

]satisfies the same conditions as those

for α; namely, both ξn, ρ′ = 1n

[(1 + κn) ξ

n−1ξ−1

]are in Z1

n. If n = 2 then

ξ = 1 + 2ξ′, ρ′ = 12

[(1 + 2κ) ξ

2−1ξ−1

]= (1 + 2κ) (1 + ξ′) and so, ρ′ ∈ Z1

2 implies

ξ = 1 + 4ξ′′.



4. Abelian groups B normalized by τ

Let B be an abelian subgroup of An normalized by τ . For a fixed β ∈ B,we define the ’state closure’ of 〈β, τ〉 as the group

H = 〈β|i (i ∈ Y ) , τ〉 .We will be dealing frequently with the following subgroups of H ,

N =⟨[β|i, τ ki ] | ki ∈ Z, i ∈ Y

⟩M = N 〈τ〉 .

When σβ = (στ )s for some integer s we will also be dealing with the subgroups

K =⟨N, β|iβ|i+sβ|i+2s · · · β|i+(m−1)s | i ∈ Y

⟩,

O = K 〈τ〉where s = n

gcd(n,s).

We show that when n is a power of a prime number pk, the activity rangeof β narrows down to a Sylow p-subgroup of Σn. This is used to restrict thelocation of an abelian group B normalized by τ , within AnProposition 5. Let n = pk, σ = (0, 1, ..., n− 1) and P be a Sylow p-subgroupP of Σn which contains σ. Then(i) P is isomorphic to ((... (...Cp)wr)Cp)wrCp, a wreath product of the cyclicgroup Cp of order p iterated k−1 times; the normalizer of P in Σn is NΣn(P ) =P 〈c〉 where c is cyclic of order p− 1;(ii) P is the unique Sylow p-subgroup P of Σn which contains σ;(iii) if W is an abelian subgroup of Σn normalized by σ then W is containedin P ;(iv) the subgroup B is contained in the layer closure L = L

(NΣp(P )

).

Proof. (i) The structure of P as an iterated wreath product is well-known.

The center of P is Z =⟨z(

= σpk−1)⟩

and CΣn(z) = P . Therefore, NΣn(P ) =

NΣn(Z) = P 〈c〉 where c is cyclic of order p− 1.(ii) If σ ∈ P g for some g ∈ Σn then zg ∈ CΣn(σ) = 〈σ〉 and therefore

〈zg〉 = 〈z〉 , P g = P . Thus, P is the unique Sylow p-subgroup of Σn to containσ.

(iii) Let W be an abelian subgroup of Σn normalized by σ. Let V = W <σ > and V0 be the stabilizer of 0 in V. Then, since σ is a regular cycle, itfollows that V = V0 〈σ〉 , V0 ∩ 〈σ〉 = e. Suppose that there exists a prime qdifferent from p which divides the order of W and let Q be the unique Sylowq-subgroup of W . Then Q is the unique Sylow q-subgroup of V and Q ≤ V0.Therefore, Q = e and W a p-group. As σ ∈ W , we conclude W ≤ P ..

(iv) Since the normal closure of 〈σβ〉 under the action of 〈στ 〉 is an abeliansubgroup, it follows that σβ ∈ P . Furthermore, as

⟨[β|u, τ k] | k ∈ Z

⟩is an



abelian group normalized by τ , it follows that [σβ|u , σ] ∈ P and thereforeσσβ|u ∈ P . Thus, we conclude σβ|u ∈ NΣn(P ) and β ∈ L.

Lemma 5. Let γ ∈ An. Conditions (i), (ii) below are equivalent:

(i) [γ, γτk] = e for all k ∈ Z;

(ii) [τ k, γ, γ] = e for all k ∈ Z.Condition (i) implies(iii)

⟨[γ, τ k] | k ∈ Z

⟩is a commutative group.

Condition (iii) implies⟨[γ|u, τ k] | k ∈ Z

⟩is a commutative group for all indices u.

Proof. First,

[γ, γτk] = γ−1

(τ−kγ−1τ k

)γ(τ−kγτ k

)= γ−1

(τ−kγ−1τ kγ

)γ(γ−1τ−kγτ k

)= [τ k, γ]γ[γ, τ k]

and so,

[γ, γτk

] = e⇔ [γ, τ k]γ = [γ, τ k].

Furthermore, since

(14) [γ, τ k1 ]τk2 = [γ, τ k2 ]−1[γ, τ k1+k2 ]

for all integers k1, k2, condition (ii) implies

[γ, τ k1 ][γ,τk2 ] = [γ, τ k1 ]γ

−1τ−k2γτk2 = [γ, τ k1 ]τ−k2γτk2

=([γ, τ−k2 ]−1[γ, τ k1−k2 ]

)γτk2

=([γ, τ−k2 ]−1[γ, τ k1−k2 ]

)τk2

= [γ, τ k1 ].

Finally, we note that by (6) and (7),

([γ, τnk])|(i)σγ = (γ−1)|(i)σγ

(τ−nk)|i (γ|i) (τnk)|(i)σγ

=(γ|−1i

)τ−k (γ|i) τ k

= [γ|i, τ k].

Since [γ, τ kn] is inactive for all k ∈ Z, we obtain [γ|i, τ k] | k ∈ Z is acommutative set for all i. The rest of the assertion follows by induction on thetree level.

Obviously,⟨[β, τ k] | k ∈ Z

⟩is normalized by τ and if condition (i) holds then

it is an abelian normal subgroup of 〈β, τ〉.

Proposition 6. Let l ≥ 1 and suppose α, γ ∈ Stab(l) satisfy [α, γτx] = e for

all x ∈ Z. Then

[α|u, γ|vτx

] = e ∀u, v ∈Mhaving |u| = |v| ≤ l and ∀x ∈ Z.



Thus, it follows that

τ δ(i,t−i)−δ(i−s,t−i)β|i−s[β|i−s, τ z]β|t= β|t−sβ|i[β|i, τ z]τ−δ(i,t−i)+δ(i+s,t−i)

for all t, i ∈ 0, 1, · · · , n− 1 and all z ∈ Z.

We develop below some properties of the ∆s function to be used in thesequel.

Proposition 8. The inductor function satisfies

(i) ∆s(i, t) = δ(i,−s)− δ(t,−s) =

0, if t, i ≥ s or t, i < s1, if t < s ≤ i−1, if i < s ≤ t

,

(ii) ∆s(i, t) = −∆s(t, i),(iii) ∆s(i+ s, t+ s) = −∆−s(i, t),(iv) ∆s(i, t) = ∆s(i, z) + ∆s(z, t),

(v)

n(s,n)

−1∑k=0

∆s(i+ ks, t+ ks) = 0,

(vi)n−1∑k=0

∆s(k, t) =

n− s, if t < s−s if t ≥ s

for all i, t, z ∈ Z.

Proof.

(i) Using the definition δ(i, j) = i+j−i+jn

we have

∆s(i, t) =i+ t− i− t

n− i− s+ t− i− t− s

n

=i+−s− i− s

n− t+−s− t− s

n

= δ(i,−s)− δ(t,−s)

=

0, if t, i ≥ s or t, i < s1, if t < s ≤ i−1, if i < s ≤ t

.

(ii) Follows from (i).(iii) Calculate

∆s(i+ s, t+ s) = δ(i+ s, t− i)− δ(i, t− i)= − (δ(i, t− i)− δ(i+ s, t− i))= −∆−s(i, t).

(iv) This part follows from (i).



(v) From the definition of the Polarizer functionn

(n,s)−1∑

k=0

δ(i+ ks, t− i) =

n(n,s)

−1∑k=0

δ(i+ (k − 1)s, t− i)

(vi) Finally, we have

n−1∑k=0

∆s(k, t) =∑s−1

k=0 ∆s(k, t) +∑n−1

k=s ∆s(k, t)

(i)=

n− s, if t < s−s, if t ≥ s

.

With the use of the inductor function notation we obtain

Proposition 9. The following relations are verified in H = 〈β|i (i ∈ Y ) , τ〉,for all x, z ∈ Z and all i, t ∈ Y :

(I) τ∆s(i,t)β|i−sβ|t = β|t−sβ|iτ∆s(i+s,t+s);

(II) [β|i−s, τ z]β|tτ−∆s(i+s,t+s)

= [β|i, τ z];(III) [[β|i, τ z], [β|t, τx]] = e.

Proof. Returning to Lemma 6, we have

τ∆s(i,t) (β|i−s) [β|i−s, τ z] (β|t)= (β|t−s) (β|i) [β|i, τ z]τ∆s(i+s,t+s).

Consequently,

(17) τ∆s(i,t)β|i−sβ|t = β|t−sβ|iτ∆s(i+s,t+s)

and

(18) [β|i−s, τ z]β|tτ−∆s(i+s,t+s)

= [β|i, τ z],for all t, i ∈ Y and all z ∈ Z.

From (18) and (14), N =⟨[β|i, τ ki ] | ki ∈ Z, i ∈ Y

⟩is a normal subgroup of

H. Moreover, by applying alternately the above equations, we obtain

[β|i, τ z][β|t,τk] = [β|i, τ z]β|

−1t τ−kβ|tτk

= [β|i, τ z](τ−∆s(i+s,t+s)τ∆s(i+s,t+s)β|−1

t τ−kβ|tτk)

(14)=([β|i, τ−∆s(i+s,t+s)]−1.[β|i, τ z−∆s(i+s,t+s)]

)(τ∆s(i+s,t+s)β|−1t τ−kβ|tτ k

)(18)=([β|i−s, τ−∆s(i+s,t+s)]−1.[β|i−s, τ z−∆s(i+s,t+s)]

)τ−kβ|tτ k(14)=

( ([β|i−s, τ−k]−1.[β|i−s, τ−k−∆s(i+s,t+s)]

)−1([β|i−s, τ−k]−1.[β|i−s, τ−k+z−∆s(i+s,t+s)]

) )β|tτ k



=([β|i−s, τ−k−∆s(i+s,t+s)]−1.[β|i−s, τ−k+z−∆s(i+s,t+s)]

)β|tτ k(18)=([β|i, τ−k−∆s(i+s,t+s)]−1.[β|i, τ−k+z−∆s(i+s,t+s)]

)τ k+∆s(i+s,t+s)

(14)= [β|i, τ z].

Corollary 3. Let β ∈ An such that [β, βτx] = e for every x ∈ Z with σβ = σsτ

for some s ∈ 0, 1, · · · , n− 1. Then

M =⟨[β|i, τ ki ], τ | ki ∈ Z, 0 ≤ i ≤ n− 1

⟩is a normal metabelian subgroup of H.

Proof. By Proposition 9 N =⟨[β|i, τ ki ] | ki ∈ Z, 0 ≤ i ≤ n− 1

⟩is abelian and

normal in H. Since Nτ ∈ Z(H/N), it follows that M = N 〈τ〉 is a normalsubgroup of H and is clearly metabelian.

We are ready to prove part (II) (i) of Theorem B.

Theorem 1. Let β ∈ An be such that [β, βτx] = e, ∀x ∈ Z and σβ = σsτ for

some s ∈ Y and H = 〈β|0, · · · , β|n−1, τ〉. Then,

(i) the group O =⟨[β|i, τx], β|jβ|j+s · · · β|j+(m−1)s, τ | i, j ∈ Y, x ∈ Zn

⟩is

an abelian normal subgroup of H;(ii) the quotient group H/O is isomorphic to a subgroup of Cm o Cn.

In particular, H is metabelian-by-finite.

Proof. (i) Recall

N =⟨[β|i, τ ki ] | ki ∈ Z, i ∈ Y

⟩,

K = N⟨β|jβ|j+s · · · β|j+(m−1)s | j ∈ Y

⟩where m = n

gcd(n,s). Then, by Proposition 9, N is an abelian normal subgroup

of H.By (18), we have

[β|i, τ z]β|jβ|j+s···β|j+(m−1)s

= [β|i+s, τ z]τ∆t(i+2s,j+s)β|j+s···β|j+(m−1)s

= [β|i+2s, τz]τ

∆s(i+2s,j+s)+∆s(i+3s,j+2s)β|j+2s···β|j+(m−1)s

= [β|i, τ z]τ∑m−1k=0

∆s(i+(k+1)s,j+ks)

Prop.8(v)= [β|i, τ z]

Thus,

(19) [[β|i, τ z], (βm)|j] = e, ∀i, j ∈ Y, ∀z ∈ Z



for all i, j ∈ Y.Further, by using equations (19),(20) (21), (24) and

(25) (βm)|i = β|iβ|i+s · · · β|i+(m−1)s,

we conclude that also K is an abelian normal subgroup of H.Now, O = K 〈τ〉 is metabelian. Moreover it is normal in H, because

τβ|i = ττ−1τβ|i = τ [τ, β|i] ∈ O

for all i ∈ Y .(ii) Consider the following Fibonacci type group

X =⟨b0, · · · , bn−1 | bibj+s = bjbi+s, bibi+s · · · bi+(m−1)s = e,∀i, j ∈ Y

⟩.

Equations (17) and (18) show thatH

Mis a homomorphic image of X. We

will prove that X is isomorphic to a subgroup ofthe wreath product Cm o Cn.As a matter of fact the group Cm o Cn has the presentation⟨

u, a | um = e, an = e, uai

uaj

= uaj

uai⟩

.

On defining b = asu−1, we have

um = e (a−sb)m = e

⇒ (a−sb · · · a−sb︸︷︷︸m terms

)a−s+i

= e

⇒ baibai+s · · · bai+(m−1)s

= e.

Also, the commutation relation

uai

uaj

= uaj

uai

implies

(b−1as)ai(b−1as)a

j= (b−1as)a

j(b−1as)a

i

⇒ (a−sb)aj(a−sb)a

i= (a−sb)a

i(a−sb)a

j

⇒ baja−sba

i= ba

ia−sba

j

⇒ bajbai+s

= baibaj+s

.

Thus, by using Tietze transformations we conclude that Cm o Cn has thepresentation

⟨a, b | an = e, ba

j

bai+s

= bai

baj+s

, bai

bai+s · · · bai+(m−1)s

= e, ∀i, j ∈ Y⟩

.

Then, on introducing bi = bai, i = 0, · · · , n − 1, the above presentation is

expressed as



⟨a, b0, · · · , bn−1 | an = e, bi = ba

i

0 , bjbi+s = bibj+s, bibi+s · · · bi+(m−1)s = e,

∀i, j ∈ Y 〉 .

The next results leads to a proof of Theorem C.

Lemma 7. Let σ = (0, 1, ..., n− 1) ∈ Σn and let L be the layer closure of 〈σ〉in An. Suppose β = (β|0, β|1, · · · , β|n−1)σβ ∈ L satisfies [β, βτ

x] = e for all

x ∈ Z. Write σβ = σs and σβ|i = σmi for all i ∈ Y . Then for all i, j ∈ Y , thefollowing congruence holds

(26) ∆s(i, t) +mi−s +mt ≡ mt−s +mi + ∆s(i+ s, t+ s) mod n,

Proof. Since σβ|i = σmi , we conclude by (17),

σ∆s(i,t)+mi−s+mt = σmt−s+mi+∆s(i+s,t+s)

and therefore, ∆s(i, t)+mi−s+mt ≡ mt−s+mi+∆s(i+s, t+s) mod n.

Lemma 8. Maintain the notation of the previous lemma and let n be an oddinteger. Then,

σ(βn)|0 = σ(β|0β|1···β|n−1) = σ.

Proof. From

∆1(i, t) +mi−1 +mt ≡ mt−1 +mi + ∆1(i+ 1, t+ 1) mod n

we concluden−2∑i=0

n−1∑t=i+1

(∆1(i, t) +mi−1 +mt

)≡

n−2∑i=0

n−1∑t=i+1

(mt−1 +mi + ∆1(i+ 1, t+ 1)) mod n.

Now,

n−2∑i=0

n−1∑t=i+1

∆1(i, t)Prop.8(i)

=n−1∑t=1

∆1(0, t)Prop.8(ii)

=n−1∑t=0

∆1(0, t)

Prop.8(ii)=

n−1∑t=0

−∆1(t, 0)Prop.8(vi)

= −(n− 1),

n−2∑i=0

n−1∑t=i+1

∆1(i+ 1, t+ 1)Prop.8(i)

=n−2∑i=0

∆1(i+ 1, 0)Prop.8(ii)

=n−1∑i=0

∆1(i, 0)

Prop.8(vi)= (n− 1),



n−2∑i=0

n−1∑t=i+1

(mi−1 +mt

)= 2(n− 1)mn−1 + (n− 2)

n−2∑k=0

mk

andn−2∑i=0

n−1∑t=i+1

(mt−1 +mi) = n

n−1∑k=0

mk.

Since n is odd, we haven−1∑k=0

mk ≡ 1 mod n

and therefore, σβ|0···β|n−1 = σ(m0+...mn−1) = σ.

We prove Theorem C below.

Theorem 2. Let n be an odd number, σ = (0, · · · , n − 1) ∈ Σn and let Lbe the layer closure of 〈σ〉 in An. Let s an integer relatively prime to n andβ = (β|0, β|1, · · · , β|n−1)σs ∈ L be such that [β, βτ

x] = e for all x ∈ Z. Then β

is a conjugate of τ in L.

Proof. We start with the case s = 1. The element

α(1) = (e, β|−10 , (β|0β|1)−1, · · · , (β|0 · · · β|n−2)−1) ∈ StabG(1)

conjugates β to

βα(1) = (e, · · · , e, β|0 · · · β|n−1)σ.

By Lemma8 we find σβ|0β|1···β|n−1 = σ. Moreover by Proposition 6,

[(βn)|0, (βn)|τx0 ] = [β|0β|1 · · · β|n−1, (β|0β|1 · · · β|n−1)τx

] = e,

for all integers x. Therefore β|0β|1 · · · β|n−1 satisfies the hypothesis of thetheorem. The process can be repeated until we obtain a sequence (α(k))k∈Nsuch that βα(1)α(2)···α(k)··· = τ, where α(k) ∈ StabG(k) satisfies α(k)|u = α(k)|vfor all u, v ∈M with |u| = |v| = k − 1.

Now, suppose more generally s is such gcd (s, n) = 1 and let k be aminimum positive integer for which sk ≡ 1 mod(n). Then βk satisfies thehypothesis of the first part and so, there exists α ∈ G such that (βk)α = τ .Since k is invertible in Zn, there exists an automorphism γ of the tree suchthat τ γ = τ k

−1. Thus, βαγ

−1= τ .

6. Solvable groups for n = p, a prime number.

We will prove in this section the case n = p of Theorem A.Let B be an abelian subgroup of Aut(Tp) normalized by τ and let β ∈ B.

By Lemma 5, σβ ∈ 〈στ 〉 and therefore in effect we have two cases, σβ = e, στ .

Proposition 10. Suppose σβ = στ . Then, σβ|i ∈ 〈στ 〉 for all i ∈ Y .



Proof. By theorem 1, O is a normal subgroup of H and HO

is isomorphic to asubgroup of Cp o Cp.

By Lemma 5, O is a subgroup of 〈στ 〉 modulo Stabp(1).Therefore, H is a p-group modulo Stabp(1) and by Lemma 5, we have σβ|i ∈

〈στ 〉.

Theorem 3. Let p be a prime number and β ∈ Aut(Tp) such that σβ = σsτ forsome integer s relatively prime to p. Suppose [β, βτ

x] = e for all x ∈ Z. Then

β is conjugate to τ in Aut(Tp).

Proof. Suppose s = 1. Recall that

α(1) = (e, β|−10 , (β|0β|1)−1, · · · , (β|0 · · · β|p−2)−1) ∈ StabG(1)

conjugates β to its normal form

βα(1) = (e, · · · , e, β|0 · · · β|p−1)σ.

By Lemma 8 we have σβ|0β|1···β|p−1 = στ . Moreover by Proposition 6,

[βp|0, (βp|0)τx

] = [β|0β|1 · · · β|p−1, (β|0β|1 · · · β|p−1)τx

] = e,

for all integers x. Therefore β|0β|1 · · · β|n−1 satisfies the condition of the the-orem.. This process can be repeated to produce a sequence (α(k))k∈N such

that βα(1)α(2)···α(k)··· = τ, where α(k) ∈ Stab(k) satisfies α(k)|u = α(k)|v for allu, v ∈M where |u| = |v| = k − 1.

Now, to the general case, s such gcd(p, s) = 1. Let k be the minimumpositive integer which is the inverse of s modulo p. Then, σ|βk = στ and βk

satisfies the hypotheses. Thus there exists α ∈ Ap such that(βk)α

= τ . Let

k−1 be the inverse of k in U (Zn); then βα = τ k−1

. There exists γ ∈ NAp< τ >

which conjugates τ to τ k−1

and so, (βα)γ−1

= τ .

Lemma 9. Let p be a prime number and β ∈ Aut(Tp) such that [β, βτx] = e

for all x ∈ Z. Then, there exists a tree level m and a conjugate µ of τ suchthat β ∈ ×pm〈µ〉 and there exists an index u of length m such that β|u = µ.

Proof. Let m be the minimum tree level such that σβ|u 6= e for some |u| = m.Therefore, σβ|u = σsτ for some integer s such that gcd (p, s) = 1 and so, µ = β|uis conjugate to τ in Aut(Tp). Since β ∈ Stab(m), by Proposition 6 [µ, β|v] = e

for all indices v such that |v| = m. Therefore, β|v ∈ 〈µ〉 for all v such that|v| = m.

Theorem 4. Let p be a prime number, σ = (0, 1, · · · , p − 1) ∈ Σp, F =NΣp (〈σ〉), Γ0 = NA(< τ > . Let G be a finitely generated solvable subgroup ofAut (Tp) which contains the p-adic adding machine τ . Then, there exists aninteger t ≥ 1 such that G is conjugate to a subgroup of

×p (· · · (×p (×pΓ0 o F )o) · · · ) o F .



Proof. We may suppose G has derived length d ≥ 2. Let B be the (d− 1)-thterm of the derived series of G. By Theorem 9, there exists a level t such thatB is a subgroup of V = ×pt〈µ〉 where µ = τα for some α ∈ Aut (Tn).

We will show that G is a subgroup of

J = ×p (· · · (×p (×p (Γ0)α o Σp) o Σp) · · · ) o Σp,

where ×p appears t times.

Let γ ∈ G\J . Then there exists an index w of length t such that γ|w 6∈ (Γ0)α.Since τ is transitive on all levels of the tree , by Theorem 9, there exists β ∈ Bsuch that β|w = µη for some η ∈ U(Zp).

Write v = wγ. Then,

(βγ) |v(9)=(β|vγ−1

)γ|vγ−1 = (β|w)γ|w 6∈ 〈µ〉,

and this implies βγ 6∈ B ≤ 〈µ〉 and γ 6∈ G. Hence, G is a subgroup of J .Now, since G is a solvable group containing τ , there exist Gi (0 ≤ i ≤ t)

solvable subgroups of Σp containing σ = (0, 1, · · · , p − 1) such that G is asubgroup of

Rt (α) = ×p (· · · (×p (×p (Γ0)α oG1) oG2) · · · ) oGt.

Since for all i, we have Gi ≤ F we may substitute the G′is by F . Finally,Rt (α) is a conjugate of Rt (1) by the diagonal automorphism α(t).

7. Two cases for n even

7.1. The case σβ = (στ )n2 .

Theorem 5. Let n be an even number, β ∈ An such that σβ = σn2τ and

[β, βτx

] = e for all x ∈ Z. Then H = 〈β|i (0 ≤ i ≤ n− 1) , τ〉 is a metabeliansubgroup of An.

Proof. Define the subgroup

R =⟨[β|t, τ k], β|iβ|i+n

2, β|2jτ−∆(j,j+n

2) | k ∈ Z and i, j, t ∈ Y

⟩.

Denote ∆n2(i, j) by ∆(i, j).

We will prove that N is an abelian normal subgroup of H.

(I) R is normal in H :

–⟨[β|i, τ k]

⟩H ≤ R :

[β|i+n2, τ k]β|j

(18)= [β|i, τ k]τ

∆(j,i)

;

–⟨β|iβi+n

2

⟩H ≤ R :

(β|iβ|i+n2)τk

=(β|iβ|i+n

2

).[β|iβ|i+n

2, τ k]

=(β|iβ|i+n

2

)[β|i, τ k]β|i+

n2 [β|i+n

2, τ k]



(18)=(β|iβ|i+n

2

)[β|i+n

2, τ k]τ

∆(i+n2 ,i+

n2 )

[β|i+n2, τ k]

Prop.8=

β|iβ|i+n2[β|i+n

2, τ k]2

(27) (β|iβ|i+n2)β|j =

(β|−1j β|iβ|i+n

2β|j)τ∆(j+n

2,i+n

2)τ−∆(j+n

2,i+n

2)

(17)=(β|−1j β|i

)τ∆(j,i)

(β|j+n

2β|i)τ−∆(j+n

2,i+n

2)

=(β|−1j β|iβ|j+n

2

)τ∆(j,i).[τ∆(j,i), β|j+n

2].β|iτ−∆(j+n

2,i+n

2)

(17)=(β|−1j

)τ∆(j+n

2,i+n

2)(β|jβ|i+n

2

).

[τ∆(j,i), β|j+n2].β|iτ−∆(j+n

2,i+n

2)

= τ∆(j+n2,i+n

2).[τ∆(j+n

2,i+n

2), β|j].

β|i+n2[τ∆(j,i), β|j+n

2]β|iτ−∆(j+n

2,i+n

2)

Prop.8= τ−∆(j,i)[τ−∆(j,i), β|j].β|i+n

2.

[τ∆(j,i), β|j+n2]β|iτ∆(j,i)

(18)= τ−∆(j,i)β|i+n

2.[τ−∆(j,i), β|j+n

2]τ

∆(j,i)

.[τ∆(j,i), β|j+n2].β|iτ∆(j,i)

(14)=(β|i+n

2β|i)τ∆(j,i)

.

–⟨β|2jτ−∆(j,j+n

2)⟩H ≤ R :

(β|2jτ−∆(j,j+n2

))τk

= β|2jτ−∆(j,j+n2

).[β|2jτ−∆(j,j+n2

), τ k]

= β|2jτ−∆(j,j+n2

).[β|2j , τ k]τ−∆(j,j+n

2 )

= β|2jτ−∆(j,j+n2

)([β|j, τ k]β|j .[β|j, τ k]

)τ−∆(j,j+n2 )

(18)= β|2jτ−∆(j,j+n

2)(

[β|j+n2, τ k]τ

∆(j,j+n2 )

.[β|j, τ k])τ−∆(j,j+n

2 )

= β|2jτ−∆(j,j+n2

)[β|j+n2, τ k][β|j, τ k]τ

−∆(j,j+n2 )

.

By Proposition 8 and 9, we can show

(28)(β|2jτ−∆(j,j+n

2))β|i

=(β|2j+n

2τ−∆(j+n

2,j)[τ−∆(j+n

2,j), β|j+n

2])τ∆(i,j)

.

(II) The subgroup R is abelian:

(29) [β|i, τ k]β|jτt Prop.9

= [β|i, τ k]τtβ|j ;

(30) [β|i, τ k]β|jβ|j+n2

(18)= [β|i+n

2, τ k]

τ∆(j,i+n2 )β|j+n

2(29)= [β|i+n

2, τ k]

β|j+n2τ∆(j,i+n

2 )



Prop.8=

(α.[τ−∆(j+n

2,j), β|j]τ

∆(j+n2 ,j))(τ∆(i,i+n

2 )[τ∆(i,j),β|i]τ−∆(i,i+n2 ))

(32)=(β|2jτ−∆(j,j+n

2)[τ−∆(j,j+n

2), β|j][τ∆(j+n

2,j), β|j]−1

)[τ∆(i,j),β|i]τ−∆(i,i+n

2 )

Prop.8=

(β|2jτ−∆(j,j+n

2))[τ∆(i,j),β|i]τ

−∆(i,i+n2 )

Prop.9 e (31)= β|2jτ−∆(j,j+n

2).

Moreover, since

R (β|i)R (β|j) = R (β|i) (β|j)Prop.5

= Rτ∆(j,i+n2

)β|j+n2β|i+n

2τ∆(j,i+n

2)

= Rβ|j+n2β|i+n

2τ 2∆(j,i+n

2) = Rβ|−1

j β|−1i τ 2∆(j,i+n

2)

= Rβ|−1j β|2jτ−∆(j,j+n

2)β|−1

i β|2i τ−∆(i,i+n2

)τ 2∆(j,i+n2

)

= Rβ|jβ|iτ−∆(j,j+n2

)−∆(i,i+n2

)+2∆(j,i+n2

)

Prop.8= Rβ|jβ|i = Rβ|jNβ|i

andRβ|i = Rβ|−1

i+n2, Rβ|2i = Rτ∆(i,i+n

2),∀i, j ∈ Y,

we concludeH

Ris a homomorphic image of

Z× C2 × · · · × C2︸︷︷︸n2

terms

.

7.2. The case σβ transposition. We prove in this section part (II ) (ii) ofTheorem B.

Theorem 6. Let n be an even number and B an abelian subgroup of Annormalized by τ . Suppose β = (β|0, β|1, · · · , β|n−1)σβ ∈ B where σβ is atransposition. Then H = 〈β|i (0 ≤ i ≤ n− 1) , τ〉 is a metabelian group.

We prove progressively that

N =⟨[β|i, τ k] | k ∈ Z, i ∈ Y

⟩,

U =⟨N, β|j | j 6= 0,

n

2

⟩,

V =⟨U, β|n

2β|0, τ (β|0)2⟩

are normal abelian subgroups of H, from which it follows that HV

is cyclic andtherefore H metabelian.



Lemma 10. The degree of the tree n is even and σβ is 〈στ 〉-conjugate to thetransposition

(0, n

2

).

Proof. On conjugating by an appropriate power of στ , we may assume σβ =(0, j). The conjugate of σβ by σiτ is the transposition (i, j + i). In particular,(j, 2j) is a conjugate which is supposed to commute with (0, j). Therefore,0, j = j, 2j, 2j = 0 modulo(n), n = 2n′ and j = n′.

We go back to part (I) of the Proposition 7,(τ v|(i)σ−vτ

)−1 (β|(i)σ−vτ

)(τ v|(i)σ−vτ σβ

)(β|(i)σ−vτ σβσvτ

)= (β|i)

(τ v|(i)σβσ−vτ

)−1 (β|(i)σβσ−vτ

)(τ v|(i)σβσ−vτ σβ

)and set in it j = (i)σ−vτ , v = kn+ r, r = v to obtain

(τ v)|−1j β|j(τ v)|(j)σββ|(j)σβσvτ(33)

= β|(j)σvτ (τv)|−1

(j)σvτσβσ−vτβ|(j)σvτσβσ−vτ (τ v)(j)σvτσβσ

−vτ σβ

.(34)

Proposition 11. The following cases hold for different pairs (j, r).

• For j = 0 there are 3 subcases– If r = 0, then

(35) [β|0, τ k]β|n2 = [β|n

2, τ k], ∀k ∈ Z;

– If r = n2, then

(36) β|0τβ|0 = β|n2τ−1β|n

2,

and

(37) [β|0, τ k]τβ|0 = [β|n2, τ k],∀k ∈ Z.

– If r 6= 0 and r 6= n2, then

(38) τ δ(n2,r)β|0β|n

2+r = β|rτ δ(

n2,r)β|0,∀r ∈ Y − 0,

n

2

and

(39) [β|0, τ k]β|r = [β|0, τ k],∀k ∈ Z.

• For j = n2

there are 3 subcases– If r = 0, then

(40) [β|n2, τ k]β|0 = [β|0, τ k], ∀k ∈ Z;

– If r = n2, then

(41) τ−1β|2n2

= β|20τ,



and

(42) [β|n2, τ k]

β|n2τ−1

= [β|0, τ k],∀k ∈ Z;

– If r 6= 0 and r 6= n2, then

(43) τ−δ(n2,r)β|n

2β|r = β|n

2+rτ

−δ(n2,r)β|n

2, ∀r ∈ Y − 0, n

2

and

(44) [β|n2, τ k]β|r = [β|n

2, τ k],∀k ∈ Z, ∀r ∈ Y − 0, n

2.

• For j 6= 0 and j 6= n2, there are 5 subcases:

– If j 6= n− r and j 6= n2− r, then

(45) β|jβt = β|tβ|j,∀j, t ∈ Y − 0,n

2

and

(46) [β|j, τ k]β|t = [β|j, τ k],∀j, t ∈ Y − 0,n

2

– If j = n− r and 0 < r < n2, then

(47) τ−1β|j+n2τβ|0 = β|0β|j, ∀j ∈ 1, 2, · · · ,

n

2− 1

and

(48) [β|j+n2, τ k]τβ|0 = [β|j, τ k],∀j ∈ 1, 2, · · · ,

n

2− 1

– If j = n− r and n2< r ≤ n− 1, then

(49) β|jβ|0 = β|0β|n2

+j,∀j ∈ 1, · · · ,n

2− 1

and

(50) [β|j, τ k]β|0 = [β|n2

+j, τk],∀k ∈ Z, ∀j ∈ 1, · · · , n

2− 1

– If j = n2− r and 0 < r < n

2, then

(51) β|jβ|n2

= β|n2τ−1β|j+n

2τ, ∀j ∈ 1, · · · , n

2− 1

and

(52) [β|j, τ k]β|n2τ−1

= [β|n2

+j, τk],∀k ∈ Z,∀j ∈ 1, · · · , n

2− 1

– If j = n2− r and n

2< r ≤ n− 1, then

(53) β|n2β|j = β|n

2+jβ|n

2,∀j ∈ 1, · · · , n

2− 1

and

(54) [β|j, τ k] = [β|n2

+j, τk]β|n

2 ,∀k ∈ Z,∀j ∈ 1, · · · , n2− 1.



Proof. We will prove just the last case. As j 6∈ 0, n2, n− r, n

2− r, we have

(j)σvτ = (j)σβσvτ = j + r,

(j)σβ = (j)σvτσβσ−vτ = (j)σvτσβσ

−vτ σβ = j.

Therefore,((τ v)|−1

j β|j(τ v)|jβ|j+r = β|j+r(τ v)|−1j β|j(τ v)j,∀v ∈ Z

)⇔

(τ−k−δ(j,r)β|jτ k+δ(j,r)β|j+r = β|j+rτ−k−δ(j,r)β|jτ k+δ(j,r),∀k ∈ Z

)⇔

(β|j[β|j, τ k+δ(j,r)]β|j+r = β|j+rβ|j[β|j, τ k+δ(j,r)], ∀k ∈ Z

),

(55) β|jβt = β|tβ|j,∀j, t ∈ Y − 0,n

2

and

(56) [β|j, τ k]β|t = [β|j, τ k], ∀j, t ∈ Y − 0,n

2.

Lemma 11. The group N =⟨[β|i, τ k] | k ∈ Z, i ∈ Y

⟩is an abelian normal

subgroup of H.

Proof. DefineNi =

⟨[β|i, τ k] | k ∈ Z

⟩for each i ∈ Y . Then, N = 〈Ni | i ∈ Y 〉, each Ni is an abelian subgroupnormalized by τ and

(57) [β|i, τ k]β|−1j = [β|i, τ k],∀k ∈ Z,∀i, j ∈ Y, j 6= 0,

n

2

We have [Ni, Nj] = 1,∀i, j ∈ Y, j 6= 0, n2, because

[β|i, τ k][β|j ,τt] = [β|i, τ k]β|

−1j τ−tβ|jτ t (57)

= [β|i, τ k]τ−tβ|jτ t

(14)=([β|i, τ−t]−1[β|i, τ k−t]

)β|jτ t(57)=([β|i, τ−t]−1[β|i, τ k−t]

)τ t(14)= [β|i, τ k]τ

−tτ t = [β|i, τ k],∀k, t ∈ Z,

∀i, j ∈ Y, j 6= 0, n2.

Furthermore, [N0, Nn2] = 1, because

[β|n2, τ k][β|0,τ

t] = [β|n2, τ k]β|

−10 τ−tβ|0τ t (37)

= [β|0, τ k]ττ−tβ|0τ t

(14)=([β|0, τ−t]−1[β|0, τ k−t]

)τβ|0τ t(37)=([β|n

2, τ−t]−1[β|n

2, τ k−t]

)τ t



(14)= [β|n

2, τ k]τ

−tτ t = [β|n2, τ k],∀k, t ∈ Z.

Therefore N is abelian.Now, equation (57) implies

(58) Ni = Nβ|ji = N

β|−1j

i , ∀i, j ∈ Y, j 6= 0,n

2;

equations (14), (35) imply

(59)Nn

2= N

β|00 , N0 = N

β|−10

n2

;

equation (40) implies

(60)N0 = N

β|0n2, Nn

2= N

β|−10

0 ;


(61)

N0 = N

β|n2

n2, Nn

2= N

β|−1n2

0 ;


(62)Nj = N

β|0j+n

2, Nj+n

2= N

β|−10

j ,∀j ∈ 1, · · · , n2− 1;

equations (14) and (50) imply

(63)Nj+n

2= N

β|0j , Nj = N

β|−10

j+n2,∀j ∈ 1, · · · , n

2− 1;

equations (14) (52) imply

(64)

Nj+n

2= N

β|n2

j , Nj = Nβ|−1n2

j+n2,∀j ∈ 1, · · · , n

2− 1;


(65)

Nj = N

β|n2

j+n2, Nj+n

2= N

β|−1n2

j ,∀j ∈ 1, · · · , n2− 1.

Thus (57)-(65) prove

N = 〈Ni | i ∈ Y 〉=

⟨[β|i, τ k] | ∀i, k ∈ Z

⟩is an abelian normal subgroup of H.

Lemma 12. The group U =⟨N, β|j | j 6= 0, n

2

⟩is a normal abelian subgroup

of H.



Proof. Lemma 11 and equations (39), (44), (45) and (46) show that U isabelian.

The fact that N is normal in H, together with the following assertions provethat U is normal in H.

Let J =⟨β0, βn

2, τ⟩. Then, for j ∈ Y − 0, n

2, we have

(I) 〈β|j〉J ≤ U :

β|τ tj = β|j[β|j, τ t];

β|β|0j

(49)= β|j+n

2;

β|β|−10

j

(47)= τ−1β|j+n

2τ = β|j+n

2[β|j+n

2, τ ];

β|β|n

2j

(51)= τ−1β|j+n

2τ = β|j+n

2[β|j+n

2, τ ];

β|β|−1n2

j

(53)= β|j+n

2;

(II)⟨β|j+n

2

⟩J ≤ U :

β|τ tj+n2

= β|j+n2[β|j+n

2, τ t];

β|β|0j+n2

(47)= β|−1

0 τβ|0β|jβ|−10 τ−1β|0

=([β|0, τ ]−1

)τ−1

β|τ−1

j [β|0, τ ]τ−1 ∈ U ;

β|β|−10

j+n2

(49)= β|j ∈ U ;

β|β|n

2

j+n2

(53)= β|j ∈ U ;

β|β|−1n2

j+n2

(51)= β|n

2τβ|−1

n2β|jβ|n

2τ−1β|−1

n2

= [β|n2, τ ]

β|−1n2τ−1

β|τ−1

j

([β|n

2, τ ]−1

)β|−1n2τ−1

.

Hence, U is a normal abelian subgroup of H.

Lemma 13. V =⟨U, β|n

2β|0, τβ|20

⟩is a normal abelian subgroup of H.

Proof. Lemma 12 together with the following assertions prove that V is anormal abelian subgroup of H.

Given j ∈ Y − 0, n2, k ∈ Z, and J =

⟨β|0, βn

2, τ,⟩, we prove

(I) β|n2β|0 ∈ CH(U) :

(β|j)β|n2β|0 (51)

= (β|j+n2)τβ|0

(47)= β|j;

(β|j+n2)β|n

2β|0 (53)

= (β|j)β|0(49)= β|j+n

2;

[β|j, τ k]β|n2β|0 = [β|j, τ k]β|

n2τ−1τβ|0 (52)

= [β|j+n2, τ k]τβ|0



(48)= [β|j, τ k];

[β|j+n2, τ k]

β|n2β|0 (54)

= [β|j, τ k]β|0(50)= [β|j+n

2, τ k];

[β|0, τ k]β|n2β|0 (35)

= [β|n2, τ k]β|0

(40)= [β|0, τ k];

[β|n2, τ k]

β|n2β|0 = [β|n

2, τ k]

β|n2τ−1τβ|0

(42)= [β|0, τ k]τβ|0

(37)= [β|n

2, τ k];

(II) τβ|20 ∈ CH(U) :

β|τβ|20

j = (β|j[β|j, τ ])β|20 = (β|β|0j [β|j, τ ]β|0)β|0

(49),(50)= (β|j+n

2[β|j+n

2, τ ])β|0 = β|τβ|0j+n

2

(47)= β|j;

(β|j+n2)τβ|

20

(47)= β|β|0j

(49)= β|j+n

2;

[β|0, τ k]τβ|20

(37)= [β|n

2, τ k]β|0

(40)= [β|0, τ k];

[β|n2, τ k]τβ|

20

(14)= ([β|n

2, τ ]−1[β|n

2, τ k+1])β|

20

(40)= ([β|0, τ ]−1[β|0, τ k+1])β|0

(14)= [β|0, τ k]τβ|0

(37)= [β|n

2, τ k];

[β|j, τ k]τβ|20

(14)= ([β|j, τ ]−1[β|j, τ k+1])β|

20

(50)= ([β|j+n

2, τ ]−1[β|j+n

2, τ k+1])β|0

(14)= [β|j+n

2, τ k]τβ|0

(48)= [β|j, τ k];

[β|j+n2, τ k]τβ|

20

(48)= [β|j, τ k]β|0

(50)= [β|j+n

2, τ k];

(III) τβ|20 ∈ CH(β|n2β|0) :

(β|n2β|0)τβ|

20 = β|−2

0 τ−1β|n2β|0τβ|20

(36)= β|−2

0 τ−1β|n2β|n

2τ−1β|n

2β|0

= β|−20 τ−1β|2n

2τ−1β|n

2β|0 = (τβ|20)−1β|2n

2τ−1β|n

2β|0

(41)= β|n

2β|0;



(IV)⟨β|n

2, β|0

⟩J ≤ V :

(β|n2β|0)τ

k

= β|n2β|0[β|n

2β|0, τ k] = β|n

2β|0[β|n

2, τ k]β|0 [β|0, τ k];

(β|n2β|0)β|0 = β|−1

0 β|n2β|20 = β|−1

0 β|n2τ−1τβ|20 = β|−1

0 β|−1n2β|2n

2τ−1τβ|20

= (β|n2β|0)−1(τβ|20)2;

β|n2β|0

(t)= (τβ|20)2((β|n

2β|0)−1)β|0 ;

(β|n2β|0)β|

−10

(u)= ((τβ|20)2)β|

−10 (β|n

2β|0)−1;

(β|n2β|0)

β|−1n2 = β|2n

2β|0β|−1

n2

= β|2n2τ−1τβ|0β|0β|−1

0 β|−1n2

(41)= (τβ|20)2β|−1

0 β|−1n2

= (τβ|20)2(β|n2β|0)−1;

(β|n2β|0)

β|n2

(x)= (β|n

2β|0)−1((τβ|20)2)

β|n2

.(V) 〈τβ|20〉

J ≤ V :

(τβ|20)τk

= τ(β|20)τk

= τβ|20[β|20, τ k] = τβ|20[β|0, τ k]β|0 [β|0, τ k];

(τβ|20)β|0 = β|−10 τβ|20β|0 = ττ−1β|−1

0 τβ|0β|20 = τ [τ, β|0]β|20

= τ [τ, β|0]τ−1τβ|20 = ([β|0, τ ]−1)τ−1

τβ|20;

(τβ|20)β|−10 = β|0τβ|0 = τβ|0[β|0, τ ]β|0 = τβ|20[β|0, τ ]β|0 ;

(τβ|20)β|−1n2

(p)=(

(τβ|20)β|−10 ([β|0, τ ]−1)β|0

)β|−1n2

= (τβ|20)β|−1

0 β|−1n2 ([β|0, τ ]−1)

β|0β|−1n2

= (τβ|20)(β|n

2β|0)−1

([β|0, τ ]−1)β|0β|−1

n2

(g)= τβ|20([β|0, τ ]−1)

β|0β|−1n2 ;

(τβ|20)β|n

2(q)= τβ|20[β|0, τ ]β|0 .



8. Solvable groups for n = 4.

Let B be an abelian subgroup of A4 = Aut(T4) normalized by τ and letβ ∈ B. Then, by Proposition 5 , σβ ∈ D = 〈(0, 1, 2, 3), (0, 2)〉, the uniqueSylow 2-subgroup of Σ4 which contains σ = στ = (0, 1, 2, 3).

The normalizer of 〈τ〉 here is Γ0 = NA4

(〈τ〉)

= 〈Λ, ι〉 where Λ is the monic

normalizer and where ι = ι(1) (0, 3) (1, 2) inverts τ .Given a group W , the subgroup generated by the square of its elements is

denoted by W 2.

Lemma 14. Let L = L (D) be the layer closure of D above. If γ ∈ L2 thenγτ is conjugate to τ .

Proof. If α ∈ L then σα2 ∈ 〈σ2〉 and the product in any order of the states(α2) |i (0 ≤ i ≤ 3) belongs to S = L2.

Let γ ∈ S. Then γτ is transitive on the 1st level of the tree and (γτ)4 isinactive with conjugate 1st level states, where the first state is

(γ|0) (γ|1) (γ|2) (γ|3) τ if σγ = e,

and

(γ|0) (γ|3) (γ|2) (γ|1) τ if σγ = σ2;

in both cases the element is contained in S2τ . Therefore, γτ is transitive onthe 2nd level of the tree. Now use induction to prove that γτ is transitive onall levels of the tree.

8.1. Cases σβ ∈ (0, 3)(1, 2), (0, 1)(2, 3). We will show that these cases can-not occur. We note that στ conjugates (0, 1)(2, 3) to (0, 3)(1, 2). Since theargument for β applies to βτ , it is sufficient to consider the first case.

Suppose σβ = (0, 1)(2, 3). Then,

βτ =(τ−1 (β|3) , β|0, β|1, β|2τ

)(σβ)στ .

On substituting α = βτ in θ = [β, α] and in (7)

(66) θ|(i)σαβ =(β|(i)σα

)−1(α|i)−1 (β|i)

(α|(i)σβ

),∀i ∈ Y .

we get θ = e and

(67) e =(β|(i)σβτ

)−1(βτ |i)−1 (β|i)

(βτ |(i)σβ

),∀i ∈ Y

and so for the index i = 0, we obtain

e = (β|3)−1 (τ−1 (β|3))−1

(β|0) (β|0) ,

e = (β|3)−2 τ (β|0)2

which is impossible.



8.2. Cases σβ ∈ (0, 2), (1, 3).

Lemma 15. Let α, γ ∈ Aut(T4) be such that

σα, σγ ∈ 〈(0, 1, 2, 3), (0, 2)〉 ,τ−1α2 = γ2τ,

[α, τ k]γ = [γ, τ k]

for all k ∈ Z. Then,

σα, σγ ∈ 〈σ〉 , σασγ = σ±1.

Proof. From the second and third equations above, we have σ−1σ2α = σ2

γσ and

[σα, σk]σγ = [σγ, σ

k].(i) Suppose σ2

γ = e. Then σ2α = σ2 and therefore, σα = σ±1, [σα, σ

k]σγ =

[σγ, σk] = e for all k; thus, σγ ∈ 〈σ〉 and σγ ∈ 〈σ2〉, σασγ = σ±1 follows.

(ii) Suppose o(σγ) = 4. Then, σγ = σ±1 and σ2α = e. Since [σα, σ

k]σγ = e forall k, we obtain σα ∈ 〈σ〉, σ2

α = e and σα ∈ 〈σ2〉 .Therefore, σασγ = σ±1.

(1) Suppose σβ = (0, 2). Then by the analysis in Section 7.2, we conclude

V =⟨[β|i, τ k], β|1, β|3, β|2β|0, τβ|20 | i ∈ Y

⟩is an abelian normal subgroup of H.

By Lemma 14 , τβ|20 = µ is a conjugate of τ . As V is abelian, there existξ, t1, t2 ∈ Z4 such that

µ = τβ|20, β|2β|0 = µξ, β|1 = µt1 , β|3 = µt2 .

Therefore,

β|2 = µξβ|−10 , τ = µβ|−2

0 .

On substituting γ = β0 and α = β2 in Lemma 15, we obtain σαγ = σβ|2β|0 =σ±1. Thus, from β|2β|0 = µξ, we reach ξ ∈ U(Z4).

By (41), we have

β|22τ−1 = τβ|20.

It follows then that

µξβ|−10 µξβ|−1

0 β|20µ−1 = µ,(µξ)β|0

= µ2−ξ.

Therefore,

(68) µβ|0 = µ2−ξξ

where 2−ξξ∈ Z1

4.

By Equation (49) we have

β|β|01 = β|3.



It follows that (µt1)β|0 = µt2 , µt1

2−ξξ = µt2 , t2 = t1

2− ξξ

.

We have reached the form of β,

β = (β|0, µt1 , µξβ|−10 , µt1

2−ξξ )(0, 2)

where µ = τα for some α ∈ Aut(T4).Now, since

β|0 =(λ 2−ξ

ξτm)α

for some m ∈ Z4, we have

µt1 = (τ t1)α,

µξβ|−10 =

(τ ξ(λ 2−ξ

ξτm)−1)α

=(λ ξ

2−ξτ (ξ−m) ξ

2−ξ

)α.

Thus

β = (λ 2−ξξτm, τ t1 , λ ξ


2−ξ , τ t12−ξξ )α

(1)

(0, 2)

andτ = µβ|−2

0

=

(τ(λ 2−ξ

ξτm)−2)α

=(λ( ξ

2−ξ )2τ(1− 2m

ξ )( ξ2−ξ)

2)αWe note that in case ξ = 1 and β has the form

β = (τm, τ t1 , τ 1−m, τ t1)α(1)

(0, 2)

where τ = (τ 1−2m)α; therefore,

β = (τm

1−2m , τt1

1−2m , τ1−m1−2m , τ

t11−2m )(0, 2).

(2) Suppose σβ = (1, 3). Then, γ = βτ satisfies [γ, γτk] = e. Therefore, the

previous case applies and

γ = (λ 2−ξξτm, τ t1 , λ ξ


2−ξ , τ t12−ξξ )α

(1)

(0, 2),

where

τ =(λ( ξ

2−ξ )2τ(1− 2m

ξ )( ξ2−ξ)

2)α= (e, e, e,

(λ( ξ

2−ξ )2τ(1− 2m

ξ )( ξ2−ξ)

2)α)στ .

Hence, β has the form

β = γτ−1

= (τ t1 , λ 2−ξξτ 1+m−ξ, τ t1

2−ξξ , λ ξ

2−ξτ (1−m) ξ

2−ξ )α(1)

(1, 3).



8.3. The case σβ = (στ )2 = (0, 2) (1, 3). We know that

V =⟨N, β|iβ|i+2, β|2jτ−∆(j,j+2) | i, j, t ∈ Y and k ∈ Z

⟩is an abelian normal subgroup of H and

(69) τ∆(i,j)β|i+2β|jτ∆(i,j) = β|j+2β|i,by analysis of the case 7.1.

From Lemmas 12 and 13, we have

τβ|20 = µ, β|2β|0 = µξ0 , β|3β|1 = µξ1 , τβ|21 = µξ2

where µ = τα and ξ0, ξ1, ξ2 ∈ U(Z4). Therefore,

(70) τ = µβ|−20

(71) β|2 = µξ0β|−10

(72) β|3 = µξ1β|−11

(73) τ = µξ2β|−21 .

Now, we let i, j take their values from Y in (69). Note that (i, j) and (j, i)produce equivalent equations and the case where i = j is a tautology. Thus wehave to treat the cases (i, j) = (0, 1) , (0, 2) , (1, 3) , (2, 3) , (0, 3) , (1, 2). Indeed,the last two cases turn out to be superfluous.

(i) Substitute i = 0, j = 2 in (69), to obtain

(74) β|22τ−1 = τβ|20Use (70) and (71) in (74) to get

µξ0β|−10 µξ0β|−1

0 β|20µ−1 = µ

and so,

(µξ0)β|0 = µ2−ξ0 .

Therefore,

(75) µβ|0 = µ2−ξ0ξ0

Since 2−ξ0ξ0∈ Z1

4, we find

(76) β|0 =

(λ 2−ξ0

ξ0

τm0

)α.

From (71),

(77) β|2 = µξ0β|−10 =

(τ ξ0τ−m0λ ξ0

2−ξ0

)α=

(λ ξ0

2−ξ0τ

(ξ0−m0)ξ0

2−ξ0

)α.



(ii) Substitute i = 1, j = 3 in (69) to get

(78) β|23τ−1 = τβ|21.

On using (72) and (73) in (78), we obtain

µξ1β|−11 µξ1β|−1

1 β|21µ−ξ2 = µξ2

and so,

(µξ1)β|1 = µ2ξ2−ξ1 .

Therefore,

(79) µβ|1 = µ2ξ2−ξ1ξ1 .

Since 2ξ2−ξ1ξ1∈ Z1

4, we have

(80) β|1 =

(λ 2ξ2−ξ1

ξ1

τm1

)α.

By (72), we find

(81) β|3 = µξ1β|−11 =

(τ ξ1τ−m1λ ξ1

2ξ2−ξ1

)α=

(λ ξ1

2ξ2−ξ1τ

(ξ1−m1)ξ1

2ξ2−ξ1

)α.

(iii) Substitute i = 0, j = 1 in (69) to get

(82) β|2β|1 = β|3β|0.

Use (76), (77), (80) and (81) in (82), to obtain

λ ξ02−ξ0

τ(ξ0−m0)

ξ02−ξ0 λ 2ξ2−ξ1

ξ1

τm1 = λ ξ12ξ2−ξ1

τ(ξ1−m1)

ξ12ξ2−ξ1 λ 2−ξ0

ξ0

τm0

and so,

λ ξ02−ξ0

2ξ2−ξ1ξ1

τ(ξ0−m0)

ξ02−ξ0

2ξ2−ξ1ξ1

+m1 = λ ξ12ξ2−ξ1

2−ξ0ξ0

τ(ξ1−m1)

ξ12ξ2−ξ1

2−ξ0ξ0

+m0 .

Therefore,

(83)

(ξ1

2ξ2 − ξ1

)2

=

(ξ0

2− ξ0

)2

and

(84) (ξ0 −m0)ξ0

2− ξ0

2ξ2 − ξ1

ξ1

+m1 = (ξ1 −m1)ξ1

2ξ2 − ξ1

2− ξ0

ξ0

+m0.

(iv) Substitute i = 2, j = 3 in (69) to get

(85) β|0β|3 = β|1β|2.



Use (76), (77), (80) and (81) in (85), to obtain

λ 2−ξ0ξ0

τm0λ ξ12ξ2−ξ1

τ(ξ1−m1)

ξ12ξ2−ξ1 = λ 2ξ2−ξ1

ξ1

τm1λ ξ02−ξ0

τ(ξ0−m0)

ξ02−ξ0

and so,

λ ξ02−ξ0

ξ12ξ2−ξ1

τm0

ξ12ξ2−ξ1

+(ξ1−m1)ξ1

2ξ2−ξ1 = λ 2ξ2−ξ1ξ1

ξ02−ξ0

τm1

ξ02−ξ0

+(ξ0−m0)ξ0

2−ξ0 .

Therefore, (ξ1

2ξ2 − ξ1

)2

=

(ξ0

2− ξ0

)2

and

(86) m0ξ1

2ξ2 − ξ1

+ (ξ1 −m1)ξ1

2ξ2 − ξ1

= m1ξ0

2− ξ0

+ (ξ0 −m0)ξ0

2− ξ0

.

We have from (83)

(87)ξ0

2− ξ0

= ± ξ1

2ξ2 − ξ1

.

(a) Ifξ0

2− ξ0

=ξ1

2ξ2 − ξ1

,

then

2ξ2ξ0 − ξ1ξ0 = 2ξ1 − ξ1ξ0,

and so,

(88) ξ2 =ξ1

ξ0

.

From (84), we get

(89) m1 =ξ1 − ξ0

2+m0.

(b) Ifξ0

2− ξ0

= − ξ1

2ξ2 − ξ1

then by (84) and (86),

m0 − ξ0 +m1 = m1 − ξ1 +m0

m0 + ξ1 −m1 = −m1 − ξ0 +m0,

which implies ξ1 = ξ0 = 0,which is impossible.Now by (88) and (89), we have

(90) β|1 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0

)α



and

(91) β|3 =

(λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α.

Therefore,

β = (β|0, β|1, β|2, β|3)(0, 2)(1, 3)

where β|0, β|1, β|2 and β|3 are described in (76),(90), (77) and (91), respec-tively, and

τ = µβ|−20

=

(τ

(λ 2−ξ0

ξ0

τm0

)−2)α

=

(λ

(ξ0

2−ξ0)2τ

(1− 2m0

ξ0

)(ξ0

2−ξ0

)2)α

.

(v) The cases (i, j) = (1, 2) , (0, 3) in (69) do not add any more informationabout β.

Summarizing, we have found

(92) β|0 =

(λ 2−ξ0

ξ0

τm0

)α, β|1 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0

)α,

(93) β|2 =

(λ ξ0

2−ξ0τ

(ξ0−m0)ξ0

2−ξ0

)α, β|3 =

(λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α,

(94) τ =

(λ

(ξ0

2−ξ0)2τ

(1− 2m0

ξ0

)(ξ0

2−ξ0

)2)α

.

In the particular case where ξ0 = 1, β has the form

β = (τm0

1−2m0 , τ

ξ1−12 +m01−2m0 , τ

1−m01−2m0 , τ

ξ1+12 −m01−2m0 )(0, 2)(1, 3)

where τ = (τ 1−2m0)α.

8.4. Cases σβ ∈ e, στ , σ−1τ . (1) Suppose σβ = e and let β stabilize the kth

level of the tree. Then by Proposition 6, we have

[β|u, β|τξ

v ] = e, for all u, v ∈M with |u| = |v| = k.

Therefore, N = 〈β|w | |w| = k, w ∈M〉 is abelian and so is its normal clo-

sure M under⟨N , τ

⟩. Also, active elements in M are characterized in 8.1,

8.2, 8.3 and 8.4. In particular, there exists κ ∈ M such that σκ = (0, 2)(1, 3)and β ∈ ×pkC(κ).



(2) Suppose σβ = στ = (0, 1, 2, 3). Then, clearly the element

β2 = (β|0β|1, β|1β|2, β|2β|3, β|3β|0)(0, 2)(1, 3)

satisfies [β2, (β2)τk

] = e for all k ∈ Z4. Therefore, by the previous analysis, wehave

(95) β|0β|1 =

(λ 2−ξ0

ξ0

τm0

)α,

(96) β|1β|2 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0

)α,

(97) β|2β|3 =

(λ ξ0

2−ξ0τ

(ξ0−m0)ξ0

2−ξ0

)α,

(98) β|3β|0 =

(λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α,

(99) τ =

(λ

(ξ0

2−ξ0)2τ

(1− 2m0

ξ0

)(ξ0

2−ξ0

)2)α

.

Therefore,

β|0β|1β|2β|3 =

(λ 2−ξ0

ξ0

τm0λ ξ02−ξ0

τ(ξ0−m0)

ξ02−ξ0

)α=

(τ

ξ202−ξ0

)α,

β|1β|2β|3β|0 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α=(τξ1ξ02−ξ0

)α.

It follows that (τ

ξ202−ξ0

)αβ|0=(τξ1ξ02−ξ0

)αand

(100) (τα)β|0 =(τξ1ξ0

)αSubstitute η = ξ1

ξ0in (100) to get

(101) β|0 = (ψητm1)α ,



where

(102) ψη =

λη, if η ∈ Z1

4

θλ−η, if − η ∈ Z14,

θ = θ(1)(e, τ−1, τ−1, τ−1)(1, 3)

(an invertor of τ). Note that

ψηλξ = ψηψξ = ψηξ = ψξη = ψξψη = λξψη

for all ξ ∈ Z14.

By (95) and (101),

(103) β|1 =

(τ−m1ψη−1λ 2−ξ0

ξ0

τm0

)α=

(ψ 2−ξ0

ηξ0

τ−m1

(2−ξ0ηξ0

)+m0

)α.

Also, by (96) and (101),

(104)β|2 =

(τm1

(2−ξ0ηξ0

)−m0ψ ηξ0

2−ξ0λ 2−ξ0

ξ0

τηξ0−ξ0

2+m0

)α=

(ψητ

[m1

(2−ξ0ηξ0

)−m0

]η+

ηξ0−ξ02

+m0

)α.

Furthermore, by (98) and (101),

(105)β|3 =

(λ ξ0

2−ξ0τ( ηξ0+ξ0

2−m0) ξ0

2−ξ0 τ−m1ψη−1

)α=

(ψ ξ0η(2−ξ0)

τ

[( ηξ0+ξ0

2−m0) ξ0

2−ξ0−m1

]η−1

)α.

Setting i = 1 and t = 2 in (17), we obtain

(106) β|0β|2 = β|21.

Use (101), (103), (104) and (105) in (106), to get

(107)ψητ

m1ψητ

[m1

(2−ξ0ηξ0

)−m0

]η+

ηξ0−ξ02

+m0

= ψ 2−ξ0ηξ0

τ−m1

(2−ξ0ηξ0

)+m0ψ 2−ξ0

ηξ0

τ−m1

(2−ξ0ηξ0

)+m0

which is the same as

(108)ψη2τ

m1η+[m1

(2−ξ0ηξ0

)−m0

]η+

ηξ0−ξ02

+m0

= ψ( 2−ξ0ηξ0

)2τ

[−m1

(2−ξ0ηξ0

)+m0

](2−ξ0ηξ0

)−m1

(2−ξ0ηξ0

)+m0 .

Therefore,



(109) η2 =

(2− ξ0

ηξ0

)2

and

m1η +

[m1

(2− ξ0

ηξ0

)−m0

]η +

ηξ0 − ξ0

2+m0

=

[−m1

(2− ξ0

ηξ0

)+m0

](2− ξ0

ηξ0

)−m1

(2− ξ0

ηξ0

)+m0

(a) Suppose

(110) η = −2− ξ0

ηξ0

(or what is the same

(111)(η2 − 1

)ξ0 = −2).

Then on substituting this in the above equation, we get

(η − 1) ξ0 = 0

contradicting the previous equation.(b) Suppose

(112) η =2− ξ0

ηξ0

.

Then,

(113) ξ0 =2

η2 + 1

and this leads to

(114) m0 = 2m1 +η − 1

2η(η2 + 1).

On substituting (113) and (114) in(103), (104), (105) and (99), we find

(115) β|1 =(ψητ

m1(2−η)+ η−1

2η(η2+1)

)α(116) β|2 =

(ψητ

m1(η2−2η+2)+ η2−1

2η(η2+1)

)α,

(117) β|3 =

(ψη−3τ

2η2+η+1

2η4(η2+1)−m1

(η2+2

η3

))α

,



(118) τ =

(ψη−4τ

η+1

2η5 −2m1

(η2+1

η4

))α

.

Substitute i = 0, t = 1 in (17), to get

(119) β|3β|1 = τβ|20.

Using (101), (115), (116), (117) and (118) in (119), we obtain

ψη−3τ2η2+η+1

2η4(η2+1)−m1

(η2+2

η3

)ψητ

m1(2−η)+ η−1

2η(η2+1)

= ψη−4τη+1

2η5 −2m1

(η2+1

η4

)ψητ

m1ψητm1 .

Thus,

ψη−2τ2η2+η+1

2η3(η2+1)−m1

(η2+2

η2

)+m1(2−η)+ η−1

2η(η2+1)

= ψη−2τη+1

2η3 −2m1

(η2+1

η2

)+m1η+m1

,

which implies

(120) (η − 1)m1 = 0

and thus,

m1 = 0 or η = 1.

• If m1 = 0 we get

(121) β = (ψη, ψητη−1

2η(η2+1) , ψητη2−1

2η(η2+1) , ψη−3τ2η2+η+1

2η4(η2+1) )α(1)στ

= τ γ,

where

(122) γ =(λ 2η2(η2+1)

)(1)

(e, ψη, ψη2τη−1

2η(η2+1) , ψη3τ2η2−n−1

2η(η2+1) )α(1)

and

(123) τ =(ψη−4τ

η+1

2η5

)α.

• If η = 1 we get

(124) β = (τm1 , τm1 , τm1 , τ 1−3m1)α(1)

(0, 1, 2, 3)

and



(125) τ =(τ 1−4m1

)α,

which produce

(126)

β = (τm1

1−4m1 , τm1

1−4m1 , τm1

1−4m1 , τ1−3m11−4m1 )(0, 1, 2, 3)

= (τm1

1−4m1 , τm1

1−4m1 , τm1

1−4m1 , τm1

1−4m1 )τ

= τ4m1

1−4m1 τ = τ1

1−4m1 = τλ 1

1−4m1

.

(3) Suppose σβ = σ−1τ = (0, 3, 2, 1). Then, β−1 satisfies the previous case.

Therefore, as θ inverts τ , we have

(127) β =(β−1)−1

= (τ γ)−1 = (τ)θγ

or

(128) β = τθλ 1

1−4m1 ,

where m1 ∈ Z4,

(129) γ =(λ 2η2(η2+1)

)(1)

(e, ψη, ψη2τη−1

2η(η2+1) , ψη3τ2η2−n−1

2η(η2+1) )α(1),

η ∈ U(Z4) and

(130) τ =(ψη−4τ

η+1

2η5

)α.

8.5. Final Step. We finish the proof of the second part of Theorem A. Inorder to treat the remaining case where the activity of β is a 4-cycle, weuse the fact that β2 ∈ B, which we have already described. Next, from thedescription of the centralizer of β2, we are able to pin down the form of β.

Proposition 12. Let β = (β|0, β|1, β|2, β|3)(0, 2)(1, 3) be such that (β|0) (β|2) =τ θ1 and (β|1) (β|3) = τ θ2 , for some θ1, θ2 ∈ Aut(T4). Then, β is conjugate toτ 2.

Proof. Let α = (e, e, β|−10 , β|−1

3 ). Then,

(131) βα = (e, e, β|0β|2 , β|1β|3)(0, 2)(1, 3).

Therefore, substituting β|0β|2 = τ θ1 and β|1β|3 = τ θ2 in the above equation,we have

βα = (e, e, τ θ1 , τ θ2)(0, 2)(1, 3).

Conjugating βα by γ = (θ−11 , θ−1

2 , θ−11 , θ−1

2 ) we produce

βαγ = τ 2.

We show below that active elements of B produce within B elements con-jugate to τ 2.



Proposition 13. Let β ∈ B with nontrivial σβ. Then

(i) If σβ = σ2τ , then β is a conjugate of τ 2.

(ii) If σβ ∈ (0, 2), (1, 3), then ββτ is a conjugate τ 2.(iii) If σβ ∈ στ , σ−1

τ , then β2 is a conjugate of τ 2.

Proof. It is enough to prove (i), since (ii), (iii) are just special cases.If σβ = σ2

τ , then

(132) β|0 =

(λ 2−ξ0

ξ0

τm0

)α, β|1 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0

)α,

(133) β|2 =

(λ ξ0

2−ξ0τ

(ξ0−m0)ξ0

2−ξ0

)α, β|3 =

(λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α,

(134) τ =

(λ

(ξ0

2−ξ0)2τ

(1− 2m0

ξ0

)(ξ0

2−ξ0

)2)α

,

where ξ0, ξ1 ∈ U(Z4), m0 ∈ Z4.Therefore,

β|0β|2 =

(λ 2−ξ0

ξ0

τm0λ ξ02−ξ0

τ(ξ0−m0)

ξ02−ξ0

)α=

(τ

ξ202−ξ0

)α= (τ)

ψξ20

2−ξ0

α

β|1β|3 =

(λ 2−ξ0

ξ0

τξ1−ξ0

2+m0λ ξ0

2−ξ0τ( ξ1+ξ0

2−m0) ξ0

2−ξ0

)α=(τξ1ξ02−ξ0

)α= τ

ψ ξ1ξ02−ξ0

α

It follows from Proposition 12, that β is a conjugate of τ 2.

Corollary 4. Suppose β ∈ B is an active element. Then, B is conjugate to asubgroup of the centralizer C(τ 2).

Proposition 14. Let γ ∈ C(τ 2). Then,

(135) γ = (τm0 , τm1 , τm0+δ((0)σγ , 2), τm1+δ((1)σγ , 2))σγ,

where m0,m1 ∈ Z4, σγ ∈ CΣ4(σ2).

Proof. Write γ = (γ|0, γ|1, γ|2, γ|3)σγ. Then τ 2γ = γτ 2 translates to

(e, e, τ, τ)(0, 2)(1, 3)(γ|0, γ|1, γ|2, γ|3)σγ= (γ|0, γ|1, γ|2, γ|3)σγ(e, e, τ, τ)(0, 2)(1, 3),

and this in turn translates to

(γ|2, γ|3, τγ|0, τγ|1)(0, 2)(1, 3)σγ

=(γ|0, γ|1, γ|2, γ|3).σγ(τ

δ(0,2), τ δ(1,2), τ δ(2,2), τ δ(3,2))(0, 2)(1, 3)

=(γ|0, γ|1, γ|2, γ|3)(τ δ((0)σγ ,2), τ δ((1)σγ ,2), τ δ((2)σγ ,2), τ δ((3)σγ ,2))σγ(0, 2)(1, 3)

= (γ|0τ δ((0)σγ ,2), γ|1τ δ((1)σγ ,2), γ|2τ δ((2)σγ ,2), γ|3τ δ((3)σγ ,2))σγ(0, 2)(1, 3)



Thus, we have γ|2 = γ|0τ δ((0)σγ ,2),γ|3 = γ|1τ δ((1)σγ ,2),τγ|0 = γ|2τ δ((2)σγ ,2),τγ|1 = γ|3τ δ((3)σγ ,2).

Hence,γ|2 = γ|0τ δ((0)σγ ,2), γ|3 = γ|1τ δ((1)σγ ,2),τ γ|0 = τ δ((0)σγ ,2)+δ((2)σγ ,2) = τ , τ γ|1 = τ δ((1)σγ ,2)+δ((3)σγ ,2) = τ

.

Therefore, there exist m0,m1 ∈ Z4 such thatγ|0 = τm0 , γ|1 = τm1 ,γ|2 = τm0+δ((0)σγ ,2), γ|3 = τm1+δ((1)σγ ,2) .

Hence, γ has the form

(136) γ = (τm0 , τm1 , τm0+δ((0)σγ ,2), τm1+δ((1)σγ ,2))σγ,

where σγ ∈ CΣ4(σ2).

Corollary 5. The centralizer of τ 2 in A4 is

C(τ 2) = 〈(e, e, τ, e)(0, 2), τ, (τm0 , τm1 , τm0 , τm1) | m0,m1 ∈ Z4〉 .

Corollary 6. Let γ ∈ C(τ 2) be such that σγ ∈ 〈(0, 2)(1, 3)〉. Then

γ ∈⟨(τm0 , τm1 , τm0 , τm1), τ 2 | m0,m1 ∈ Z4

⟩.

Proposition 15. Let H = 〈(τm0 , τm1 , τm0 , τm1), τ 2 | m0,m1 ∈ Z4〉. Then thenormalizer NA4(H) is the group⟨

C(τ 2), (ψ2m0+1, ψ2m1+1, ψ2m0+1τm0 , ψ2m1+1τ

m1) | m0,m1 ∈ Z4

⟩,

where, for each η ∈ U(Z4), ψη is defined by (102) and

τψη = τ η.

Proof. Note that H is an abelian group. Let α ∈ NA4(H). Then,

(τ 2)α = (τm0 , τm1 , τm0+1, τm1+1)(0, 2)(1, 3),

where m0,m1 ∈ Z4.Suppose α is inactive. Then,

(τm0 , τm1 , τm0+1, τm1+1)(0, 2)(1, 3)= (α|−1

0 , α|−11 , α|−1

2 , α|−13 )(e, e, τ, τ)(0, 2)(1, 3)(α|0, α|1, α|2, α|3)

= (α|−10 , α|−1

1 , α|−12 , α|−1

3 )(e, e, τ, τ)(α|2, α|3, α|0, α|1)(0, 2)(1, 3)= (α|−1

0 α|2, α|−11 α|3, α|−1

2 τα|0, α|−13 τα|1)(0, 2)(1, 3)

which produces α|−1

0 α|2 = τm0 , α|−11 α|3 = τm1 ,

α|−12 τα|0 = τm0+1, α|−1

3 τα|1 = τm1+1 .



Therefore, α|2 = α|0τm0 , α|3 = α|1τm1 ,α|−1

0 τα|0 = τ 2m0+1, α|−11 τα|1 = τ 2m1+1.

Thus,

α = (α|0, α|1, α|2, α|3) = (ψ2m0+1, ψ2m1+1, ψ2m0+1τm0 , ψ2m1+1τ

m1)

satisfies(τ 2)α = (τm0 , τm1 , τm0+1, τm1+1)(0, 2)(1, 3).

Theorem 7. Let G be a finitely generated solvable subgroup of Aut(T4) whichcontains τ . Then, G is a subgroup of

(137) ×4 (· · · (×4 (×4NA4(H)α o S4) o S4) · · · ) o S4

for some α ∈ A4.

Proof. As in the case n = p, we assume G has derived length d ≥ 2 and letB be the (d − 1)th term of the derived series of G. Then, B is an abeliangroup normalized by τ . On analyzing the case 8.4 and the final step, thereexists a level t such that B is a subgroup of V = ×4kC(µ2),where µ = τα forsome α ∈ A4 and where σµ2 = (0, 2)(1, 3). There also exists β ∈ B such thatβ|u = µ2 for some index u ∈M.

Moreover, if T is the normalizer of C(τ 2), then clearly, Tα is the normalizerof C(µ2).

We will show now that G is a subgroup of

J = ×4 (· · · (×4 (×4NA4(H)α o S4) o S4) · · · ) o S4

where the cartesian product ×4appears t times..Let γ 6∈ J . Since γ 6∈ J , there exists w ∈ M having |w| = t and γ|w 6∈ Tα.

Since τ is transitive on all levels of the tree, by Corollary 6 we can conjugateβ by an appropriate power of τ to get θ ∈ B such that

θ|w = µ2 or θ|w =(µ2)τ

=((τm0 , τm1 , τm0+1, τm1+1)(0, 2)(1, 3)

)α,

where m0,m1 ∈ Z4. Thus, for v = wγ we have

(θγ) |v(9)= θ|

γvγ−1

vγ−1 = θ|γ|ww 6∈ C(µ2)

which implies θγ 6∈ B ≤ V and γ 6∈ G. Hence, G is a subgroup of J .

References

[1] Bass H., Espinar O., Rockmore D., Tresser C., Cyclic Renormalization and the Au-tomorphism Groups of Rooted Trees, Lecture Notes in Mathematics 1621, Springer,Berlin, 1995.

[2] Sidki, S., Regular Trees and their Automorphisms, Monografias de Matematica, No¯56,

Impa, Rio de Janeiro, 1998.



[3] Brunner, A., Sidki S., Vieira A. C., A just-nonsolvable torsion-free group defined on thebinary tree, J. Algebra 211 (1999), 99-114.

[4] Grigorchuk R. I.,. Nekrachevych, V., I. Suschanskii, I., Automata, dynamical systems,and groups, Proc. Steklov Institute 231 (2000), 128-203.

[5] Sidki, S., Automorphisms of one-rooted trees: growth, circuit structure and acyclicity,Journal of Mathematical Sciences, Vol. 100, no

¯1 (2000),

[6] Sidki, S., Silva, E.F., A family of just-nonsolvable torsion-free groups defined on n-ary

trees, In Atas da XVI Escola de Algebra, Brasılia, Matematica Contemporanea 21(2001).

[7] Sidki, S., The Binary Adding Machine and Solvable Groups, International Journal ofAlgebra and Computation, Vol. 13, no

¯1 (2003), 95-110.

[8] Jones, G. A., Cyclic regular subgroups of primitive permutation groups. J. GroupTheory 5 (2002), no. 4, 403–407.

[9] Sidki, S., Just-Non-(abelian by P-type) Groups, Progress in Math., 248, (2005) 389-402.[10] Nekrashevych, V., Self-similar groups, volume 117 of Mathematical Surveys and Mono-

graphs. American Mathematical Society, Providence, RI, 2005.[11] Vorobets, M., Vorobets,Y., On a free group of transformations defined by an automaton,

Geom. Dedicata 124 (2007) 237–249.

E-mail address: [email protected]

E-mail address: [email protected]

Instituto Federal de Educacao, Ciencia e Tecnologia de Goias, CampusFormosa, 73800-000, Formosa - GO, Brazil

Departamento de Matematica, Universidade de Brasılia, 70910-900, Brasılia-DF, Brazil


1

Three–dimensional compact manifolds and the Poincare conjecture

Alexander A. Ermolitski

IIT–BSUIR, Minsk, Belarus E-mail: [email protected]

__________________________________________________________________

Abstract: The aim of the work is to prove the following main theorem. Theorem. Let 3M be a three–dimensional, connected, simple connected,

compact, closed, smooth manifold and 3S be the three–dimensional sphere. Then

the manifolds 3M and 3S are diffeomorphic. Keywords: Compact manifolds, Riemannian metric, triangulation,

homotopy, algorithms MSC(2000): 53C21, 57M20, 57M40, 57M50

__________________________________________________________________ 0 Introduction We can fix some Riemannian metric g on a manifold nM of dimension n

which defines the length of arc of a piecewise smooth curve and the continuous function (x; y) of the distance between two points x, y nM . The topology defined by the function of distance (metric) is the same as the topology of the manifold nM , [5].

We should mention that it will suffice to prove that 3M and 3S are

homeomorphic since the existence of a homeomorphism between nM and nS (n=dim nM , n6, n4) implies the existence of a diffeomorphism between them. If n=7 then there exist such 28 smooth manifolds that every one from them is

homeomorphic to 7S but any two from them are not diffeomorphic. The proof of the main theorem is based on some notions from [1], [2] and

that will be considered step by step in the following sections. Some results can be useful in the case when 3M is not simply connected or can be generalized for manifolds of dimension n3.

In section 1, using a smooth triangulation and a Riemannian metric we see that every compact, connected, closed manifold nM of dimension n can be represented as a union of a n–dimensional cell Сn and a connected union Kn–1 of some finite number of (n–1) –– simplexes of the triangulation. A sufficiently small closed neighborhood of Kn–1 we call a geometric black hole. In dimension 3 we

have 233 KCM .


2

In section 2, we get some technical results permitting to retract 2–dimensional and 1–dimensional simplexes from K2 having boundaries i. e. to

obtain a decomposition 233 ~~KCM where 2~

K contains less simplexes than K2

does. In section 3, we consider the proof of the main theorem consisting of the

realization of several algorithms. The number of 2–dimensional simplexes of the complex K2 becomes less every step and finally we have a decomposition

133 KCM where K1 is a connected and simply connected union of some 1–dimensional simplexes i. e. K1 is a tree. Using the section 2 we can retract complex

K1 to a point х0 therefore a decomposition 033 xCM is obtained and 3M is

homeomorphic to sphere. 1. On extension of coordinate neighborhood

1°. Let nM be a connected, compact, closed and smooth manifold of

dimension n and Cn be a cell (coordinate neighborhood) on nM . A standard simplex n of dimension n is the set of points x=(x1, x2, ..., xn) nR defined by conditions

0xi1, i= n,1 , x1+x2+...+xn1.

We consider the interval of a straight line connected the center of some face of n and the vertex which is opposite to this face. It is clear that the center of n belongs to the interval. We can decompose n as a set of intervals which are parallel to that mentioned above. If the center of n is connected by intervals with points of some face of n then a subsimplex of n is obtained. All the faces of n considered, n is seen as a set of all such subsimplexes. Let U(n) be some open

neighborhood of n in Rn. A diffeomorphism φ : nU Мп nn is called a singular n–simplex on the manifold M n. Faces, edges, the center, vertexes of the

simplex n are defined as the images of those of n with respect to . The manifold M n is triangulable, [6]. It means that for any nll 0, such

a finite set Фl of diffeomorphisms φ : l Мп is defined that

a) M n is a disjunct union of images llInt Ф, ;

b) if lФ then

1Ф li for every і where i : kk 1 is the

linear mapping transferring the vertexes 10 ,..., kvv of the simplex 1k in

the vertexes ki vvv ,...ˆ,...,0 of the simplex k .

2°. Let n0 be some simplex of the fixed triangulation of the manifold Мп.

We paint the inner part nInt 0 of the simplex

n0 white and the boundary

n0 of

n0 black. There exist coordinates on

nInt 0 given by diffeomorphism φ0. A

subsimplex nn0

101

is defined by a black face nn0

101

and the center с0 of


3

n0 . We connect с0 with the center d0 of the face

101n and decompose the

subsimplex n01 as a set of intervals which are parallel to the interval с0d0. The face

101n is a face of some simplex

n1 that has not been painted. We draw an interval

between d0 and the vertex 1v of the subsimplex n

1 which is opposite to the face 1

01n then we decompose

n1 as a set of intervals which are parallel to the interval

d0 1v . The set nn

101 is a union of such broken lines every one from which consists of two intervals where the endpoint of the first interval coincides with the

beginning of the second interval (in the face 1

01n ) the first interval belongs to

n01

and the second interval belongs to n

1 . We construct a homeomorphism (extension) 101 : nnn IntInt 10101 . Let us consider a point х

nInt 01 and let x belong to a broken line consisting of two intervals the first interval is of a length of s1 and the second interval is of a length of s2 and let x be at a distance of s from the beginning

of the first interval. Then we suppose that x101 belongs to the same broken line

at a distance of ss

ss

1

21 from the beginning of the first interval. It is clear that

101 is a homeomorphism giving coordinates on nnInt 101 . We paint points of nnInt 101 white. Assuming the coordinates of points of white initial faces of

subsimplex n01 to be fixed we obtain correctly introduced coordinates on

nnInt 10 . The set nn

101 is called a canonical polyhedron. We paint

faces of the boundary 1 black. We describe the contents of the successive step of the algorithm of extension

of coordinate neighborhood. Let us have a canonical polyhedron 1k with white inner points (they have introduced white coordinates) and the black boundary

1 k . We look for such an n–simplex in 1k , let it be n0 that has such a black

face, let it be 1

01n that is simultaneously a face of some n–simplex, let it be

n1 ,

inner points of which are not painted. Then we apply the procedure described

above to the pair n0 ,

n1 . As a result we have a polyhedron k with one simplex

more than 1k has. Points of kInt are painted in white and the boundary k is painted in black. The process is finished in the case when all the black faces of the last polyhedron border on the set of white points (the cell) from two sides.

After that all the points of the manifold Мп are painted in black or white,

otherwise we would have that Мп = nn MM 10 (the points of

nM 0 would be painted

and those of nM1 would be not) with

nM 0 and nM1 being unconnected, which

would contradict of connectivity of Мп. Thus, we have proved the following


4

Theorem 1. Let Мп be a connected, compact, closed, smooth manifold of

dimension n. Then Мп = 11, nnnn KCKC , where Сп is an п–dimensional cell and Кп–1 is a union of some finite number of (п–1)–simplexes of the triangulation.

3°. We consider the initial simplex n0 of the triangulation and its center с0.

Drawing intervals between the point с0 and points of all the faces of n0 we obtain

a decomposition of n0 as a set of the intervals. In 2° the homeomorphism :

nInt 0 Сп was constructed and evidently maps every interval above on a piecewise smooth broken line in Сп. We denote

nM~

=Мп \c0. nM

~ is a connected and simply connected manifold if Мп is that. Let

І=[0;1], we define a homotopy F:nM

~×І

nM~

: (х; t) у=F(x;t) in the following way

a) F(z; t)=z for every point zKn-1; b) if a point x belongs to the broken line in Сп and the distance between x

and its limit point zKn-1 is s(x) then у=F(x; t) is on the same broken line at a distance of (1–t)s(x) from the point z.

It is clear that F(x;0)=х, F(x;1)=z and we have obtained the following

Theorem 2. The spaces nM

~ and Кп–1 are homotopy–equivalent, in

particular, the groups of singular homologies Hk nM

~ and Hk

1nK are isomorphic for every k.

Corollary 2.1. The space Кп–1 is connected and if Мп is simply connected then Кп–1 is simply connected too.

Remark. The white coordinates are extended from the simplex n0 in the

simplex n

1 through the face 1

01n hence

101nInt has also the white coordinates.

On the other hand there exist two linear structures (intervals, the center etc) on n01 induced from

n0 and

n1 respectively. Further, we set that the linear structure

of 1

01n is the structure induced from

n0 .

2. On the complex 2K For a three–dimensional, connected, compact, closed, smooth manifold М 3

we consider a decomposition 233 KCM obtained in theorem 1. We call simplexes of dimension 3, 2, 1 by tetrahedrons, triangles, edges

(intervals) respectively. 1°. Definition 1. a) A triangle from the complex 2K is called a

f–triangle (free) if it has at least one free edge i. e. such an edge that it is not an edge of any other triangle from 2K .


5

b) A triangle from the complex 2K is called a m–triangle if it has such an edge that is an edge of more than two triangle from 2K . By definition, a m–triangle can not be a f–triangle.

c) A triangle from the complex 2K is called a s–triangle (standard) if every its edge is an edge of only one other triangle from 2K .

Let us have a f–triangle 2 2K with some free edge 1 . We consider such

a polyhedron which is a set of all the tetrahedrons with 1 as their edge.

Among them we have exactly two tetrahedrons, let they be 3

1 and 3

l with 2 as

their face. We call the output of 3

1 the face 2

1 with 1 as its edge. Inner points

of the triangle 2

1 are white because the edge 1 is free. The face 2

1 is a face of

another tetrahedron 3

2 that has only one another face 2

2 with the edge 1 ,

moreover, all inner points of the triangle 2

2 are white. The faces 2

1 and 2

2 are

called respectively the input and output (conversions) of the tetrahedron 3

2 . The

face 2

2 is called the input of some tetrahedron 3

3 etc. Taking a finite number of

steps we come to the tetrahedron 3

l with an input 2

1l with 1 as its edge and all

inner points of the triangle 2

1l are white. Thus, we obtain l

ii

1

3

(minimal

possible meaning is l=3). We have to note that all inner points of the faces of

conversions 2

1 , ..., 2

1l in the tetrahedrons of the polyhedron are white. It is

clear that (Int)/ 2 is a cell. 2°. We consider the closed cube Cu3 in the three – dimensional coordinate

space R3 having the vertexes A(1; 1; 1), B(1; -1; 1), C(-1; 1; 1), D(-1; -1; 1), A1(1; 1; -1), B1(1; -1; -1), C1(-1; 1; -1), D1(-1; -1; -1). Let Rc be the intersection of

Cu3 with the semiplane ; ;M x y z R3 0; 0z x and τ be the intersection of with the square ABB1A1. It is easy to construct such a homeomorphism

3 31 : \ \Cu Rc Cu that 1 i d on 3 \Cu .

Proposition 3. We can redistribute coordinates of white points of the polyhedron and introduce white coordinates of points from Int 12 (construct the corresponding homemorphism ) in such way that the following conditions are fulfilled

a) all the points of Int are painted in white i.e. have white coordinates, b) white coordinates of points of boundary faces of the polyhedron are not

changed.

Proof. There exists a homeomorphism 3

2: Cu , 2

2 Rc . Then 1

2 1 2o o is a required homeomorphism.


6

QED. Remark. Further, we set that the linear structure (intervals, the center etc.)

of 2 is the structure induced from 1

3 where 1

3 was equipped with white

coordinates earlier than 3l .

3°. Definition 2. An edge 1 =x0x1 is called semi-isolated if it is not an edge

of any triangle from 2K . A semi-isolated edge 1 is called isolated if one of the

endpoints of the interval 1 (let it be x1) is free i.e. it is not an endpoint of any edge from 2K .

An isolated edge 1 can appear as a result of painting white some

neighboring f-triangles containing 1 . We consider a polyhedrons where is the set of all tetrahedrons with x1 as their vertex. It is clear that all the points of

Int are white with the exception of black points of 1 \x0. Proposition 4. We can redistribute coordinates of white points of the

polyhedron and introduce white coordinates of points from Int 1 x1 (construct the corresponding homeomorphism) in such a way that the following condition are fulfilled)

a) all the points of Int are painted in white i. e. have white coordinates, b) white coordinates of points of boundary faces of the polyhedron are

not changed.

Proof. It is clear that 1\Int is a cell. There exists a homeomorphism

32 : Cu , 1

2 , where Cu2 was defined in 20 and

; 0; 0 0;1x x is a closed interval in Cu2. It is easy to construst such a

homeomorphism 3 31: \ \Cu Cu E that 1 id on 3 \Cu E where

E , E (1; 0; 0). Then 1

2 1 1o o is a required homeomorphism.

QED.

4°. We assume that in the process of painting f-triangles white by the proposition 3 all the triangles from 2K are white i.e. that we have a representation

133 KCM , 13 KC , where 3C is a three–dimensional cell and 1K is a connected union of finite number of black edges of the triangulation. Since the process of painting f–triangles white does not influence simple connectedness of a space that is been obtained after every step then 1K is a tree if the complex 2K is simply connected. Painting isolated edges of 1K in white by the proposition 4 as a result we have unique black point x0. Thus, we obtain a representation

);( 033 xBCM , where );( 0 xB is an open geodesic ball with the center in


7

x0 and of radius . The manifold 3M is homeomorphic to sphere 3S by the following lemma 5.

Lemma 5 [5]. If a topological manifold Mn is a union of two n–dimensional

cells then Mn is homeomorphic to sphere nS . 3. Proof of the main theorem The proof has a combinatorial nature and assumes the realization of a

number of algorithms. We consider that step by step. The initial complex 2K is assumed to be connected, simply connected and without free triangles.

1°. We call a sequence of tetrahedrons (triangles, edges) a simple chain (s – chain) if every such a simplex participates in the sequence only one time and if every subsequent tetrahedron (triangle, edge) has a common face (edge, vertex) with the previous one. The number of elements of a s–chain is called the length of the s–chain.

Let 20 be a triangle from the complex 2K with

10 = x0x1 as its edge. The

edge 10 can also be an edge of some m–triangles other than

20 .

Lemma 6. We can rebuild the complex 2K in such a way that as a result we

have got a black triangle 20 with the free edge

10 = x0x1. A new rebuilt complex

2K is connected and simple connected.

Proof. We consider the s–chain of tetrahedrons with 10 as their edge the

first of which has the upper part of 20 as its face and the last of which has the

lower part of 20 as its face. In this s–chain we can find a tetrahedron, let it be

31 ,

which is the first from the s–chain to have a black m–triangle 21 with the edge

10

as its face. The face 21 is the common face of 3

1 and 32 . Thus, we obtain a s-

chain 1

3 31, , l of tetrahedrons (some of them have

m-triangles as their faces) 1

3l has the lower part of

20 as its face.

We consider the graph G connecting by intervals the centers of all the tetrahedrons of the triangulation via the centers of all the white faces. There exists the maximal tree L connecting by intervals all the centers of the tetrahedrons of the triangulation via centers of some white faces. The tree L defines the maximal cell C3. Really, if we consider a maximal tree L and some tetrahedron 3 then we can

extend white coordinates from 3 on the maximal cell C3 along the tree L as it was

shown in section 1. We assume that the centers of all the tetrahedrons of the s-chain from the first to the 3

1 are connected by the broken line via the centers of their common white faces and the broken line is a part of L.


8

We cancel the white painting of points of 32 and paint the tetrahedron

32

black. The repainting of 32 black brings to a gap of L on three subtrees L1, L2, L 3

or less where the center of 3

1 belongs to L1. Further, we repaint inner parts of 2

1 and 32 white (extend white

coordinates from 31 through the face 2

1 as it was shown in section 1) and

connect the centers of 31 , 2

1 , 32 by intervals. Those centers belong to the

subtree L1. Other faces of 32 are black and they are simultaneously faces of other

tetrahedrons. We consider the following cases. a) L1 = L. The black faces of 3

2 remain black. b) We have got two subtrees L1 and L2 where L2 defines a cell called a dead end. We repaint the closure of the dead end black. Then we are looking for a black face of 3

2 which is simultaneously a face of other tetrahedron with the center from

L2. We extend white coordinates from 32 through this face along the tree L2 as it

was shown in section 1 and repaint inner points of this face and inner points of the dead end white. Then we connect by intervals the center of this face with the centers of 3

2 and other tetrahedron obtaining a new maximal tree L defining a new

maximal cell C3. Two other faces of 32 remain black.

c) We have got three subtrees L1, L2, L3 where L2 and L3 define two cells called dead ends. We repaint the closure of each the dead end black. Then we are looking for a black face of 3

2 which is simultaneously a face of other tetrahedron with the center from L1. This face remains black. We extend white coordinates from 3

2 through two other black faces of 32 along the trees L2 and L3 as it was

shown in section 1 and repaint inner points of this faces and inner points of the dead ends white. Then we connect by intervals the centers of this faces with the centers of 3

2 and two other tetrahedrons obtaining a new maximal tree L defining a new maximal cell C3.

Further, we apply the process above to the tetrahedrons 32 ,

33 etc. All the

centers of the tetrahedrons 31 , 3

2 , 33 ;… are connected by broken line which is a

part of the subtree L1 at every step. As a result we have got a black triangle 20 with

the free edge 10 = x0x1.

QED.

Remark. We have obtained the s-chain of the tetrahedrons (with white inner

part) having 10 as their edge and all the centers of the tetrahedrons are

connected by the broken line via the centers of their common white faces. The broken line can be considered as a part of a subtree L1 of some maximal tree L i.e.


9

this broken line can be extended to L. Really, we can apply the method considered in the proof of the lemma 6 to the s-chain above sequentially repainting the

tetrahedrons from the second ( 31 ) to the last ( 1

3l ) to get the broken line in the

end. 2°. We choose a small ball with the center in x0 which is diffeomorphic to a

small ball in R3 and call a trace of a simplex with a vertex or an endpoint in the point x0 its intersection with the sphere which is the surface of the ball where the sphere is supposed to be transversal to all the triangles with the vertex x0 . Such a sphere exists because of the smoothness of the triangulation. All other vertexes of the triangles do not belong to the ball. The ball can be choosed in such a way that every edge with the endpoint x0 has only one point of the intersection with the sphere and every triangle with the vertex x0 is intersected with the sphere by only one segment of a curve. There exists one to one correspondence between the set of simplexes having a vertex (endpoint) x0 and the set of their traces therefore all steps of the algorithm below can be illustrated on the sphere.

We continue a consideration of the edge 10 =x0x1 and a set Bt(x0) of black

triangles with x0 as their vertex. We extract pyramids from the set Bt(x0). The trace of the surface of a pyramid formed by some black triangles is a maximal loop on the sphere i.e. a curve that divides the sphere into two parts. Any exterior white

point of the sphere close to the loop can be connected with the trace of 10 (a black

point) by a white curve and any interior white point with respect of the loop cannot. Such loops can be connected among themselves by segments of black curves. See Fig.1 as a possible picture of such traces.

Further, we consider one of the pyramids and any s-chain (with the white inner part) of tetrahedrons having x0 as their vertex the first of which has 1

0 as its edge and last of which (the first in the s-chain) has a black triangle from the surface of the pyramid as its face. All the centers of the tetrahedrons of any such a s-chain are supposed belonging to a subtree L1 of some maximal tree L. Really, we can apply the method considered in the proof of the lemma 6 to any such a s-chain above sequentially repainting tetrahedrons from the second to the last to get a


10

broken line in the end. The subtree L1 is the union of all such broken lines and the initial broken line considered in the remark in 1°, 3. The subtree L1 can be extended to L and L1 cannot get a gap by the procedure below. In the set of all possible similar s-chains we look for a s-chain of the minimal length. In the last

tetrahedron 3l of the s–chain we consider the subtetrahedron

30l

with the center

of 3l as its vertex and the mentioned above black triangle as its face. The latter

belongs to tetrahedron 31l

. The inner points of 1

3l are simultaneously inner points

of the pyramid. By definition, x0x1 cannot be an edge of such a tetrahedron 1

3l .

Canceling white painting of those inner points and painting the tetrahedron 31l

black we extend white coordinates from 30l

into 31l

through their common face as it was described in section 1 and paint those inner points white again. A new one more length s–chain has been obtained (see Fig.2). If we obtain a gap of the maximal tree L then we eliminate it by the procedure described in lemma 6 using introduced above the subtree L1. Further, we iterate the algorithm above and so on.

It is clear that we cannot get a new black triangle having 10 as its edge by the

procedure above. In the end any tetrahedron with a vertex in x0 can be considered as an element of some s-chain with the white inner part connecting this tetrahedron with the edge 1

0 i.e. all the loops on the sphere are annihilated and we have got a number of trees on the sphere (see Fig.3). Any endpoint of a tree is simultaneously the trace of a free edge of some f-triangle and we can paint the f-triangle white by proposition 3. As it has been noted in the proof of this proposition the painting of boundary points of a polyhedron containing a black f-triangle is not changed. Sequentially painting all those f-triangles white we retract all the trees on the sphere to a number of black points which are traces of some semi-isolated (isolated) edges. As a result we have got a situation when the set Bt(x0) becomes empty.

Really, othervise if we have only one f-triangle 20 in Bt(x0) and 2

1 is an other triangle in Bt(x0) then we can construct some s-chain 2

1 , 22 , … , 2

n of triangles from Bt(x0) where 2

n has some a common edge x0 xl with a previous triangle from this s-chain i.e. we have got a pyramid. There exists a tetrahedron containing inner white points of the pyramid which can not be connected by any s-chain (with white inner part) of tetrahedrons having x0 as their vertex with the edge

10 i.e. the contradiction to the situation above has been obtained.

It is obvious that the set of all the white points is a three–dimensional cell at every step. It is clear that the last rebuilt complex K2 is connected and simple connected because of a homotopy–equivalence.

Remark. Further, a structure consisting of a semi–isolated edge and a black subcomplex joined to it is called a «black flower» growing from the point x0. Let bf1 and bf2 be any two black flowers connected by a system of semi-isolated edges. The simple connectivity of K2 implies that if we paint the semi-isolated edge of bf1


11

white then the black subcomplex obtained can not be connected with bf2 by a black curve.

10

3°. We consider a black flower consisting of a semi-isolated edge 1 with the endpoints x0 , y0 and a black two-dimensional subcomplex of some black triangles having y0 as their vertex. Further, we apply the procedure considered in 2° to the

point y0 and the edge 1 . The simple connectivity of K2 implies that we cannot get a black loop in K2 having a semi-isolated edge as its part therefore the annihilation of black triangles of Bt(y0) cannot bring to an appearance of a black triangle in Bt(x0) and Bt(x0) remains empty. Similarly, if we have a s-chain of semi-isolated edges 1

1 ,…, 1k = xk yk then the process of the annihilation of black

triangles in Bt(yk) cannot bring to an appearance of a black triangle having a generic point with 1

i (i<k). Really, otherwise such a black triangle gives an opportunity to connect the endpoints xk and yk of 1

k by a black curve which is different from 1

k . As a result a black loop with the semi-isolated edge 1k has been

obtained and the loop is not contractible that is a contradiction to the simple connectivity of K2. Thus, a number of the black isolated and semi-isolated edges is increased and the sets Bt(x0), Bt(y0 ), … remain empty. It follows that a number of black triangles becomes less at every step. Finally, at some step of our algorithm the set of black triangles must be exhausted i.e. we come to 4°, 2.


12

Remark. The obtained complex can be imagined as a «tree with flowers» growing in the endpoints of the branches of the tree. An iteration of the algorithm can be interpreted as a sequential transformation of those flowers into branches to get a tree in the end.

The main theorem is completely proved. References

[1] A.A. Ermolitski: Riemannian manifolds with geometric structures, BSPU, Minsk, 1998 (in Russian), (English version in arXiv: 0805.3497).

[2] A.A. Ermolitski: Big handbook on elementary mathematics. Universal manual, Harvest, Minsk, 2003 (in Russian).

[3] A.T. Fomenko, D.B. Fuks: Kurs gomotopicheskoj topologii, Nauka, Moscow, 1989 (in Russian).

[4] D.B. Fuks, V.A. Rohlin: Beginner’s course in topology/ Geometric chapters, Nauka, Moscow, 1977 (in Russian).

[5] D. Gromoll, W. Klingenberg, W. Meyer: Riemannsche geometrie im grossen, Springer, Berlin, 1968 (in German).

[6] J.R. Munkres: Elementary differential topology, Princeton University Press, Princeton, 1966.


Five-Dimensional Tangent Vectors in Space-TimeIII. Some Applications

Alexander KrasulinInstitute for Nuclear Research of the Russian Academy of Sciences∗

[email protected]

Abstract

In this part of the series I show how five-tensors can be used for describing in a coordinate-independent way finite and infinitesimal Poincare transformations in flat space-time. As anillustration, I reformulate the classical mechanics of a perfectly rigit body in terms of theanalogs of five-vectors in three-dimensional Euclidean space. I then introduce the notion ofthe bivector derivative for scalar, four-vector and four-tensor fields in flat space-time andcalculate its analog in three-dimensional space for the Lagrange function of a system ofseveral point particles in classical nonrelativistic mechanics.

A. Preliminary remarks

Before developing the mathematical theory of five-vectors further, it will be useful to consider some oftheir applications in flat space-time. At the sametime I will say a few words about the analogs of five-vectors in three-dimensional Euclidean space and willconsider some of their applications.

In the case of vectors of the latter type therearises a problem with terminology. Since the analogsof five-vectors in three-dimensional space are four-dimensional, by analogy with five-vectors one shouldterm them as four-vectors. This, of course, is un-acceptable since the term ‘four-vector’ is tradition-ally used for referring to ordinary tangent vectors inspace-time. To avoid confusion, in the following theanalogs of five-vectors in three-dimensional space willbe called (3+1)-vectors. It makes sense to use thesame terminology for all other manifolds as well. Theonly exception from this convention will be made forfive-vectors in space-time, which will still be calledfive-vectors and not (4+1)-vectors.

Let me also say a few words about the notations.Ordinary tangent vectors in three-dimensional Eu-clidean space will be denoted with capital Romanletters with an arrow: ~A, ~B, ~C, etc. Three-plus-one-vectors will be denoted with lower-case Romanletters with an arrow: ~a, ~b, ~c, etc. It will be takenthat lower-case latin indices run 1, 2, and 3 and thatcapital Greek indices run 1, 2, 3, and 5 (the value “5”corresponds to the additional dimension of the spaceof (3+1)-vectors). Finally, the Euclidean inner prod-

∗Former affiliation.

uct will be denoted with a dot or with the symbol δ,which should not be confused with the unit tensor ofrank (1, 1).

In conclusion, let me write out the formulae thatexpress the result of transporting parallelly an activeregular five-vector basis associated with some systemof Lorentz coordinates, from the origin of the latterto the point with coordinates xµ:

[eα(0)]transported to x

= eα(x) + xα e5(x)

[e5(0)]transported to x

= e5(x).(1)

The analogs of these formulae for (3+1)-vectors are[~ei(0)]

transported to x= ~ei(x) + xi ~e5(x)

[~e5(0)]transported to x

= ~e5(x),(2)

where xi ≡ δijxj and xj is the system of Cartesian

coordinates with which the considered active regularbasis of (3+1)-vectors is associated.

B. Active Poincare transformationsin the language of five-vectors

The replacement of any given set of scalar, four-vector and four-tensor fields in flat space-time withan equivalent set of fields, as it was discussed in sec-tion 4 of part I, and a similar replacement of five-vector and five-tensor fields, are nothing but activePoincare transformations. Though the procedure forconstructing the equivalent fields, described in thesame section of part I for four-vector and four-tensorfields and in section 3 of part II for five-vector and

1


five-tensor fields, makes use of Lorentz coordinatesystems, the latter are only a tool for performing sucha construction. In itself, any such active field trans-formation can be considered without referring to anycoordinates and in this sense is an invariant opera-tion. Let us now see how one can give it a coordinate-independent description.

From the very procedure for constructing theequivalent fields it follows that any active Poincaretransformation can be characterised by the parame-ters Lαβ and aα that specify the transition from theselected initial Lorentz coordinate system xα to thecorresponding final Lorentz coordinate system x′α:

x′α = Lαβ xβ + aα. (3)

Since parameters Lαβ and aα explicitly depend on thechoice of the initial coordinate system, such a descrip-tion will naturally be non-invariant. To reduce thedependence on the choice of the coordinate system,instead of Lαβ and aα one speaks of the displacementvector and rotation tensor. Namely, in the selectedinitial system of Lorentz coordinates, from the cor-responding transformation parameters Lαβ and aα,basis four-vectors Eα, and dual basis four-vector 1-forms Oα one constructs the four-vector A ≡ aαEα

and four-tensor B ≡ (Lαβ − δαβ) Eα ⊗ Oα. By usingthe transformation formulae for Lαβ and aα derivedin section 5 of part I, one can easily show that B willbe the same at any choice of the initial Lorentz co-ordinate system and that A will be the same for anytwo systems with the same origin. For systems withdifferent origins A will in general be different, so byusing the four-vector quantities A and B one cannotget rid of the dependence on the choice of the initialLorentz coordinate system completely.

To obtain a completely coordinate-independent de-scription of a given active Poincare transformation,let us construct in the selected Lorentz coordinatesystem a five-tensor field TT which in the P -basis as-sociated with these coordinates has the following con-stant components:

T αβ = Λαβ , T α5 = 0

T 5β = aβ , T 5

5 = 1,(4)

where aβ ≡ gβσaσ and Λαβ ≡ (L−1)αβ is the inverse ofthe 4×4 matrix Lαβ . As it has been shown in section 5

of part I, the quantities T AB transform as componentsof a five-tensor of rank (1, 1), and consequently thefield TT will be the same at any choice of the initialLorentz coordinate system.

To understand why the components of TT are con-structed from the transformation parameters for co-variant coordinates and not from the transformation

parameters for the Lorentz coordinates themselves,let us introduce this five-tensor in a slightly differ-ent way. Instead of the two coordinate systems xα

and x′α, let us consider the corresponding P -bases,pA and p′A. Since at each point the vectors p′A areexpressed linearly in terms of pA, the former can beobtained by acting on the latter with some linear op-erator, TT . Since all P -bases are self-parallel by defi-nition, TT regraded as a five-tensor field of rank (1, 1)will be covariantly constant. If qA and q′A are thebases of five-vector 1-forms dual to pA and p′A, re-spectively, one can present the tensor TT as

TT = p′A ⊗ qA,

for in this case

TT (pA) ≡ p′B < qB ,pA > = p′A.

Furthermore,

TT (q′A) = < q′A,p′B > qB = qA,

so when acting on 1-forms, the operator TT performsthe reverse transformation. The inverse of TT is ap-parently TT −1 = pA ⊗ q′A.

It is not difficult to prove that the field TT definedthis way will be the same at any choice of the ini-tial P -basis. Indeed, when acting on the fields pAof the initial P -basis, the operator TT performs theconsidered active field transformation, and since thistransformation is linear, TT will act the same way onall other covariantly constant five-vector fields, in-cluding the basis fields of any other P -basis.

Let us now find the components of TT and TT −1 inthe basis pA ⊗ qB . Since

p′α = pβΛβα + aαp5 and p′5 = p5,

one has

TT = (pβΛβα + aαp5)⊗ qα + p5 ⊗ q5

= pα ⊗ qβ · Λαβ + p5 ⊗ qβ · aβ+ pα ⊗ q5 · 0 + p5 ⊗ q5 · 1,

so in this basis the compoents of TT are given by for-mulae (4). In a similar manner one can find the com-ponents of TT −1:

(T −1)αβ = Lαβ , (T −1)α5 = 0

(T −1)5β = − aγLγβ , (T −1)5

5 = 1.

One should not mix the latter up with the quantitiesT αβ = Lαβ , T α5 = aβ

T 5β = 0, T 5

5 = 1(5)

2


constructed from the transformation parameters forthe Lorentz coordinates themselves. What the latterare is explained in Appendix.

Let us now consider an infinitesimal Poincaretransformation. In this case the matrix Lαβ in for-mula (3) can be presented as

Lαβ = δαβ + ωαβ ,

where ωαβ are infinitesimals that satisfy the condition

ωαβ + ωβα = 0, (6)

where ωαβ ≡ gασωσβ . The parameters aα in formula

(3) are also infinitesimals and accordingly the field TTthat describes this infinitesimal Poincare transforma-tion can be presented as

TT = 1− SS,

where 1 is the unity five-tensor and tensor SS has thecomponents

Sαβ = ωαβ , S5β = − aβ , SA5 = 0, (7)

and therefore

SS = (pαωαβ − p5aβ)⊗ qβ . (8)

It is evident that the latter expression for SS will bevalid in any P -basis.

In view of condition (6), it is more convenient todeal not with SS itself, but with the corresponding five-tensor whose indices are either both lower or both up-per. It is essential that this latter five-tensor shouldbe related to SS by the map ϑg and not by ϑh (see theirdefinition in section 3 of Part II), for the quantitiesωαβ in the left-hand side of equation (6) are obtainedfrom parameters ωαβ by contraction with the matrixgαβ , not hAB . (Besides, if one lowers or raises anindex of SS with hAB , one will obtain a covariantlynonconstant tensor field.) Since

ϑg(pα) = qβηβα and ϑg(p5) = 0, (9)

the completely covariant tensor related to SS by ϑg is

ϑg(pαωαβ − p5aβ)⊗ qβ = ωαβ qα ⊗ qβ , (10)

and is apparently the five-vector equivalent of thecompletely covariant four-tensor of infinitesimal ro-tation: R = ωαβ Oα ⊗ Oβ . However, tensor (10) hasonly six independent components (which is a conse-quence of ϑg being noninjective) and therefore doesnot describe the considered infinitesimal Poincaretransformation completely.

Let us now observe that according to formulae (8)and (9), SS can be obtained from a completely con-travariant five-tensor of rank 2 by lowering one of theindices of the latter with gαβ . Indeed, denoting thiscontravariant five-tensor as RR, by virtue of formulae(9) one has

RAB pA ⊗ ϑg(pB) = (pαRαβ + p5R5β)⊗ qβ ,

where RAβ ≡ RAξgξβ , and comparing the right-handside of the latter equation with formula (8) one findsthat

Rαβ = ωαξ gξβ ≡ ωαβ and R5β = − aβ . (11)

The components RA5 are not fixed by the compo-nents of SS and can be selected arbitrarily. A partic-ularly convenient choice is

Rα5 = −R5α = aα and R55 = 0. (12)

In this caseRR becomes completely antisymmetric andconsequently has only 10 independent components,i.e. exactly as many as does the five-tensor SS. Sinceantisymmetry is an invariant property, equations (12)will hold in any five-vector basis. Moreover, RR willbe a covariantly constant field. By analogy with theformula for generators of Lorentz transformations forfour-vectors, the relation between RR and SS can bepresented as

SAB = − 12 RKL (MKL)AB = −R|KL| (MKL)AB , (13)

where, as usual, the vertical bars around the indicesmean that summation extends only over K < L, andthe quantities

(MKL)AB ≡ δAL gKB − δAK gLB (14)

are the analogs of the Lorentz generators (Mµν)αβ ≡δαν gµβ − δαµ gνβ .

As any other five-vector bivector, RR can be in-variantly decomposed into a part made only of five-vectors from Z and a part which is the wedge productof a five-vector from E with some other five-vector.In the following these two parts of RR will be calledits Z- and E-components, respectively. Since Z isisomorphic to V4, at each space-time point to the Z-component of RR one can put into correspondence acertain four-vector bivector, which, as is seen fromequation (11), is the completely contravariant formof the infinitesimal rotation four-tensor R introducedabove and which for this reason I will denote withthe same letter. It is easy to check that the four-tensor field R obtained this way is covariantly con-stant and that if Eα is some four-vector basis and eA

3


is the associated active regular five-vector basis, thenthe components of R in the basis Eα ⊗Eβ equal thecomponents Rαβ of RR in the basis eA ⊗ eB .

In a similar way, since the maximal vector spaceof simple bivectors over V5 with the direction vectorfrom E is isomorphic to V4, at each space-time pointto the E-component of RR one can put into correspon-dence a certain four-vector A. For practical reasons,it is more convenient to establish the isomorphismbetween the above two vector spaces not as it hasbeen done in section 3 of part II, but in a slightlydifferent way: supposing that the space of bivectorsis endowed not with the inner product induced by h,but with the inner product differing from the latterby the factor ξ−1. In this case the components of Ain the basis Eα will equal the components Rα5 of RRin the basis eA⊗eB , so A itself will coincide with theinfinitesimal displacement four-vector. At R 6= 0 thefour-vector field A will not be covariantly constant,which is in agreement with the fact that the values ofR and A at any given point Q determine the rotationand translation of a Lorentz coordinate system withthe origin at Q that one has to make to perform theconsidered active Poincare transformation.

C. Motion of a perfectly rigit bodyin the language of three-plus-one-vectors

Everything that has been said above about the de-scription of active Poincare transformations in flatspace-time in terms of five-tensors can be applied,with obvious modifications, to the case of flat three-dimensional Euclidean space. Instead of five-vectorsone should now speak about (3+1)-vectors and in-stead of Poincare transformations, about transforma-tions from the group of motions of three-dimensionalEuclidean space. I will now show how the formal-ism developed in the previous section can be appliedfor describing the motion of a perfectly rigit body inclassical nonrelativistic mechanics. In order not tointroduce new notations, I will denote the analogs oftensors TT and RR in three-dimensional space with thesame symbols.

Owing to the absolute rigidity of the body in ques-tion, its motion can be viewed as an active trans-formation of the fields (discrete or continuous) thatdescribe the distribution of matter inside the body—a transformation that develops in time. Accordingly,the change in the position of the body that occursover a finite time period t can be described invari-antly with a certain (3+1)-tensor, TT (t), and the rateof this change can be described with a certain anti-symmetric (3+1)-tensor, WW, equal to the ratio of the(3+1)-tensor RR(dt) that describes to the first order

the infinitesimal transformation that corresponds tothe change in the body position over the time dt tothe magnitude of this time interval:

WW ≡ RR(dt)/dt. (15)

In the following, WW will be referred to as the velocitybivector of the body.

As in the case of RR, at every point in space onecan put into correspondence to WW a pair consistingof a three-vector, which will be denoted as −~V , andof a three-vector bivector. Since the space is three-dimensional, it is more convenient to deal not withthe latter bivector itself, but with the three-vectordual to it, which I will denote as ~Ω. If ~eΘ is someactive regular basis of (3+1)-vectors and ~Ei is theassociated three-vector basis, then the components of~V and ~Ω in the latter are related to the componentsof WW in the basis ~eΘ ⊗ ~eΣ in the following way:

W 5i = −W i5 = V i

W ij = −W ji = εijk Ωk.(16)

It is not difficult to show that at each point in space~V coincides with the translational velocity of a framerigitly fixed to the body and with the origin at thatpoint. Similarly, ~Ω can be shown to coincide with theangular velocity of this frame.

Let us introduce in space an arbitrary system ofCartesian coordinates and consider the vectors ~V and~Ω at the point with coordinates xi. Since the field RRis covariantly constant, so is the field WW, and conse-quently the value ofWW at the considered point can beobtained by transporting parallelly to this point thevalue of WW at the origin. By using formulae (2) onecan easily find that in the O-basis associated with theselected coordinates,

W 5j(x) = W 5j(0) + xiW ij(0)W ij(x) = W ij(0),

and substituting the components of ~V and ~Ω for thoseof WW, one obtains

V i(x) = V i(0) + εijk Ωj(0)xk

Ωi(x) = Ωi(0).

In view of the meaning the vectors ~V and ~Ω have at agiven point, from the latter formulae follows the well-known rule for transformation of translational andangular veclocities as one transfers the origin of themoving frame for which they are defined to anotherpoint:

~V ′ = ~V + ~Ω× ~X and ~Ω′ = ~Ω, (17)

4


where ~X is the position vector that connects the oldorigin with the new one.

Let us now suppose that there is a particle of thebody at the point with coordinates xi. Since at anygiven moment, the velocity of this particle coincideswith the translational veclocity of the frame con-nected to the body with the origin at that point, fromequations (17) follows another well-known relation:

~v = ~V + ~Ω× ~r,

where ~v is the particle velocity and ~r is its positionvector relative to the frame for which ~V and ~Ω aredefined.

Let us now consider the expression for kinetic en-ergy. As one knows, in the general case the latter canbe presented as a sum of three terms: (i) a term bi-

linear in ~V and independent of ~Ω, (ii) a term bilinear

in ~Ω and independent of ~V , and (iii) a term linear

both in ~V and in ~Ω. Since with respect to the three-dimensional space the kinetic energy is a scalar, theabove means that in terms of (3+1)-tensors it can bepresented in the following form:

Ekin. = 12 IΓ∆ΘΣW |Γ∆|W |ΘΣ|, (18)

where IΓ∆ΘΣ are components of some (3+1)-tensorof rank 4, which by definition have the following sym-metry properties:

IΓ∆ΘΣ = IΘΣΓ∆

andIΓ∆ΘΣ = −I∆ΓΘΣ = −IΓ∆ΣΘ.

It is obvious that the kinetic energy of a single particlecan be presented in the same form. By comparingthe right-hand side of formula (18) with the usualexpression for the kinetic energy of a point particlein classical nonrelativistic mechanics, and consideringthat at the point where the particle is located, ~Vcoincides with the particle velocity vector, one findsthat in this case

I 5i5j = −I i55j = −I 5ij5 = I i5j5 = m · δij , (19)

where m is the particle mass, and all other compo-nents of II are zero (here and below I omit the indicesthat numerate the particles). To express the kineticenergy of the body as a whole in terms of the veloci-ties ~V and ~Ω that correspond to some moving framewith the origin at point O, let us make use of the factthat the contraction of (3+1)-tensors is conserved byparallel transport, so the kinetic energy of a givenparticle of the body equals the contraction of twosamples of tensor WW at O with the tensor II corre-sponding to this particle, transported from the point

where the particle is located to O. Thus, for everyparticle

Ekin. = 12 (II transported) 5i5j V

i V j

+ (II transported) 5ijk Vi ε|jk|

l Ωl

+ 12 (II transported) ijkl ε

|ij|m ε|kl|n Ωm Ωn,

(20)

and the kinetic energy of the body as a whole is thesum of the expressions in the right-hand side of thisformula, taken over all the particles. If the consideredparticle is located at the point with coordinates xi,then by using formulae (2) one can easily find that

(II transported) 5i5j = mδij

(II transported) 5ijk ε|jk|

l = mε jil xj

(II transported) ijkl ε|ij|m ε|kl|n

= m (δmn xixi − xmxn),

(21)

and substituting these expressions into formula (20),one obtains the usual expression for the kinetic energyof a rigit body in terms of ~V and ~Ω:

Ekin. = 12 (∑

m) (~V )2 + ~V ·~Ω×(∑

m~r)+ 12 Iij Ωi Ωj ,

where the sum goes over all the particles and

Iij ≡∑

m (δij xkxk − xixj).

Thus, the moments of inertia of the body with respectto O can be found by dualizing the ijkl-componentsof the summary (3+1)-tensor

∑II at O with respect

to the indices i and j and with respect to the indicesk and l by using the three-dimensional ε tensor. Tofind the moments of inertia relative to any other pointO′, one should simply transport the tensor

∑II from

O to O′ according to the rules of parallel transportfor (3+1)-tensors and dualize its ijkl-components atthat point.

Tensor II can be contracted with only one sample oftensor WW. One will then obtain a completely covari-ant antisymmetric (3+1)-tensor of rank two, which Iwill denote as MM, with the components

MΓ∆ = IΓ∆ΘΞW |ΘΞ|. (22)

As in the case of RR and WW, at each point in spaceone can put into correspondence toMM a certain pairconsisting of a three-vector 1-form and a three-vector2-form. For practical reasons it is convenient first toreplace the 2-form with the 1-form dual to it, therebyobtaining instead of the original pair a pair consistingof two three-vector 1-forms, and then replace theselatter 1-forms with the corresponding three-vectors.As a result, toMM there will correspond a pair consist-ing of two three-vectors, which I will denote as −~P

5


and ~M . If oΣ is the basis of three-plus-one-vector1-forms dual to the basis ~eΘ introduced above, thenthe components of ~P and ~M in the associated three-vector basis ~Ei will be related to the components ofMM in the basis oΘ ⊗ oΣ as follows:

P i = δijM5j and M i = 12 εijkMjk.

Let us now calculateMM for a single point particle.According to equations (19), at the point where thelatter is located

M5i = mδijvj = mvi and Mij = 0,

so in this case ~P coincides with the particle momen-tum three-vector, and ~M = 0. Let us now supposethat in some system of Cartesian coordinates the con-sidered particle has the coordinates xi. Let us trans-portMM to the origin of this system. Then, accordingto formulae (2), in the O-basis associated with thesecoordinates,

M5j(0) = M5j(x) = mvjMij(0) = xiM5j(x) + xjMi5(x)

= m (xivj − xjvi),

and consequently the pair (−~P , ~M) corresponding to

the transportedMM is such that ~P coincides with theparticle momentum three-vector transported to theorigin according to ordinary rules of parallel trans-port for three-vectors, and ~M coincides with thethree-vector of the particle angular momentum rel-ative to the origin. Since the latter can be selectedarbitrarily, this correspondence between MM and theparticle momentum and angular momentum will existat every point. Naturally, each of the three-vectorsin the pair (−~P , ~M) can be transported to any otherpoint in space according to the rules of parallel trans-port for three-vectors, however, the pair as a wholewill correspond to the (3+1)-tensor MM only at thepoint with respect to which the angular momentumis defined.

We thus see that the momentum and angular mo-mentum in classical nonrelativistic mechanics can bedescribed by a single geometric object—by the an-tisymmetric (3+1)-tensor MM. It is natural to calledthe latter the momentum–angular momentum tensor.One of the advantages of such a description comparedto the description in terms of three-vectors is thatMMcan be defined in a purely local way, with no refer-ence to any other point in space. For example, inthe case of a single point particle, at the point wherethe latter is located the E-component of MM is ex-pressed in terms of the particle momentum and theZ-component of MM is zero. Having defined the ten-sor MM this way, one can then transport it to any

other point in space. This transport will result inthat MM will acquire a nonzero Z-component, whichwill be exactly the angular momentum of the particlerelative to the point where MM has been transportedto.

In order to calculate the momentum–angular mo-mentum tensor for a system of point particles, oneshould first transport the tensors MM correspondingto all the particles to one point in space. It is atthis stage that the tensors MM of individual parti-cles will acquire nonzero Z-components, which, whensummed up, will give the total angular momentum ofthe system relative to the selected point. In the par-ticular case where the system is a rigit body, its to-tal momentum and total angular momentum can beexpressed in terms of the velocities ~V and ~Ω corre-sponding to some frame connected to the body, withthe origin at point O. To do this, one should followthe same procedure that has been used above for cal-culating the kinetic energy: one should transport thetensors II corresponding to individual particles of thebody to O, sum them up there, and then contractthe sum

∑II with the velocity bivector of the body

at that point. As a result, one will obtain the usualexpressions for momentum and angular momentumof the rigit body in terms of ~V and ~Ω, which I willnot present here.

Let us now discuss the equations of motion. In thecase of a single point particle one has:

d~P/dt = ~F , (23)

where ~P is the particle momentum and ~F is the actingforce. Since ~P corresponds to the E-component of the(3+1)-tensor MM, one may suppose that the above

equation of motion corresponds to the E-componentof some three-plus-one-tensor equation. One shouldexpect that the left-hand side of this latter equation isthe time derivative ofMM and that its right-hand sideis some antisymmetric (3+1)-tensor of rank 2, whichI will denote as KK. Thus, the three-plus-one-tensorequation will have the form

dMM/dt = KK, (24)

and now we should determine how KK is related to theknown three-vector quantities.

Let us introduce in space some system of Cartesiancoordinates and let x(t) denote the trajectory of theparticle. To evaluate the time derivative in the left-hand side of equation (24), one should take the tensorMM(t+ dt) at the point x(t+ dt), transport it accord-ing to the rules of parallel transport for (3+1)-tensorsto the point x(t), subtract from it the tensorMM(t) at

6


that point, and divide the difference by dt. Follow-ing this procedure, one will find that in the O-basisassociated with the selected coordinates,

K5i(t) =mvi(t+ dt)−mvi(t)

dt=d(mvi)

dt= δijF

j(t).

Similarly, since at any t the Z-component ofMM(t) atthe point x(t) is zero, one will have

Kij(t) = vi(t)M5j(t) + vj(t)Mi5(t)= mvi(t)vj(t)−mvj(t)vi(t) = 0.

As MM, KK can be represented by a pair of three-vectors. The results we have just obtained mean thatat the point where the particle is located, the paircorresponding to KK is (−~F ,~0). In the particular casewhere no forces act on the particle, one obtains

dMM/dt = 0,

which is nothing but the conservation law for mo-mentum and angular momentum of a free particle,written down in the language of (3+1)-tensors.

Suppose now that ~F 6= 0. Let us transport thetensors in both sides of equation (24) to some otherpoint O. As we know, the pair corresponding to thetransportedMM will consist of the particle momentumthree-vector with the minus sign and of the three-vector of particle angular momentum relative to O.Similarly, one can find that the pair corresponding tothe transported KK will consist of the three-vector −~Ftransported to O according to the rules of paralleltransport for three-vectors, and of the three-vector~K of the force moment relative to O. Thus, whentransported to the indicated point, the three-plus-one-tensor equation (24) is equivalent to the followingtwo three-vector equations: equation (23) and theequation

d ~M/dt = ~K.

In the case of a system of point particles, one cansum up equations (24) corresponding to all the parti-cles in the system, provided one first transports themall to some point O, and obtain the three-plus-one-tensor equation

dMMtot/dt = KKtot, (25)

which is apparently equivalent to two three-vectorequations that equate the time derivatives of the to-tal momentum three-vector and of the three-vector oftotal angular momentum relative to O respectively tothe three-vector of total force evaluated in the usualway and to the three-vector of the total force momentrelative to O.

D. Bivector derivative

Let us consider the group of active Poincare tranfor-mations of scalar, four-vector and four-tensor fieldsin flat space-time. Let us distinguish in it some one-parameter family H that includes the identity trans-formation. Let us denote the parameter of this fam-ily as s and the image of an arbitrary field G under atransformation from H as ΠsG. It is convenient totake that the identity transformation corresponds tos = 0.

For the selected one-parameter family H and forany sufficiently smooth field G from the indicatedclass of fields, one can define the derivative

DHG ≡ (d/ds)ΠsG|s=0, (26)

which is a field of the same type as G. It is apparentthat for every type of fields, the operators DH cor-responding to all possible one-parameter families Hmake up a 10-dimensional real vector space, which isnothing but the representation of the Lie algebra ofthe Poincare group that corresponds to the consid-ered type of fields.

Let us introduce in space-time some system ofLorentz coordinates xα and select a basis in thespace of operators DH consisting of the six opera-tors Mµν that correspond to rotations in the planesxµxν (µ < ν) and of the four operators Pµ that cor-respond to translations along the coordinate axes. Ifone parametrizes the indicated transformations withthe parameters ωαβ and aα introduced in section B,then for an arbitrary scalar function f one will have

Pµf(x) = ∂µf(x)Mµνf(x) = xν∂µf(x)− xµ∂νf(x),

(27)

for an arbitrary four-vector field U one will have

(PµU)α(x) = ∂µUα(x)

(MµνU)α(x) = xν∂µUα(x)− xµ∂νUα(x)+ (Mµν)αβ U

β(x),(28)

where the components correspond to the Lorentzfour-vector basis associated with the selected coor-dinates; and so on.

With transition to some other system of Lorentzcoordinates with the origin at the same point, thederivatives MµνG transform with respect to the in-dices µ and ν as components of a four-vector 2-form,and the derivatives PµG transform with respect to µas components of a four-vector 1-form. Consequently,if one constructs out of these quantities the fields

PG ≡ PµG · Oµ and MG ≡ M|µν|G · Oµ ∧ Oν , (29)

7


where Oµ is the basis of four-vector 1-forms associ-ated with the selected coordinate system, these fieldswill be the same at any choice of the latter. Fromdefinition (29) it follows that at every point in space-time

PµG = < PG , Eµ >MµνG = < MG , Eµ ∧Eν >,

(30)

where Eµ is the four-vector basis corresponding to theLorentz coordinate system with respect to which theoperators PµG and MµνG are defined. If the fields PGand MG were completely independent of the choice ofthe Lorentz coordinate system, then basing on rela-tions (30) one could regard MµνG as a special kind ofderivative whose argument is a four-vector bivector:

MµνG ≡ DEµ∧EνG,

and PµG as a derivative whose argument is a four-vector:

PµG ≡ DEµG,

and then at any point in space-time, the family H forwhich the derivative DH is evaluated could be identi-fied in a coordinate-free way by indicating the four-vector and the four-vector bivector that correspondto this family. The field PG is indeed independentof the choice of the coordinate system, and it is easyto see that for all types of fields from the consideredclass of fields the operator DEµ coincides with theoperator of the covariant derivative in the directionof the four-vector Eµ (I am talking about flat space-time only). However, the field MG does depend onthe choice of the origin, since under the translationxα → xα + aα it transforms as

MG → MG + PG ∧ A,

where A ≡ aαOα, so one cannot regard MµνG as aderivative whose argument is a four-vector bivectorirrespective of the choice of the coordinate system.

Since derivative (26) is associated with activePoincare transformations, basing on the results ofsection B one may expect that at any point in space-time the derivatives DH corresponding to various one-parameter familiesH can be parametrized invariantlywith five-vector bivectors. To see that this is indeedso, one may observe that with transition from oneLorentz coordinate system to another, the quantitiesMµνG and PµG transform respectively as the µν- andµ5-components of a five-vector 2-form in the P -basis.Consequently, the field

DG ≡ M|µν|G · qµ ∧ qν + PµG · qµ ∧ q5,

where qA is the basis of five-vector 1-forms dual tothe P -basis, pA, associated with the selected Lorentz

coordinate system, will be the same at any choice ofthe latter. Similar to equations (30), one will havethe relations

PµG = < DG , pµ ∧ p5 >MµνG = < DG , pµ ∧ pν >,

basing on which one can regard PµG and MµνG asparticular values of the derivative whose argumentis a five-vector bivector, and which in view of thisI will call the bivector derivative. For any Lorentzcoordinate system one will apparently have

PµG = Dpµ∧p5G and MµνG = Dpµ∧pνG, (31)

where pA is the P -basis associated with these coordi-nates. Comparing the latter formulae with formulae(27) at the origin, one can see that for any activeregular basis eA and any scalar function f ,

Deµ∧e5f = ∂eµf and Deµ∧eνf = 0. (32)

From these equations it follows that at the point withcoordinates xα,

Dpµ∧p5f = DpZµ ∧p5

f = Deµ∧e5f = ∂eµf = ∂µf

and

Dpµ∧pνf = Deµ∧eνf + xνDeµ∧e5f + xµDe5∧eνf

= xν∂µf − xµ∂νf,

which is in agreement with formulae (27) in the gen-eral case (in the latter two chains of equations and inequations (33) and (34) that follow, eA denotes theO-basis associated with the considered coordinates).Comparing formulae (31) with formulae (28) at theorigin, one can see that for any Lorentz four-vectorbasis Eα,

Deµ∧e5Eα = 0 and Deµ∧eνEα = Eβ (Mµν)βα, (33)

so for any such basis

DpA∧pBEα = DeA∧eBEα (34)

at allA andB. From the properties of Poincare trans-formations and from definition (26) it follows that forany scalar function f and any four-vector field V,

DpA∧pB (fV) = DpA∧pBf ·V + f · DpA∧pBV, (35)

which together with equations (33) and (34) gives for-mulae (28) for an arbitrary four-vector field U. Simi-lar formulae can be obtained for all other four-tensorfields.

One can now consider a more general derivativethan DH by allowing the one-parameter family H to

8


vary from point to point. For any field G from theconsidered class of fields, such a derivative is a fieldwhose value at each space-time point coincides withthe value of one of the fields DHG for some familyH, which at different points may be different. Every-where below, when speaking of the bivector derivativeI will refer to this more general type of differentiation.

According to the results obtained above, any suchderivative can be uniquely fixed by specifying a cer-tain field of five-vector bivectors. Therefore, by anal-ogy with the covariant derivative, for any type offields D can be formally regarded as a map thatputs into correspondence to every pair consisting ofa bivector field and a field of the considered type an-other field of that type. For example, the bivectorderivative for four-vector fields can be viewed as amap

FF∧FF ×DD → DD, (36)

where FF∧FF is the set of all fields of five-vector bivec-tors and DD is the set of all four-vector fields. Fromthe definition of the bivector derivative it follows thatmap (36) has the following formal properties: for anyscalar functions f and g, any four-vector fields U andV, and any bivector fields AA and BB,

D(fA+gB)U = f · DAU + g · DBU (37a)DA(U + V) = DAU + DAV (37b)DA(fU) = DAf ·U + f · DAU. (37c)

In the third equation, the action of D on the functionf is determined by the rules:

D(A+B)f = DAf + DBf,DAZf = 0, DAEf = ∂Af,

(38)

where A denotes the four-vector field that corre-sponds to the E-component of AA.

The properties of D presented above are similar tothe three main properties of the covariant derivativethat are used for defining the latter formally. Usingproperties (37) for the same purpose is not very con-venient, since to define the bivector derivative com-pletely one has to supplement them with the formulaethat determine the relation of D to space-time metric,and usually from such relations one is already able toderive part of the properties expressed by equations(37). As an example, let us consider the formulae thatexpress the operator D in terms of the operator ∇· ofthe torsion-free g-conserving covariant derivative and

of the linear local operator M defined below, both ofwhich are completely determined by the metric. Foran arbitrary four-vector field U one has:

DAEU = ∇·AU and DAZU = MBU, (39)

where A, as in definition (38), denotes the four-vectorfield corresponding to the E-component of AA, B de-notes the field of four-vector bivectors corresponding

to the Z-component of AA, and the operator M, whichdepends linearly on its argument, has the followingcomponents in an arbitrary four-vector basis Eα:

MEα∧EβEµ = Eν(Mαβ)νµ.

It is easy to see that properties (37b) and (37c) followfrom formulae (39) and property (37a), and property(37a) itself follows from equations (39) and the fol-lowing simpler property:

D(A+B)U = DAU + DBU,

which is similar to the first equation in definition (38)and which, together with equations (39), can serve asa definition of the bivector derivative for four-vectorfields.

The action of operator D on all other four-tensorfields can be defined either independently—accordingto formula (26), or as in the case of the covariantderivative—according to the equations that expressthe Leibniz rule in application to the contraction ofa four-vector 1-form with a four-vector field and tothe tensor product of any two four-tensor fields. Thecorresponding formulae are quite obvious and will notbe presented here.

For the bivector derivative one can define theanalogs of connection coefficients. Namely, for anyset of basis four-vector fields Eα and any set of basisfive-vector fields eA one can take

DABEµ = EνΓνµAB , (40)

where DAB ≡ DeA∧eB . According to equations (33),for any Lorentz four-vector basis and any standardfive-vector basis associated with it, one has

Γµνα5 = −Γµν5α = 0Γµναβ = −Γµνβα = (Mαβ)µν .

(41)

The bivector connection coefficients for any otherchoice of the basis fields can be found either by usingthe following transformation formula:

Γ′µνAB = (Λ−1)µσΓστSTΛτνLSAL

TB

+ (Λ−1)µσ(DSTΛσν)LSALTB ,

(42)

which corresponds to the transformations E′α =EβΛβα and e′A = eBL

BA of the four- and five-vector

basis fields, or by using formulae (39). In particular,for an arbitrary four-vector basis and the correspond-ing active regular five-vector basis one has

Γµνα5 = Γµνα and Γµναβ = (Mαβ)µν , (43)

9


where Γµνα are ordinary four-vector connection coef-ficients associated with ∇· .

Let me also observe that the quantities DABG foran arbitrary field G can be presented as derivativeswith respect to the components of the bivector RR in-troduced in section B. Indeed, according to the defi-nition of the operators Pµ and Mµν , one has

D5µG = −(∂/∂aµ)ΠG|ωαβ=aα=0

DµνG = (∂/∂ωµν)ΠG|ωαβ=aα=0 (µ<ν),(44)

where it is assumed that the image ΠG of fieldG under an infinitesimal Poincare transformation isa function of parameters ωαβ and aα. In view ofequations (11), the latter formulae can be rewrittenas

D5µG = (∂/∂R5µ)ΠG|Rαβ=R5α=0

DµνG = (∂/∂Rµν)ΠG|Rαβ=R5α=0,(45)

and since Dµν = −Dνµ and Rµν = −Rνµ, the secondequation will be valid at µ > ν as well. Since Dµ5 =−D5µ, one can write that

Dµ5G = (∂/∂(−R5µ))ΠG|Rαβ=R5α=0,

so if one takes Rµ5 = −R5µ, as it has been done insection B, one will have

DABG = (∂/∂RAB)ΠG|R=0, (46)

for all A 6= B, which is one more argument in favourof the choice (12).

E. Bivector derivative of the Lagrange function

In the previous section I have defined the bivectorderivative for scalar, four-vector and four-tensor fieldsin flat space-time. In a similar manner the bivectorderivative can be defined for more complicated ob-jects. As an example, I will now consider the defini-tion of the analog of the bivector derivative in three-dimensional Euclidean space for the Lagrange func-tion of a system of several point particles, in classicalnonrelativistic mechanics.

As is known, the state of motion of such a system atevery moment of time can be fixed by specifying theposition of each particle in space and its velocity. TheLagrange function L for this system can be viewed asa map that puts into correspondence to each allowedstate of motion C a real number, L(C). In the generalcase this map may be explicitly time-dependent.

The bivector derivative of the Lagrange functioncan be defined according to an equation similar toformula (26). For that one apparently has to de-fine first how L changes under active transformations

from the group of motions of three-dimensional Eu-clidean space. This can be done in a standard way ifone knows how such transformations affect the statesof motion of the system. Namely, for any transfor-mation from the indicated group the image ΠL ofthe Lagrange function is such a function of the stateof motion that for any C

ΠL(ΠC) = L(C), (47)

where ΠC is the image of state C under the consid-ered active transformation. It is apparent that equa-tion (47) can be presented in the following equivalentway:

ΠL(C) = L(Π−1C), (48)

where Π−1 is the transformation inverse to Π. Bas-ing on definition (47), one can define the derivativeof L relative to some one-parameter family of trans-formations H as follows: DHL is such a real-valuedfunction of the state of motion that for any C

DHL(C) = (d/ds)ΠsL(C)|s=0, (49)

where s is the parameter of the considered family.If one proceeds from definition (48), then instead of(49) one will have the following equivalent definition:

DHL(C) = (d/ds)L(Π−1s C)|s=0. (50)

By using some particular set of variables for char-acterizing the state of motion of the system, for ex-ample, the coordinates of all the particles in someCartesian coordinate system and the components oftheir velocities in the corresponding three-vector ba-sis, it is not difficult to show that derivatives (49) forall one-parameter families H are correlated with thederivatives of scalar, three-vector and three-tensorfields relative to all these families in the followingsense: if families H, H′, and H′′ are such that for anyfield G from the indicated class of fields,

DHG = a · DH′G + b · DH′′G,

where a and b are some real numbers, then for anystate of motion C,

DHL(C) = a · DH′L(C) + b · DH′′L(C).

This fact enables one to construct another functionof the state of motion, which, as L, may explicitlydepend on time, but whose values will be covari-antly constant fields of antisymmetric (3+1)-tensorsof rank two rather than numbers. To this end, let usobserve that as in the case of space-time, to every one-parameter family H one can put into correspondence

10


a certain covariantly constant field AA of three-plus-one-vector bivectors, such that the derivative DHG ofany scalar, three-vector or three-tensor field G can bepresented as the contraction < DG,AA >, where DGis a three-plus-one-vector 2-form independent of H,whose values are quantities of the same kind as thoseof G. In a similar way one can define the scalar-valued three-plus-one-vector 2-form DL, which willbe a function of the state of motion of the system(and of time), by taking that for any C

< DL(C),AA > = DHL(C),

where AA is an arbitrary covariantly constant fieldof three-plus-one-vector bivectors and H is the one-parameter family that corresponds to it. SinceDHL(C) is simply a number, from the fact that AAis covariantly constant follows that at any C the fieldDL(C) will be covariantly constant, too.

Characterizing the state of motion of the systemwith coordinates xi` of all the particles in some Carte-sian coordinate system and with the componentsvi` ≡ dxi`/dt of their velocities in the correspondingthree-vector basis (the index ` numerates the parti-cles), it is not difficult to evaluate the components ofDL in the basis of three-plus-one-vector 2-forms cor-responding to the P -basis of (3+1)-vectors associatedwith the selected coordinates. One obtains:

Di5L =∑`

∂L/∂xi` (51)

and

DijL =∑`

(xj ` · ∂L/∂xi` − xi ` · ∂L/∂xj`

), (52)

where xi ` ≡ δijxj` . Since the quantities ∂L/∂xi` are

covariant components of the force that acts on the`th particle relative to the basis of three-vector 1-forms associated with the selected coordinate system,equations (51) and (52) mean that at the origin O of

this system the E-component of DL corresponds tothe three-vector 1-form of the total force that acts onthe system and the Z-component of DL correspondsto the three-vector 2-form of the total force momentrelative to O taken with the opposite sign. Therefore,−DL is exactly the three-plus-one-vector 2-form KKtotintroduced in section C.

Among other thing, from the latter fact follows theresult we have obtained earlier: that momentum andangular momentum of a system of particles in clas-sical nonrelativistic mechanics can be described bya single local object—by a three-plus-one-vector 2-form. Indeed, according to the equations of motion,the force that acts on the particle and the moment

of this force relative to an arbitrary point O are totaltime derivatives respectively of the particle momen-tum and of its angular moment relative to O. Conse-quently, the three-plus-one-vector 2-form KK is also atotal time derivative of some (3+1)-tensor. Since inthe nonrelativistic case time is an external parameter,the rank of this (3+1)-tensor should be the same asthat of KK, and according to what has been said above,at an arbitrary point in space the E-component ofthis (3+1)-tensor will correspond, with the oppositesign, to the three-vector 1-form of the particle mo-mentum, and its Z-component will correspond to thethree-vector 2-form of the particle angular momen-tum relative to that point.

Appendix: Contravariant basis

As in the case of any other vector space endowed witha nondegenerate inner product, to any basis Eα in V4

one can put into correspondence the basis

Eα ≡ gαβEβ , (53)

which will be called contravariant. Here, as usual,gαβ denote the matrix inverse to gαβ ≡ g(Eα,Eβ).Definition (53) is equivalent to the following relation:

< Oα,V > = g(Eα,V) for any four-vector V,

where Oα is the basis of four-vector 1-forms dualto Eα. The latter relation means that the four-vectors Eα are inverse images of the basis 1-formsOα under the map V4 → V4 defined by the innerproduct g. From definition (53) it also follows that

g(Eα,Eβ) = δ βα , so the four-vector 1-forms Oα that

make up the basis dual to Eα are images of the ba-sis four-vectors Eα under the indicated map. It isevident that when Eα is a Lorentz basis in flat space-time, associated with some Lorentz coordinate sys-tem xα, one has Eα = ∂/∂xα, where xα are the cor-responding covariant coordinates. It is also evidentthat at any affine connection relative to which themetric tensor is covariantly constant, the relation be-tween the bases Eα and Eα is preserved by paralleltransport, and it is easy to see that in this case

∇µEα = − ΓαβµEβ ,

where Γαβµ are the connection coefficients correspond-ing to the basis fields Eα.

In a similar manner, by using the nondegenerateinner product h, one can define the contravariant ba-sis eA corresponding to an arbitrary five-vector basiseA:

< oA,v > = h(eA,v) for any five-vector v,

11


where oA is the basis of five-vector 1-forms dual toeA. However, such a definition of the contravari-ant basis is inconvenient in two ways. First of all,the contravariant basis corresponding to an arbitrarystandard basis eA will in general not be a standardbasis itself (this will be the case only if eA is a reg-ular basis). Secondly, since the inner product h isnot conserved by parallel transport, the latter willnot preserve the correspondence between eA and eA

either. In view of this, in the case of five-vectors it ismore convenient to define the contravariant basis inanother way. Namely, for any standard basis eA onetakes that

eα = gαβeβ and e5 = e5. (54)

It is a simple matter to see that the first four vectorsof the contravariant basis defined this way satisfy therelation

< oα,v > = g(eα,v) for any five-vector v,

which, however, is not equivalent to the first equa-tion in definition (54) since it does not fix the E-components of the vectors eα. It is not difficult tosee that the basis of five-vector 1-forms dual to eA,which will be denoted as oA, is expressed in terms ofthe basis 1-forms oA as

oα = oβgβα and o5 = o5.

The first of these equations is equivalent to the rela-tion

< oα,v > = g(eα,v) for any five-vector v,

which means that oα are images of the basis five-vectors eα with respect to the map ϑg : V5 → V5

defined in section 3 of part II. It is also evident thatfrom eα ∈ Eα follows eα ∈ Eα.

According to definition (54), the contravariant P -basis associated with some system of Lorentz coordi-nates xα in flat space-time is expressed in terms ofthe corresponding contravariant O-basis as

pα = eβ + xαe5 and p5 = e5. (55)

Among other things, from the latter equation it fol-lows that in such a P -basis, the five-vector 1-form xintroduced in section 5 of part I has the components(xα, 1). Since the correspondence between pA andpA is preserved by parallel transport, from formula(55) one finds that for the contravariant O-basis

∇µeα = − δαµe5 and ∇µe5 = 0.

Finally, it is a simple matter to show that in the ba-sis of five-tensors of rank (1, 1) associated with the

P -basis (55), the components of the tensor TT thatdescribes the active Poincare transformation corre-sponding to transformation (3) of Lorentz coordi-nates, are given by formula (5).

12


Fractal-Сluster Theory and Thermodynamic

Principles of Biological Systems Control and Analysis

V.Volov

Samara Institute of Fundamental Research, Samara, Russia

e-mail: [email protected]

Abstract

In this article we suggest the results of using of the fractal-cluster theory and thermodynamics for

the biological organism evolution and protein-water dehydration analysis. It has received tfree fractal-

cluster laws for biological organisms: energy law, probability law and evolutional law. It was shown that

self-organized energy of water-protein system approximately equals 1% percent from ATF (АТP) energy

per one of weight.

By using the mathematical tools of the fractal-cluster theory [1] we have

determined the values of fractal-cluster entropy criteria (H), F-criterion and the

effectiveness criterion D for the biological organisms [2] (table 1). It has allowed to

formulate three fractal-cluster laws: energy, evolutional and stochastic for the biological

organisms:

1) stochastic law, which determines the probability of biological organisms appearance

(as shown in Fig. 1, there is a bijection - the most ancient biological organisms -

hlamidomonas, Hydra, had the highest probability of ~ 0.01);

2) the evolutional law (Fig. 2), illustrating the increasing complexity and perfection of

the emerging organisms;

3) energy law (Fig. 3), which characterizes the energy perfection of biological

organisms (the dependence of the fractal-cluster entropy H or F- criterion on the energy

consumption per 1 kg body weight per day).


Besides it the article presents the results of researches of biological systems

protein-water [3] genesis on the basis of the thermodynamical tools.

Professor E.G. Rapis’ work first showed that the protein study exclusively by

classical crystallographic methods is absolutely unjustified. As it is shown in her study,

the process of liquid evaporation in protein-water system (in vitro) leads to the

processes of protein structures self-assembly in the form of complex fractal structures

for micro- (protein nano films), as well as for macro-level. In this case there exists the

identity of the structural pattern of protein self-assembly in vivo and in vitro (Fig. 4).

In connection with this fact the study of "protos" (protein + H2O) system self-

organization should be carried out as an object representing an open thermodynamic

system. To analyze the processes of protein self-organization as an open thermodynamic

system, it is convenient to use the developed apparatus of fractal-cluster theory (FCT)

[1] based on the fundamental laws and theorems of non-equilibrium thermodynamics.

The paper shows that in the "protos" system a solid residue is, in terms of FCT, the

energy clusterэК . This allows us to calculate the conditional entropy (FC-entropy of

the system), D-criterion and the F-criterion. By measuring the content of H2O in the

"protos" system at different points of time, you can analyze the stability of protein self-

organization processes during the evaporation.

To estimate the upper energy limit of protein self-organization (self-assembling

three-dimensional macro protein structures, [3]) the respective work Aselforg is calculated

on the basis of Carnot theorem:

sub

sur

CarnotТ

Т1η −= , 1

where Tsur, Tsub- surrounding environment temperature and substrate temperature.

The total energy of "protos" system is


TmcW ii

2

1

∑=

=

i

, 2

where ci (i = 1, 2) - heat capacity of protein and water respectively, T - temperature of

the "protos" system, mi (i = 1, 2) - mass of the protein and H2O respectively.

The work of self-assembly (self-organization) of three-dimensional protein

structures from nano-to macro-level will be equal

Carnotselforg ηWА ⋅= , 3

The calculations show that the specific energy of protein structures self-

organization is commensurate with Adenosine-5'-triphosphate (ATP) specific energy.

When the "protos" system was investigated as an open thermodynamic system, ATP

was absent. From this fact, given the identity of the protein structure in vivo and in vitro

(Fig. 4), we can assume that in the living there is not one energy information system

(ATP), but two (ATP and energy information system of protein structures self-

organization).

Table 1

Species of Biological

organism D

H

J Kэ F

1 Chlamidomonas 0,361 0,914 0,622 0,324 0,939 0,387 -0,235

2 Hydra vulgaris 0,636 0,953 0,637 0,619 0,538 0,4 -0,237

3 Scorpiones mingrelicus 0,833 0,957 0,561 0,726 0,333 0,338 -0,223

4 Oligochaeta 0,549 0,858 0,342 0,281 0,911 0,19 -0,152

5 Anisoptera libellula depressa 0,832 0,965 0,832 0,827 0,252 0,4 -0,238

6 Micromys minitus 0,882 0,973 0,639 0,886 0,177 0,4 -0,239

7 Rona ridibunda 0,849 0,971 0,639 0,852 0,22 0,4 -0,239

8 Testudo horsefieldi 0,624 0,94 0,614 0,579 0,571 0,38 -0,234

9 Cucules canorus 0,882 0,973 0,639 0,886 0,177 0,4 -0,239

10 Procellariida 0,872 0,968 0,662 0,904 0,177 0,42 -0,242

11 Larus argentatus 0,883 0,974 0,627 0,872 0,185 0,39 -0,237

η χ


12 Heroestes edwardsi 0,879 0,973 0,639 0,882 0,182 0,4 -0,239

13 Ciconia ciconia 0,868 0,97 0,664 0,902 0,18 0,421 -0,243

14 Lepus timidus 0,871 0,973 0,639 0,875 0,192 0,4 -0,239

15 Grus grus 0,868 0,97 0,664 0,902 0,18 0,421 -0,243

16 Paralithodes camtchatica 0,553 0,859 0,359 0,282 0,9 0,2 -0,159

17 Pelecanida onocrotalus 0,871 0,973 0,639 0,875 0,192 0,4 -0,239

18 Vulpes vulpes 0,895 0,974 0,639 0,9 0,16 0,4 -0,239

19 Castor fiber 0,868 0,97 0,664 0,902 0,18 0,421 -0,243

20 Acinonyx jubatus 0,811 0,968 0,639 0,81 0,278 0,4 -0,239

21 Canis lipus 0,9 0,975 0,639 0,905 0,153 0,4 -0,239

22 Pan troglodytes 0,907 0,976 0,627 0,897 0,152 0,39 -0,237

23 Orycturopus afer 0,881 0,976 0,633 0,879 0,182 0,395 -0,238

24 Homo sapiens 0,932 0,977 0,614 0,906 0,13 0,38 -0,234

25 Ursus arctos 0,879 0,973 0,639 0,882 0,182 0,4 -0,239

26 Cervina nippon 0,852 0,971 0,639 0,855 0,218 0,4 -0,239

27 Sus scrofa 0,878 0,973 0,639 0,881 0,183 0,4 -0,239

28 Pongo pygmaeus 0,883 0,974 0,627 0,871 0,186 0,39 -0,237

29 Gorilla gorilla 0,922 0,976 0,627 0,913 0,133 0,39 -0,237

30 Equida burchelli 0,849 0,969 0,636 0,845 0,228 0,4 -0,239

31 Tursiops 0,806 0,965 0,659 0,827 0,272 0,42 -0,242

32 Equus caballus 0,849 0,969 0,636 0,845 0,228 0,4 -0,239

33 Galeocerdo cuvieri 0,811 0,968 0,639 0,81 0,278 0,4 -0,239

34 Camelus bactrianus 0,851 0,971 0,639 0,853 0,221 0,4 -0,239

35 Giraffa cameleopardalis 0,852 0,972 0,639 0,854 0,219 0,4 -0,239

36 Hippopotamus amphibius 0,878 0,973 0,639 0,881 0,183 0,4 -0,239

37 Loxodonta africana 0,878 0,973 0,639 0,881 0,183 0,4 -0,239

38 Balaena mysticetus 0,849 0,969 0,663 0,881 0,206 0,42 -0,243

39 Balaenoptera musculus 0,849 0,969 0,663 0,881 0,206 0,42 -0,243


Fig. 1. Probability law of biological Fig. 2. Evolutionary FC law for biological

organisms appearance organisms

Fig. 3. Energy FC law for biological Fig. 4. Non-equilibrium film of "protos" protein

organisms (lysozyme – water system ) in vivo and

in vitro. Optical microscop. zoom 200.

References

1. V.T. Volov, Fractal-Cluster Theory of Resource Distribution in Socio-Economic

Systems. The "IntellectualArchive", Vol.1, Number 2, ISSN 1929-4700, Toronto, June

2012. pp. 30-51

2. Burdakov V.P. The Effectiveness of Life. M: Energoizdat, 1997.

3. Rapis E.G. Protein and Life. Jerusalem: "Filobiblon, Moscow: ZAO Milta- PCP GIT, 2003.

Deff


Survival is the only moral goal of life

J.C. Hodge1∗

1Blue Ridge Community College, 100 College Dr., Flat Rock, NC, 28731-1690

Abstract

What morals are required for survival and life? The principles of

science are applied to the nature of society and religion. The measure of

moral systems is suggested to be their survival ability. The data are the

natural life of animals and plants and the history of human societies. The

result is a view of nature and a set of morals that are very different from

current thought. We are again facing a crisis. What must we change to

survive?

Human societies have many different moral systems. Each claims to havethe ultimate moral structure in competition with the others. Yet, history hashad many other moralities that have become extinct. Each of the modernsystems has evolved from prior systems. Moral systems seem subject to naturalevolutionary selection.

The priest and the philosopher are engaged in determining the morals andstructure of a society that produce survival. The current testing of social moralsis largely by trial-and-error. That is, the test related to natural evolutionaryselection is lacking. Failure results in war, wholesale collapse, and death.

Why do scientific societies win, survive, and grow? Science ultimately judgesmodels on results not their elegance or intention.

Science theoretically is tolerant of many models. The key decider betweenscientific models is the ability to predict observations. Prediction allows creationof survival tools that are also tools of conquering others. The model is true onlyto the extent of how far into the future the predictions are valid. Unfortunately,many of the postulates our society uses have already been proven false. Yet,the postulates continue to be used.

Another key idea of scientific pursuits is that observations are required to bereproducible. Reproducibility requires the ability to measure results that othersmay observe. Both of these concepts are not currently applied to social ideas.

Ayn Rand in her book The Virtue of selfishness, 1964 edition said: “I quotefrom Galt’s speech[Atlas Shrugged, 1957]: ‘there is only one fundamental al-ternative in the universe: existence or nonexistence ... It is only the conceptof ‘life’ that makes the concept of ‘value’ possible. What standard determineswhat is proper in this context? The standard is the organism’s life, or: that

∗E-mail:jc [email protected]

1


2

which is required for the organism’s survival.’ ” This quote suggests the valueof science, philosophy, and religion is the ability of followers to obtain survival.Rand (an atheist) suggests in Atlas Shrugged and in The Virtue of selfishness

that the Objectivist philosophy provides greater survival ability.The theme of Atlas Shrugged is of a producer (John Galt) struggling with

a collectivist society that takes resources from him by force of guns. Thoseresources are redistributed to non-producers and to the politicians in the nameof altruism. Resources were spent on tasks that failed to contribute to sur-vival. The society collapsed because the innovators and industrialists disap-peared (went on strike, left the collective, or joined the welfare class). Atlas,the supporter (the producers) of the world, shrugged (left). This has been called“going Galt”.

Rand appears to have achieved a cause-and–effect model that suggests thefuture of a collectivist society. The model predicted the crisis we are witnessingin Europe and in the US. Perhaps this is the reason the Objectivist philosophyand Atlas Shrugged is becoming more popular. The sales increase has beenattributed to the uncanny similarities between the plot line of the book and theevents of our day. However, the model fails to suggest how the problems areto be solved other than a total collapse caused by nature followed by a restart.Human history and the natural world support this.

When Galileo was born, Catholic Europe had been tolerant and trading withMuslim countries for several popes. The trial of Galileo was a result of a changeto intolerance. The result of the trial was that other scientists voted with theirfeet and moved to the tolerant north. The same occurred when producers movedto the new world and when scientists left Nazi Germany. Greece today is notingthe brain drain of producers to Germany.

How did we get into this crisis? The Civil War was a result of the Southwanting Northern states to return fleeing slaves. A few people in the Northernstates were attempting to impose their views on the South. Both groups ap-pealed to the federal government to enforce their intolerant views. The 1950sand 1960’s saw the trend toward big federal programs yield improvement inpeoples lives. “Big” in biology and corporate America has efficiency advan-tages. This improvement trend reversed in the 1990s and 2000s while federalprograms continued to increased. Politicians and the electorate viewed the nor-mal, short term, economic downturn and stagnation of improvement as a crisis.What started as a few small disturbances was allowed to accumulate into amajor pressure. The major pressure must eventually release into a catastrophe.Politicians invariably respond to crises by spawning new government programs,laws, and regulations. These, in turn, generate more havoc, poverty, and intol-erance, which inspire the politicians to create more programs. The downwardspiral repeats itself until the productive sectors of the economy collapse underthe collective weight of taxes, regulation, and other burdens imposed in thequest for fairness, equality, and altruism. This results in a quest for similar be-liefs and morals that becomes intolerant of other views that includes regulationof what views must be accepted such as anti–discrimination regulation. That is,the tolerance of allowing individuals to decide their morals and reap the result


3

is reduced. This is the Tragedy of Tolerance of choice. Nature favors diversityfor selection.

The problem is that the federal government rather than the states becomesubject to nature’s rules. If the federal government enacts regulations inconsis-tent with nature’s rules, all suffer.

Hold on a minute. Current family values suggest a sickly child should besupported, for example. Technology a hundred years ago did not support theoption of keeping the child alive. These are the same values the liberal leftwants for their collective. Rand argued against both the liberal left and theconservative right.

However, the family as a unit has survived for a long time. Is this a paradox?The family unit survived during a time when nature solved the problem of thesickly child. Technology changed that. Further, many societies had a moralcode which “exposed” (killed) the weak or unsupportable child. Others that hadreached a population limit left the family heritage to the oldest while ostracizingthe younger. Our society must solve the problem or nature will select againstour society and most of us.

The problem is not that poverty, over population, etc. must be addressed.The problem is that the electorate elects those that make them feel good andthose that support their intolerant views rather than those that understand(predict the outcome of actions) the problem. Trying to save the non-producersspawns a greater percentage of non-producers. The politicians’ interest is tobe elected. Their interest becomes to cater to the increasing number of non-producers. Businesses are investing in offshore plants and moving from statessuch as California to states such as Texas. Investment money moves easier thanpeople. All who are left behind are soon sacrificed.

The development of a society starts with the shared need by diverse groupsto meet external pressures trying to kill them. The groups were formed by ex-pansion or by a major catastrophe such as famine and war. The new confederacyrequires tolerance of diverse views.

Wealth for Objectivists is a creation of the producers. The successful in-dividuals have won in a natural environment where survival or death is theoutcome (the “leaders”) which excludes politicians. Therefore, they are deemedbest qualified to distribute the wealth for the survival of the group. Therefore,the producers and leaders are left to value and direct the use of their products.Thus, successful individuals are left to reap the gains of their actions. However,individuals must also reap the losses. The Objectivists seem silent about theexample of a baby born sickly and the need for a military. Why should the weakor those on welfare be kept alive?

The leaders’ actions are often in conflict with the values necessary for thefollowers to form the cohesive and organized group. The leaders find they mustprotect the more numerous followers from nature’s selection process. The mil-itary’s job is to kill others. This is in conflict with a group’s cohesion. This iswhy military bases are segregated from the population.

The individuals may become rich by overuse or misuse of the common re-


4

sources1 (the “commons”) of a society such as rivers and air. This injures thesociety. Because the cost to society of commons use is miscounted, the methodto determine the qualifications for leadership produces false results. For exam-ple, the industrialist who dumps waste into a river ruins the water downstream.The Objectivists society results in the “Tragedy of the Commons”. Regulationsare introduced to reduce the Tragedy of the Commons. The regulations shouldat least charge the use or misuse of the commons to those who profit. Thenjudge the qualifications for leadership. However, because regulations from thegroup often exceed the minimum requirement, over regulation harms the society.Regulations based on follower group morals encourage a collective mentality.

Wealth for Collectivist is a fixed pool and is part of the “commons”. Wealthis achieved by producers but must be “fairly” divided according to family values.The people who decide on the distribution must be political rather than a wealthcreator. This introduces family values and a need for “fairness”. The demandfor fairness is really a demand for special privilege from nature’s judgment atsomeone else’s expense. Those people and businesses needing special privilegeare those that have failed the survival test. The concept of “fairness” is a conceptof the Collectivists. The leaders and producers2 are seen as taking more from thefixed pool than they need. That is, the leaders and producers in the Collectivistsview must have done something wrong to accumulate a larger share of the fixedwealth. The Collectivist views the action of the producers as the root cause ofthe need for regulation. Therefore, the leaders and producers are the oppressorsand the others are the victims. Because wealth can direct power, the “fair”distribution must be accomplished by regulation and by force. Regulationstend to treat all in a similar moral manner. This Tragedy of Tolerance resultsin the internal collapse of the society.

The cycle of catastrophe, Tragedy of the Commons, Tragedy of Tolerance,and catastrophe in a changing environment must be stopped if human kind isto advance. The problem is that some balance between the Collectivist and theObjectivists must be achieved.

Consider the social organization as another form of a species. The rules ofnatural selection and extinction must be obeyed. Goals that are inconsistentwith survival or that are wasteful of resources cause the extinction of a socialorganizational concept.

A scientific Theory of Everything should include more than the physics ofcosmology and quantum mechanics. “Everything” in human experience andsurvival should also include the organization of the interaction of individuals.Such a model was suggested in the Scalar Theory of Everything (STOE)3.

This paper presents a framework to judge the survival potential of moral sys-

1Entities that are not of advantage to human life such as swamps, mountains, and endan-gered species are not resources.

2The rich includes the producers and non-producers. Non-producers are those gainingwealth through bailouts, government support, “fairness” arguments, and “tragedy of the com-mons” actions.

3Hodge, J.C., Theory of Everything: Scalar Potential Model of the big and the small (ISBN9781469987361, available through Amazon.com, 2012).


CONTENTS 5

tems. The STOE principles are applied to the moral responses for the problemof survival. The STOE principles are the Reality Principle, the Change Princi-ple, the Competition Principle, the Limited Resources Principle, the AnthropicPrinciple, the Negative Feedback Principle, and the Fractal Principle.

Contents

1 The purpose of life is life 5

2 The nature of nature 14

3 Biological to Social Mechanisms 23

4 The Vital Way to life 28

5 Discussion and conclusion 38

1 The purpose of life is life

Who are you? How will you survive the next 1000 years?An Individual is a Life Spirit that is the unit of action within a larger In-

dividual Spirit. An Individual can exist in its environment within the timethe strength of the Spirit allows. A child is an Individual only because of theHope for future contribution. Except where energy is the only component, eachlarger Individual has components that consist of other component Individuals ofa different type and energy. The component Individual Spirits of a larger Indi-vidual are interdependent to the degree that a failure of a component causes theself–destruction of the Individual. An Individual can act on other parts of theenvironment to a greater degree than the sum of the components. For the largerIndividual to continue the components can Compete and Change and they arenot acting destructively toward each other. Businesses in the U. S. Competeand the weak businesses are Disintegrated.

The current relationship of the U.S. and Japan is not an Individual. TheU. S. provides a defensive umbrella for Japan. Japan produces products lessexpensively because of the lower military cost. In the end, this relationship willcause the U. S. to self-destruct. Japan must pay the cost or the U. S. mustwithdraw the defense.

A person on welfare is not an Individual and not an Individual componentof the U.S. Individual. Welfare is received from the exploitation (taxation) ofproducers. Because the Resource return to producers in minimal, this taxationis a type of Murder of the producer.

Such understanding to predict in scientific terms is in a very primitive state.What should we tally? What should we measure against what standard? Tal-lying the number of people has been shown to be a false measure of survival.Consider the societies that increased their numbers during times of plenty, only

84

93

102

107

117


1 THE PURPOSE OF LIFE IS LIFE 6

to have the society as a whole crash when a lean period arrived. How do wedetermine what is good and bad? The rise of Greece, the Chin dynasty, Rome,Briton, and the US was done with brutality and violence. Authoritarian andbrutal Sparta survived longer than democratic Athens.

Science uses theories that are tested by measurement of observations andexperiments. The goals of science are Understanding and Wisdom. To Under-stand is to predict experimental outcomes and events. To have Wisdom is tobe able to cause experimental outcomes and events. Prediction is the measureof the correctness of a scientific model.

Philosophy and religion are concerned with “truths” that are consideredmorally “good”. However, the differentiating feature between philosophy andscience is that the “truths” have no known concrete test that can decide betweenrival philosophies. As soon as such a test is devised, the thought becomes atheory of science.

What is proposed herein is that the test of correctness of religious and philo-sophical concepts is measured in the years of survival of the people practicingthe concepts. Conversely, the thriving people and groups are practicing thebetter religious concepts. The test is in the practice and not in the ideals.

History has shown societies fail because of loosing a war with another societyor by internal decay and collapse. War has been the only method to determinewhich moral and political systems are better. Truly, when war erupts betweentwo societies, the Gods of each society are Competing. The impending perilof life today is that man’s society has developed weapons powerful enough todestroy all. This can be good news because the scope of the weapons has usu-ally preceded the development of larger organizations and population supportcapability. Technology helps survival.

Another possible peril is that the sizes of governing bodies and the size ofpopulations are too large for the morals or for the food production capability.Such systems collapse under their own weight by internal disorder. Only a majorrevolution in the governing and religious codes can avoid a major cataclysm. Thekind of morality in the U.S. today is failing the test.

History also shows us that an organization without Competition falls tointernal disorder and internal war. A world organization of humanity mustallow the necessary Competition without war and must have a means to matchthe population with the food supply. The Way suggests the models on whichto organize humanity are present today. The present governing systems andmoralities appear to have reached their limits. The life Spirit requires a newmorality and governing system to progress to the next step. The new moralitymust cause Change that will result in abolishing war. The universe is changing.The Spirit that Changes the universe of physics also Changes the universe ofthe Spirit.

The Fundamental Principles of life, social structures, and physics are thesame.



Basis of morality

Survival is not preordained. Humanity’s philosophy and morals have not solvedthe survival problem for either Individuals or our species. Of all species knownto have lived, less than one percent still survives. Nature is against us.

Survival is a difficult goal to achieve. If it is a simple goal, why hasn’t itbeen achieved? Why haven’t philosophers agreed? Why is there disagreementon how to survive? Why isn’t there only one religion?

If there is an underling moral of nature, it is to survive. Morals in conformityto nature help their observers’ survive. The morals that allow survival are notto be made by humans. Rather, they are to be discovered by humans.

The test of any moral proposition is “does it aid or hinder survival?” If itdoes not aid survival, then it hinders survival because it consumes Resources.

Knowledge, mind and brain

Biological life is an extension of the chemical world via the chemical DNA. Thisadaptation allowed faster Change and better use of Resources than the chemicalscould do on their own. Animals in societies are also a continuation of the limitsDNA imposes. Fewer genes are required to meet the challenge of a wider range ofenvironments. Instinctive or DNA Changes to do this would be of unimaginablygreater complexity than currently exists. Thus, group organization is the nextstep in a long chain of increasing life Spirit.

A mind is group of cells in a brain arranged in a particular order. Modifythe order and the mind is modified. The mind is a Spirit, an arrangement ofrelationships.

Feelings, sensations, and perceptions are more than just the brain. All in-volve the release of chemicals and Change in the Spirit of the whole body.Therefore, we are made of 2 distinct elements - body and relationships. Thebody, through chemicals, atoms, and particles, is ultimately energy. The rela-tionship among chemicals and energy is Spirit.

Life is a combination of energy and Spirit. Spirit is the relationship betweenenergy and energy. As energy combines with even more complex Spirits, moreSpirit and more life are created.

Therefore, knowledge in the mind is a vague term to describe particulararrangements. At best, knowledge is true in the sense that action taken basedon a particular knowledge produces survival. However, actions in a particularenvironment can have different outcomes in other environments. The knowledgeof one may differ from the knowledge of another.

The concept of knowledge seems to have little practical use. This chapter isto use concepts of Truth and Understanding rather than knowledge.

Hypothesizing beyond this seems to have no survival enhancing value.



Universe beyond our minds

If nothing really exists or if the universe is destined to end, then there is nopoint to survival. However, if there is any point, we must try. Therefore, thosewithout the goal of survival will not survive and their ideas are meaningless.

If a group is allowed to achieve any goal it desires, survival is necessary toachieve any other goal. Whether a second goal can be allowed by nature isuncertain and is doubtful.

If humanity, in our egotism or philosophy, decides that ending life is themoral policy, the universe will continue. Less than one nanosecond after hu-manity commits suicide, the heroic sacrifice will be meaningless and unknown.Pascal’s retorted to the cosmos “When the universe has crushed him, man will

still be nobler than that which kills him, because he knows that he is dying, and

of its victory the universe knows nothing” (Pascal, Pensees, No. 347). This issheer nonsense. Man would die and be nothing. The Universe would survive.

Nature of death

Death is an adaptation of life to develop and gain strength. The members ofthe mineral world are very long lived. This means that Change occurs overmillions if not billions of years. Life’s adaptation of death allows Change tooccur much faster. Death, therefore, helps life meet nature’s conditions betterthan everlasting life for an Individual.

That species or Individuals die is neither tragic nor immoral. The death orcrippling of a child may be shocking to human sensibilities. So humans thinkthe laws of nature are villainous. However, nature is neither good nor bad.Nature merely is. Nature does not need to follow mere human ideas of morality.Change will occur. That species die is part of the increase of the life Spirit of alllife. As the death (selection) of Individuals helps make room and strengthen theSpirit of groups and species, the continued modification of and death of speciesstrengthens life.

What can be the goal of an Individual or a group if its destiny is to die andbe replaced? The basic argument for a supreme being (God) is that only a beingof superabundant Wisdom and Resources could have the leisure and humor tocreate people and set them on the path of becoming Gods. The goal is to beone more step along a very long path.

Life must follow nature’s rules if it is to continue to exist. This applies tothe big such as the dinosaur and to the beautiful such as the passenger pigeon.

Nature’s rules are at least very difficult for us to know. They may be un-knowable. Current physics holds that the early seconds of the universe had verydifferent forces than now. As new organizations of societies are created, newrules of behavior are created. Therefore, nature’s rules are constantly Changing.



Meaning of life

Life and all the alleged political rights people have claimed for themselves arenot options for us. Life itself is not an option. Life has to strive for life. Lifecosts energy. Death is the only option.

Nature has neither read nor endorsed the American Declaration of Indepen-dence, nor the French Declaration of the Rights of Man. We are born unequaland confined. As civilization grows, inequality also grows. Nature smiles on theprocess of selection from diversity.

Current physics holds that the earth, the solar system, and the universemust eventually die. The only questions preoccupying physicists are when andhow the universe will end. Human’s nobler goal should be to identify that pathof Spiritual development leading to a Spirit that outlives the universe.

This message is the exhilarating imperative to live. The only question ishow?

Free will and problem solving

Homo sapien is just one species among many that have evolved and surviveduntil now. There are other species that have survived longer and, therefore, aremore worthy. The study (man’s method of survival) of the methods of thesemore worthy species can offer insights about the nature of nature and about themoral methods we may use.

The study of survival methods yields human ideas of morals, philosophy andreligion. If any conclusion can be drawn from human history, it is that thedevelopments of some of human views of morality of grace, of beauty, of faith,of good deeds, etc. do not yield survival. Sometimes immoral behavior of war,of unfairness, of Change, etc. yield survival. There are only a few concepts thathave survival value. Homo sapiens must act consistent with such concepts tosurvive. These concepts are called the 7 Great Ideas. The pursuit of all theGreat Ideas will yield conflicts between them. Therefore, the pursuit must alsobalance the effort among them.

There are many methods recognized as methods to solve problems. Thestudy or formal approach so cherished by homo sapiens is only one means ofsolving relatively low complexity, slow changing problems. So homo sapienshave slowly found some Truths. These helped him reduce the complexity ofproblems and so yield the survival problem up to formalistic approaches.

People can plan only to the limit of their forecast ability, of their Under-standing. Beyond that limit, all species approach the survival problem by trial-and-error. So homo sapiens establish morals and religions. Some survive, mostdie. Mystically, the greater glory of God is discovered.

Humans cope by determining the horizon of forecast ability, and should planwithin this horizon. Should a plan require a time longer than their forecastingability (Understanding), they are really using an experimental method and runthe risk of failure through unforeseen causes. Consider the economic and polit-ical crises afflicting the U.S. and the world during the past centuries. All have



Table 1: Methods of problem solving used by humans.

Complexity Rate of change

slow fast

high Experimental Analogy Serendipity

Focus Representation Synergy

low Formal Piecemeal Parallelism

taken us unaware. World War II could have been prevented had the ending ofWorld War I been approached with more Wisdom.

A problem is complex when the time over which the problem is to be solvedis longer than the forecast ability. As the process of Change becomes betterunderstood, predictability improves.

The problem of survival is a slow rate of Change problem. Sometimes homosapiens has seen the problem with a simple viewpoint in a fast changing environ-ment and attempted to use analogous, synergistic, or serendipitous approaches.Such was Nazi German’s approach. Nature would have none of this.

The problem of survival is highly complex. There are many methods to helpsolve the survival problem. Care must be exercised to remain within the limitof use of any partial solution.

Species have used a natural selection approach as Darwin noted. A highrate of Change can occur when a new species is introduced into an otherwisestable ecosystem. Homo sapiens have attempted to reduce the complexity ofthe problem faced by life by creating models and morals.

Study is man’s way but is not nature’s way. For instance, termites havebeen building perfect arches much longer than humans. Termites use a methodof construction that works. Even the mineral world has been building archeslonger than humans. Study and logical thought are only one of several methodsnature does not prohibit (as opposed to allow). Study succeeds for humansbecause the Understanding of basic principles allows humans to Change fasterthan termites or minerals.

The survival problem is very complex and of long-standing existence witha slow rate of Change. Like humans building of the arch, or his approach tothe movement of the planets, the reduction of complexity requires redefinitionof terms, new perspectives, and an increase in the type of problems that canbe simplistically assembled together. A redefinition of a problem often includesthe solution and points the way to opportunity.

A group establishes its morals and philosophy. It then Competes with othergroups. Thus, the Spirit of humanity wins by experimentally solving the prob-lem of survival.

As human Understanding and Wisdom improves, his survival problem isextended in time. Man is slowly rising from the dust (mineral) to what he callshis God.



Free will is merely the choices Individuals make in the heuristic approachto problem solving. Biology allows apparently random Changes in DNA andthen allows the Justice and Mercy of nature to select for or against the Change.Nature has allowed this because it does imply progress toward a greater lifeSpirit.

Life requires energy. Life cannot survive without energy. Because life takesenergy and creates its own order, it causes the universe to run down fasteraccording to the current cosmology paradigm. Life is a parasite on the universe.

A paradox is that as life advances it must become a more efficient user ofResources.

Life must be active. It must Change and Compete. Without activity, lifecan wait only for death. Nothing lives outside death.

Life is not a collection of static substances, mechanisms or patterns. Life isalways in the process of emerging, of becoming. Even things we’ve regarded asmineral or inorganic are not static substances. Energy is the only unchangingthing in the universe.

Einstein emphasized the necessity of the time dimension in the physics ofthe mineral universe. Life’s Spirit must also have the time dimension as part ofits being. Life is at every instant in the process of having been, of being, andof becoming.

The species and Individuals alive today evolved from previous life. Currentthought holds that all life evolved from matter and that matter evolved fromenergy. Did energy evolve from nothing?

Part of growing is to Change. Change is hard. The loss of a parent or onewho has helped you is a Change, not a loss of Love. Change and losses arepart of life, part of growing. Those Individuals who cannot overcome a loss orChange cannot grow or survive.

Humanists and philosophers see the loss of a group identity as akin to a lossof sanity. The Individual in the Competition–Change system has not progressed.Likewise, the submission–dominance system is only part of the Competition–Change balance.

An Individual must be one with life and be part of the life. The “ideal state”,the feeling of living in a state of oneness with your mothers, your religion, yourGod, the universe is part of Change–Competition.

The “I’m in the milk and the milk is in me” analogy is known to lovers insex (not Love) with each other, saints, psychotics, druggies, and infants. It iscalled bliss. I think it no mistake that some of the people in bliss are very lowon the life scale (infants, druggies, psychotics, lovers seeking procreation). Thelonging for union is a longing to return, not to progress.

Life progression is the becoming of an Individual Spirit. Adam and Eve werethe infants who by their own act chose to leave Eden. God arranged it thus. Godspent considerable effort to plant the apple tree, to tell Adam and Eve of this,and to both forbid and entice them. He didn’t have to tell them of the benefits.So they could feel guilt and know there was no return - a prime requirementfor growth. Therefore, Eden can be regarded not as connoting paradise but as



connoting irresponsible dependence. Eden symbolizes a life Spirit incapable ofsurvival.

The loss of paradise and dependence can be suffered positively if the lifeforce (Spirit) itself initiates the Change. To be thrown out before one is readyis to be permanently in an environment where the Spirit’s survival mechanismdoesn’t function. Thus, the Spirit has a permanent psychosis. The cure is to goback and begin again. Such restarting is common in species, Individuals, andgroups. The progress of Rome was halted and the dark ages followed.

The progress of a life Spirit demands that the Individual leave the samenessand progress to a union with different Individuals. Electrons and protons unitedto form the atom Spirit. First an electron must be an electron and a protonmust be a proton.

Maslow (“Motivation and Personality”, Abraham H. Maslow, New York,Harper and Row 1954) defined a set of needs that form in a hierarchy. Whena lower need is satisfied, a stress for a higher need predominates in an Individ-ual’s concern. If a higher need is not satisfied, frustrating stress occurs. Thisfrustration is likely to endanger the Individual’s survival in a last effort to gainsatisfaction.

The bottom two needs (physiological and security) of Maslow’s need hier-archy are easily seen to be directly survival oriented. The survival orientationof social affiliation and self–esteem are less clearly related to survival. The dif-ference is one of time orientation. The lower two needs affect survival for theextent of months. The latter two needs effect survival from months to decades.The highest need, for self-actualization, may require generations to effect thesurvival of a group. The need to increase the likelihood of survival for longertime is the driving force to the higher stages of Maslow’s hierarchy.

The group’s need for survival over generations has been internalized into theIndividual’s need for self-actualization. Those groups whose Individuals did notsatisfy the group’s long term interest became less able to Compete.

The length of time over which a set of needs is satisfied is defined as “survivalhorizon”. This definition also includes the set of actions that Individuals orgroups use to obtain survival. The set of actions or morals of the group inrelation to its resulting survival horizon depends on the Competition. NaziGermany might well have had a longer survival horizon if its Competitors weretribes or feudal states as the Europeans found in Africa two centuries ago.

As Maslow’s needs are related to the survival horizon, so is the survivalhorizon related to Individual’s needs. Thus, if an Individual’s need is self–esteem, he need only begin to live his life with concern for the effect of eachdecision of each moment on his well being for the next few years.

Increasing the survival horizon is not as easy as it sounds. The Individualmust live each moment with an Understanding of the results of his every action.He must Understand the reactions of others. Note the stress on “each moment”rather than on long term plans. Knowledge of the interaction of conflicting re-lations (Spirits) is required to Understand reactions. The Individual must knowhis place in his environment. An Understanding of the Spirit of an environmentis mostly unknowable.



Beyond Understanding and self-actualization there is a Godlike need to cre-ate. Creation requires the Individual Spirit must not only have Understandingbut also have Hope and goals, and the Wisdom to Change the Spiritual inter-actions.

The turnaround or crisis manager in the business management world has areputation for rapid action. If this were his only requirement, competent crisismanagers would be more common. This survival manager must also have anuncommon Understanding of the likely outcome of decisions.

What an Individual of a Spirit group does in life has a continuing impact onthe Spirit of the group life. Individuals need not appear in the history booksnor engage in world–shaking enterprises. Each Individual lives in the Spirit asthe Individual contributes to the Spirit.

Living adds to greater Spirituality. Upon the death of the Individual, theability of that Individual to add to the group’s Spirit ends. Dying ends thecontribution the Spirit the Individual has made. The Spirit of romantic Love ofTristan and Isolde is gained in life and fixed, never to Change after death. Whatwould have happened had the lovers lived? Their Love would have Changed.Therefore, Wagner had them die at the moment they reached their highestSpirituality. The idea of gaining Spirituality in death is not convincing. Throughliving Love, parts of a Spirit give to the greater survival horizon of the group.Thus, living increases a Spirit’s strength.

Who you are is not the Individual now. Tomorrow that Individual will learnand Change. Rather, who you are is what process you have chosen to continueand to advance. This is how you survive for 1000 years.


2 THE NATURE OF NATURE 14

REALITY

TOO LATE to care about the pastTOO LATE to flee the presentTOO LATE to change our sympathiesTOO LATE to live

TODAY we help the spirit we loveTODAY we hurt the spirit we hateTODAY we change fateTODAY we fight and die

TOMARROW the spirits that strengthen are goodTOMARROW the spirits that weaken are badTOMARROW the spirits’ changes judges our lifeTOMARROW the spirits will not care about us

OUR PROGENY must rememberOUR PROGENY must changeOUR PROGENY must struggleOUR PROGENY may survive

2 The nature of nature

Nature is an unfolding of events over time instead of a being or a Spirit. Thisviewpoint leaves little to react against, little to alter, and little to worship.

All entities are subject to a set of rules imposed by nature. What theserules are is still a subject of debate. However, violation of these rules results innon-survival. The force imposing the rules is irresistible.

Societies are made of people. People are made of chemicals. Chemicals aremade of atoms. Atoms are made of protons, electrons, and neutrons. Theseparticles are made of quarks. Perhaps, with greater energy equipment, thequarks will be found to be made of sub-quarks. But the pattern is the same.Sub-particles are energy and are “glued” together by energy. Democritus usedfood analogies to conclude there must be one type of particle that composes allother matter.

If energy is the fundamental component of all matter and is the “glue”, thenthe relationship is the definition of “Spirit”. This relationship between energy“glue” and energy is a Spirit of matter. The universe as we know it is onlyenergy (measurable) and Spirit (immeasurable).

The source of both energy and Spirit (the ability of energy to relate andinteract) is suggested by the STOE. Even a definition of 3-dimensional space issomething. Thus, a perfect vacuum cannot be made because the confining wallwill carry a space definition and, therefore, will have both a Spirit and energy.



Energy and Spirit must be independent of each other. That is, energy hasno component that is Spirit. Spirit has no component that is energy. Spirit, toexist, must relate energy. Energy, to exist, must be related by Spirit to otherenergy and must be unrelated to remain energy. The combination of energy andSpirit forms matter, space and time.

The current popular models of the universe imply that there exists a limitedamount of energy. Whether energy can be created out of nothing is unknownbut may be possible. After all, the universe exists. Limited need not implyconstant. For all we currently know, the condition of limited Resources is, atleast locally, valid and is a dominant factor in the life of all species and groups.The STOE suggests energy is continually being injected into our universe. Thecooling flow suggests there is excess energy available for Spirit development. Ifthere were not excess energy, the universe would not have life.

If the Spirit could be strengthened, larger forces could exist. We observelarger forces do exist. A Change must have occurred. Time and space began.The relationship of energy–to–energy Changed.

Another response to create a larger life Spirit is to repeat successful imple-mentations. The Change that started life must be more than a singular event.The life Spirit developed. Life began.

A Resource is the energy of which a life entity is composed. To be composedmeans the life force unilaterally directs its use. To be stable and survive, theSpirit must be able to respond to Limited Resources with Change and Compe-tition. This is the unity of a life. This is an Individual.

The brain of an animal is less than an Individual because it cannot respondwithout the body. Neither is any tissue part of the body an Individual. However,the chemicals of the body are Individuals.

Change and Competition require Resources to continue. Other Resourcesavailable are other energy and Spirit groupings. Matter draws other mattertogether to form larger bodies. The Spirit of the smaller body still exists and islarger because of the other. The (gravitational) force of the new Spirit is largerthan either of the parts. As this process continues, life Individuals becomelarger. The larger new life exerts more force.

The Spirit is stronger if it can encompass more energy. Bringing togethergreater amounts of energy means the Spirit becomes bigger. We observe that,in this state, the Spirit binds the energy less. Suns are created. If the massis large enough, it explodes. Therefore, we observe an opposing mechanism ofenergy to prevent the Spirit from becoming infinitely large in the mineral world.Balance is required.

The unity of the atom is that the proton and electron serve to define onlyone atom at a time. The atom contracts to share an electron with another atomand a chemical is formed.

The unity of a society is that the people serve only that society. An Indi-vidual may attempt to serve many societies. If the societies come into conflict,choices will be made. These choices determine which society survives.

A society Individual includes the animals, plants, minerals and land overwhich it can achieve dominion. A society can kill animals for food for other



members. A society can trade with another society any of its Resources withoutregard to the survival of the Spirit of the Resource. A society may commit itsResources to war with another society in an attempt to acquire more Resources.

When a Spirit reaches a limit, Change allows the formation of another Spiritthat may overcome the limitation of previous Spirits. When a sun or an atombecomes too large, it explodes. When a society becomes larger than its Spirit(organizational structure) can contain, it explodes.

Why should any arrangement of matter, any Spirit, be favored over anyother Spirit? The Spirit that survives must be more efficient in the use ofenergy. A Spirit that is more efficient must be able to acquire more energy withless waste than Competing Spirits. With limited Resources, the more efficientSpirit eventually has more energy and, therefore, has the potential of survivinglonger.

If a Spirit is to survive, it must grow faster than the available Resources orgain new uses for existing Resources. If a lower growth rate were allowed, evena minor misfortune would kill the life.

The life Spirit in societies also grows faster than the Resources. Therefore,societies always appear to be more complex than the models can comprehend.

Therefore, for all life there is one condition of existence and two responsesof existence. The condition is that there are only limited Resources available.The two responses are Change and Competition.

Change

A constant of nature is that Change occurs. Even in the world of matter, energyis being used, the universe is expanding, and matter is continually changingform.

Change is a building upon previous Spirits and death of Spirits. A babyChanges into an adult as time passes and it acquires more Resources. Theadult dies.

The types of Change are creation, evolution, revolution, and disintegration.Change is time related. The speed of Change is different for different Spirits.The speed that DNA can Change is revolutionary for minerals. The speed thatsocieties can Change is revolutionary for DNA.

Lack of Change is not an option. A Spirit must grow or die because otherSpirits are growing and need Resources.

Creation

Creation is the formation of a new Spirit from other Spirits that will become apart of the new Spirit. Several people may meet and from a new social groupsuch as the formation of a corporation. Asteroids may attract each other andbecome one bigger asteroid, a planet, or a sun.



Evolution

Evolution is the formation of a new Spirit by modifying a small part of a Spirit.The Change from one to another is such that the majority of the new Spirit isthe same as the old Spirit. A planet attracts one asteroid at a time to becomea sun. Part of a baby growing to be an adult is eating. Little Change occurseach day. The corporation hires new people one by one.

The faster a Spirit can Change, the more able it is to acquire Resources.As a Spirit grows, it will become too big. It must either undergo a revolution

or disintegration. A caterpillar must become a butterfly through a revolution.As a corporation grows, it becomes too big for the management style and struc-ture. Reorganization is required.

Revolution

A revolution is the formation of a new Spirit by rapidly incorporating otherSpirits. This doesn’t include killing the old Spirit. A revolution is rapidlybuilding on the old. The time required to Change differs from one Spirit toanother.

The revolution can occur from within by reorganization or from without bybeing conquered by another or conquering another. Mass grows by attractingother mass. It evolves until the mass is big enough to become a sun. Then arevolution from within occurs and a sun is born. A corporation may be boughtand become a subsidiary of another corporation.

The danger of a rapid Change is that the new Spirit may not be able toincorporate the new size. If the new size is too big, the Spirit disintegrates.

Disintegration

Disintegration is the death of a Spirit by its components becoming related inless than an Individual. Disintegration may also occur from within or fromwithout. A sun explodes if it becomes too large. A corporation may be boughtand dissolved.

Competition

Change adds new forms and uses of Resources. The limited Resource conditionimplies there isn’t enough for all. Therefore, some process to direct Resourcesto the more efficient use is necessary. This is Competition.

Competition’s place is to select and enlarge the forms of life that are efficientusers of Resources. Those less efficient must cease.

One form of being less efficient is to grow at a rate less than or equal to therate available Resources will allow. Other Spirits will kill such a Spirit.

The power of Competition is selection rather than disintegration or death.If no choice exists, there is no selection and no death of less suitable Individuals.Nature abhors stagnation. Death then is a part of life.



The varieties of Competition are reproduction, repetition, cooperation, andwar.

Reproduction

Reproduction is growth of an existing Spirit by duplication. Because theyuse the same type of Resources, they must occupy different areas. Growthis achieved by increasing the numbers of a Spirit. This is why the word Com-petition is chosen rather than reverberation for this response.

War

Competition occurs when a limited Resource condition has been experiencedand the Individuals’ response is taking Resources without compensation fromother Individuals. Other Resources that occupy the same niche and use thesame Resources are prime targets. Acquisition can be by destroying the otherIndividuals or by taking the other’s Resources. Each of the planets attractsasteroids. The planets are in war to gain more mass. The effect of asteroidimpacts on organic life is on little concern to the earth Spirit. It’s only themass of the organic life that earth needs. Lions and cheetahs will kill and noteat each other’s young. War is a way for different homo sapiens societies toarrive at a decision as to which form of society uses Resources more efficiently.War has been used by societies to grow and expand. Because the victorioussocieties have the opportunity to become stronger and survive longer, war isin accordance with nature’s values. All the horrors of war abound routinely innature.

Resources must be used to strengthen a Spirit. If Resources are used forpurposes other than to strengthen a Spirit, soon that Spirit will become weak.If the Resources expended in war are not at least replaced from the vanquished,the society will be weakened. Often the Resources are restricted from use byother Spirits. A weak Competing Spirit may not have enough Resources tosurvive. A strong Competing Spirit must have excess Resources. The excessResources must be productively used to promote Change, to war, to reproduce,or to cooperate.

From the moment they first came into Competition with one another, west-ern culture, whether any like it or not, has predominated over other cultures.The key reason for western culture’s ability to conquer has been the pursuit ofChange. Even in western culture, attempts by otherwise strong political groups(e.g., the Catholic Church of the middle ages) to inhibit Change have resultedin the group’s loss of power.

War is a part of Competition. Resources must be used to accomplish thevictory in war. Resources must be gained in war. Nature allows war as a wayto further life. Therefore, the only way to abolish war is to further life by othermeans.

Trying to simply abolish war unilaterally will be abhorrent to nature. Naturewill advance life by using the Resources of those it eliminates for other Spirits.



This may mean the warriors will ultimately kill those Spirits unwilling to engagein Competition and Change.

The economics of predation and war and their place as an acceptable formof Competition are harsh relative to the morals associated with cooperation.A common theme of human history is to have a “less cultured” group destroya “more cultured” group. Thus, the Mongol herder society conquered manyagricultural groups. This is because the agricultural groups had a Spirit thatfailed to fully comprehended Competition and Change.

Within a society, litigation is a form of war. Resources are spent for theprimary purpose of seizing other’s Resources.

Peoples’ sensibilities may disapprove of war being categorized as a positiveprocess toward life. Life’s advance requires diversity and the culling of weakerSpirits. People can be assured that if they don’t perform life’s goal, nature willwith war. So, if people wish to avoid war, then they must do as life dictates.This is difficult to do in society, as we know it.

If a society has no warrior class, both its male and female members mustdie during famine or be conquered by a neighbor. Societies with a distinctsoldier/policeman/lawyer class from the producer class can survive only if itsneighbors are subject to the same ratio of soldiers to producers or are veryisolated. The versatility of requiring all males be subject to military servicecould produce more in times of plenty, and form larger military might whenrequired. Standing armies and police forces are tremendous burdens. This hastremendous influence on society’s organization and laws. Serf and slave stateshave tended to fail in war against states that allow most males to be armed.

If war is to be abolished, a method must be found to allow weaker societiesto Change or die. The problem of war is the Murder of Individuals. Cooperationand Competition must allow the weaker Individuals to dissolve without taxingthe more vital Individuals.

As war is a part of Competition, Murder is not. Murder is the destruction ofcontributing Individual’s ability to Compete. The slaughter of cattle Individualsis not Murder if the meat is eaten and if the cattle Spirit is maintained bybreeding replacements. The slaughter of a steer is not the Murder of the cattleSpirit in a society. The killing of a pray animal is not the Murder of the praySpirit if the pray Spirit is encouraged to reproduce and thrive.

History has many examples of one society conquering another through war.So far, so good. Then some conquerors salt the fields, destroy the plants, and killthe cattle to extinction. After conquering a society, the destruction of Resourcesis Murder. Although this procedure eliminates a future Competitor, it fails toreturn Resources to the victor. Societies that engage in this type of activity areshort lived.

Isn’t it curious that in international, Competitive relations, nations are ac-quisitive, pugnacious, prideful, and greedy? Violent and destructive war iscommon. Even the victors of war loose Resources. Victors often plant the seedsof the next war in an attempt to profit or, at least, regain the lost Resources.The only exceptions have been where the victors totally exterminated the con-quered and occupy their territory or where the victors forget the war loss and



incorporate the conquered into a new organization of Spirit. These same nationsvalue internal peace and cooperation and punish those citizens who act towardtheir fellow citizens in the same manner their governments act toward othernations. Religions and other groups share this behavior. War is a nation’s wayof acquiring Resources, of eating. But within a nation, such warlike behavior isoutlawed. Frequently, domestic outlaws in a society make good soldiers becausethey know how to cooperate. Thou shalt not kill a member of one’s own group(murder), but thou shall kill a member of an opposing group and be commendedfor it.

Cooperation

Cooperation occurs when a limited Resource condition has been experiencedand the Individuals’ response is to allocate the Resources to as many as pos-sible. The goal is to obtain a large Spirit in numbers of Individuals by usingResources most efficiently. Individuals do not war or Compete if they have theminimum necessary for survival. Individuals who have less than the minimumdie. Individuals who have the minimum are expected to support other’s warand Competition. Individuals with more than the minimum are expected togive the excess to other Individuals within the larger Individual. Note this doesnot include members that do not contribute to the larger individual and haveno Hope of contributing. Such a system has few Resources for expansion or fordeveloping other Resources. Cooperation makes Change difficult

The planets and Sun cooperate in allocating matter and keeping the planetsseparate. The sun remains a size that won’t explode. The planets exist asIndividuals. The method of cooperation is the revolving forces counter thegravity forces. As the planets and Sun are well defined, the rate of Change hasslowed.

Life for all can be made stronger if there is a mechanism to cooperate infinding the most efficient niche positions. Cooperation reduces the Resourcesused in the Competition and selection process. So long as cooperation helpseach Spirit find his most efficient, contributing niche, the group that is cooper-ating will be stronger. Therefore, cooperation aids Competition. A system notpursuing the goal of more efficient use of Resources, is not cooperating and isbecoming a weak Competitor. For instance, the systems of welfare and subsi-dies are not systems of cooperation although there is organized action becausenon-contributing members are not Resources.

However, Competition with or without cooperation will proceed. It mayproceed from within the group or from without the group.

Predator, prey, parasite, victim

All Spirits feed on other Spirits. Plants feed on minerals. Animals feed on plants.Societies feed on minerals, plants, animals and other societies. Individuals areincorporated into the predator’s Spirit. The Individual’s Spirit is destroyed sothe predator can acquire the Resources of the Individual.



Each Spirit must try to survive. Whether a Spirit survives depends on thelarger Spirit of which it is a part, the Spirits to which it is a potential victim,and the Spirits it eats.

Rabbits and foxes cooperate. Predation is a cooperative process betweendifferent Spirits occupying different niches in the use of Resources. The non-cooperation Competition is when Competing Spirits are attempting to occupythe same niche.

Humans, as in other Spirits, at the top of a food chain are less fortunate thanrabbits and foxes. Humans must be held in check by Malthus’s agents (“Essayon Population” (1798) Thomas Malthus) of famine, pestilence, and war. Hopeand greater Understanding may refute Malthus.

There are cycles in nature of abundance and scarcity. A population withabundance can expand to require Resources beyond those available in scarceperiods. The storage of Resources and continued efficient use of Resources forthe scarce periods is one approach to having a larger population. Anotherapproach is to allow the weaker part, which is determined by Competition inthe ability to contribute, of the population to die during scarcity. If the weakerpart is allowed to continue during abundant periods, then more than only theweaker will die during scarce or revolutionary periods. Indeed, the toleranceof the weaker may cause Disintegration from within or without. The storagesolution is viable for short cycles. Eventually the latter approach must be used.

Profitable human management of some wildlife species involves people hunt-ing (harvesting) just before winter. During the following summer the populationwill expand. Several years of maintaining a healthy population results in a rel-atively large population that is easily able to recover from a disaster short oftotal extermination and that is greater than would occur otherwise. Therefore,continued harvests are assured. Prohibition against a harvesting goal may leadto extermination of some species because of human encroachment. Harvestingto extinction leads to less Resource availability. If the wildlife species can useland the humans cannot, hunting is better.

Note the distinctions being made between an Individual’s fate and a life pro-cess (or Spirit). The Spirit embodied in any one Individual mineral or life formmay fail to survive because of Competition with a pre-existing form. Thus, bac-teria may kill baby humans. But, the Spirit will exist again and may eventuallysucceed. Wheat, a heavy grain not suited for natural selection, was probablycreated many times before man noticed it and caused its survival. Human andwheat cooperated and both created a new Spirit that helped both survive. Bothretain their unique identification. Diversity is the propeller of evolution throughChange ability.

The male character rewards the more bellicose and greedy. The male role inrearing young is reduced. Because the male has become the most expendable ofthe group, he is the one placed at risk to gain outside Resources for the group.He is the warrior whose role is to seize Resources from other groups duringscarce periods such as through war, hunting, and farming. This requires anExpansive or penetrating of current Resources.

The females must be able to perform all the Resource use functions. The



female role is then more oriented to cooperation and reproduction. This requiresNurturing of current Resources.

A Spirit must engage in the responses of Change and Competition in orderto continue to exist. A Spirit failing this is abhorrent to nature for it has nopurpose in the continuation of life.

For instance, a deer Individual possesses Resources other Spirits may use.It also consumes Resources other Individuals of the deer Spirit may use. If adeer Individual is failing to Change or Repeat, the Resources it has are wastedand the Resources it consumes are preventing the larger deer Spirit (other deerIndividuals) from Changing and Repeating. Nature will engage the predatorSpirit to eliminate the wasteful deer Individual. If this fails to happen, thewasteful deer Individuals would soon be very numerous. The limited Resourceswould soon starve the larger deer Spirit. Nature will have none of the wasteful.If the wasteful or inefficient Individuals are allowed to exist, nature will wipeout the whole deer Spirit and start again on another Spirit.

If an Individual is not contributing to a society, it is not Murder to kill thatIndividual.

Balance the responses

The Competition concept leads to the concept of a life cycle - birth, growth,death, birth. The same Spirits exist with differing Individuals. Life needsprogress. Change provides that progress, that breaking of the life cycle.

The Change concept leads to the concept of advancing toward a better state.If the Change is inefficient, nature starts again.

Competition is the repeating of past occurrences, without Change. WhileChange implies an expansion, a penetrating, a new usage. Competition is aNurturing of already existing Spirits.

To acquire more energy, a Spirit may Change or may Repeat itself. Eachrequires energy. Each can further a Spirit’s ability to survive. If all energyis devoted to Change, another Spirit may conquer through sheer size. If allenergy is devoted to Competition, another Spirit may conquer through efficiency.Therefore, the responses of Change and Competition must be balanced.

At the simplest level, the need to keep order in the universe or in a societyis a need for Competition. The tendency to disorder is the tendency to Change.Thus, the order/disorder conflict is the Change/Competition conflict that needsto be balanced for survival.

Indeed, the need to balance the forces of Change and Competition createsthe fertile basis for a more efficient approach than mineral and physical forcesallow.

The same laws determine life processes. This is not because the laws orprocesses of life are equal in different fields - organisms, minerals, physical,religions, groups, or societies. Rather, because the condition of life processesare the conditions of the functioning of all forms of Spirit and energy. Theprocess of each level of Spirit is different. Speaking of parallel processes forminerals, bacteria, and humans is not justified. However, parallelism is present


3 BIOLOGICAL TO SOCIAL MECHANISMS 23

in the sense that all Spirits must obey the same rules and overcome the sametype of forces.

Understanding a physical or Spiritual condition of a life form (process) whichwe call lower (more primitive) or higher means (1) determining the significanceof the processes in the development of the life (past), (2) determining the rolethat other life phenomena play with the totality of the experiences of the lifeprocess in study (present), and (3) determining how that life form is best ableto realize its nature to the degree that the life process survives (future). Notethe distinctions being made between an Individual’s fate and a life process (orSpirit).

This Understanding, however advanced, is limited. Just determining thepast, present, and future does not mean survival in the presence of Change. Afourth, timeless factor of a Spirit is necessary. This timeless factor is the abilityto cause Change.

The causing of Change begins with an alteration in the way a life formrealizes its nature. This causes a Change in the role of that life form thatcauses a Change in the processes of development of the life form. Out of thesereorientations evolve new niches and Competition results. Bison hooves cut thegrass that helps the grass roots reproduce and that kills other plants. The bisoncause the grassland and the grassland feeds the bison. This timeless factor thatcauses the alteration is called Wisdom.

3 Biological to Social Mechanisms

Solving the survival problems of groups presents very complex issues. The long–term impact of any philosophical approach can be tested only in the span ofmillennia. Even then there are limitations.

Existing biological mechanisms have passed the survival test over very longperiods. Biological systems of survival can show the Spiritual characteristics’groups must have to survive. A group is as much a living, breathing biomass asbiological organisms are. The progression from Energy to mineral Life to bio-logical Life to ideologies is not each a separate step. The growth of a worldviewof human kind and human ideologies is merely the continuation of the increaseof the Spirit of Life. Each is fundamentally Energy and Spirit.

Individuals do only three things

They eat, they breed, and they die. These are biological words. Eating is theacquiring of Energy into the Spirit. This may be directly acquired as chemicalsdevelop chemical bonds to make larger molecules. War is one way for a nationto eat. The U. S. showed another way when it bought the Louisiana Purchaseterritory, allowed emigration, and killed Indians. The acquired Energy may beused to Repeat or to build a larger body (Spirit).

Breeding in biological Spirits involves direct lineage. Breeding in other typesof Spirits may not be direct lineage. The important concept is that new Spirits



are formed. The new Spirits may be similar or evolved. A hydrogen and oxygenmixture can form water molecules. Each molecule doesn’t necessarily providedirect lineage for the other molecules. However, the Energy released by theformation of one molecule may trigger a chain reaction to form other molecules.The Spirit of water molecules becomes larger. Breeding also occurs when peoplefrom a new group or organization.

Because the world of matter has Limited Resources, Change, and Competi-tion, the forces of order created a structure of matter that could reproduce itselfwith minimum Energy. This copying ability allowed the storing of additionalEnergy.

One of the characteristics of the Spirit of Life is that it requires constantEnergy input to continue. Thus, order has been increased, as has been the rateof increasing disorder. No system that does not achieve a balance of order anddisorder is stable. Life was born. The rules of nature were expanded. Theprohibitions of nature were not violated.

This new Life Spirit, biology, must obey the same rules as matter. Theconditions of Limited Resources, Change, and Competition apply. BecauseEnergy was needed continually and could not be obtained by all the new Life,some of the Life could not continue. Death became necessary for Life to satisfynature’s conditions. The matter in the Life did not Change. Only the Spirit(the relation between the material components) of the Individual ceased. TheSpirit of other Life increased and grew.

There is no significant, distinguishing feature of biological Spirits and mineralSpirits. Each is a Life force. Each is an addition to the previous Life stage. Ourancestors were correct. Trees and rocks do have Spirits of Life. Mankind’ssocial development is another Life Spirit. Mankind’s further advance dependson developing a view of the unification of all Life Spirits.

As Change occurs, as one Spirit becomes strong or another is created, thanother Spirits (Individuals, groups, species, religious beliefs, governments) areplaced in a state of disorder and anxiety generated by Competition. A catas-trophe often results. The Life process of one or all is threatened. A Life formmay not be able to continue according to its nature.

Such pressure creates a state of anxiety. The conflicting forces are thoseoriented to restore the Life process back to pre-anxiety conditions and thoseattempting to realize a more advanced nature for further Life. If each anxietyis dealt with in such a manner that a relative balance occurs before the nextdisturbance, then a slow evolving will occur as Darwin suggests.

As more Spirits come into being, as each Changes, or as the environmentChanges, the frequency of catastrophic disorders increases. If each disturbanceis not dealt with before the next disturbance, then the Spirit must completelyrevolt in a major upheaval oriented toward changing the environment. Thisis done by eliminating other Spirit’s influence, by shrinking his own role, byadjusting to the new order, or by dying. This is a growth crisis.

The only option Life has within its own control is death. A destructive al-ternative is that a Life Spirits may choose to self-destroy itself and destroy itsCompetitor with it. This “Samson syndrome” is seen frequently in history. A



strong Life Spirit must allow for and overcome the possible evil of the Competi-tor’s death wish.

Charles S. Elton, a British scientist, wrote a wonderful, sensible and infor-mative book on this subject, “The Ecology of Invasions by Animals and Plants”.The biological world has been consistently disproving mankind’s view of moral.The various invasions by plant and animals into mankind’s ordered world havebeen very savage. Even the wildest fiction of doom cannot compare with reality.Life is not for the squeamish. The infestations cannot be stopped with poisonsor bombs. The only sensible course of action is to Understand the basics of thefight for survival and meet these invasions on their own terms.

Examples in just the last few decades in the Americas are terrifying. Thereare fire ants, killer bees, chestnut fungus, the sea lamprey of the Great Lakes,and many more.

There are also many lessons. Lesson one: very often, a successful invasion isirrevocable. Once a new Life force is established, it will require a new Life force(not poison or bombs) to supplant it. The new species find crannies to hidein and tricks of survival so eradication is prohibitively expensive in Resourcerequirement.

Lesson two: having shed its past and left its own enemies behind, an invaderwill often explode to extreme population densities. Often the native species andolder Life forms are victims. The most benign invasion is when the invader ismerely Competitive.

Lesson three: most invaders that achieve calamitous population explosionshave come from ecosystems that are more complex and rich in species. Theyinvaded a habitat that is less rich, less ecologically tangled, and more simplified.So mankind’s effort to simplify his environment to his level of Understanding isreally creating an opportunity for invasion.

Lesson four: every species or Life Spirit requires its own ecological niche.If all the niches are occupied, the invader will meet resistance. Many will notsucceed. There is a valid reason for tolerance of difference.

Lesson five: the cause of catastrophe is not that the invasion occurs, but thatthe invasion is rapid and causes rapid Change. The rate of Change producedby an already established Life Spirit is the result of slow evolution. The vacantniche the invader fills would be filled by slow Change.

Lesson six: the successful invasion of a new species will ultimately result ina richer Life Spirit. This is the fundamental force of all Spirit. The character ofinvader or destroyer is man’s view and not how nature views the transaction.

Lesson seven: human effort and Resources spent toward a reduction of di-versity or an inhibition to Change can only be, ultimately, self destroying. TheU. S. is strong not because it is a “melting pot”, but because it allows a mixtureof social groups to exist Competitively without war. Indeed, the melting potconcept is a false description. A stronger Life force could be developed if apolitical diversity could exist without war.

Lesson eight: the boundary between two Spirits is the place where the great-est amount of creation of new Spirits occurs. Where there is isolation, there issameness.



Life organisms in trying to meet Competition have found it useful to developlarger, effective biomass. The invention of the Spirit of society has allowed theCreation of larger effective biomasses.

To be effective means that parts of a biomass must help the other parts tosurvive and to gain survival longer than they would alone. This is coupling.For instance, the heart pumps blood but cannot survive without the rest of abody as the other body parts cannot survive without a heart. A worker antcannot survive long without the rest of the colony. The Spirit of the colonycannot survive or reproduce without the worker. Coupling has occurred. Theant is not an Individual because it cannot meet the response of Competition.The colony is the Individual. The ant lives for the greater survival of the largerSpirit and larger biomass.

A society is a group of Individuals who can Repeat and Change to furthertheir own survival. In furthering their survival, they create an Individual Spiritthat seeks to survive.

An Individual consists of all component Individuals. A country consists ofIndividual businesses, Individual organizations, Individual people, Individualanimals (e.g., cattle and wildlife), Individual mineral Spirits, etc.

Groups and societies are a type of Individual and distinct from biologicalIndividuals as biological Individuals are distinct from mineral Individuals. Cat-tle are Individuals in a group. Traditional thinking in the U.S. has people in anation holding power. Private plunder or public graft is equally unhealthy. So-cieties have different allocations of power. Societies are a matter of organizationand relationships of Individuals. Therefore, groups and societies are Individualsof a Life force. Societies must obey the same natural laws and respond to thesame conditions as the mineral and biological worlds. Therefore, the societyis a Life Spirit as is the mineral and biological Spirits. Further, because moreEnergy is controlled by the society, it is potentially an advance of the Life Spirit.

There are several organizational types of societies. One type is the totali-tarian society where its members are viewed as members of the state and littlemore. Its members are as organs of a body. Each has his function. Resourcesare used by the state. If one function fails, all die. Its strength is that it canbe single-mindedly directed to react rapidly to an emergency. The good of thestate must prevail over the Individual. The problem is that the perceived goodof the state does not always mean the survival of the state. Thus, ants in onecolony are one effective Individual rather than a collection of ants. Competi-tion is restricted and the state looses its Understanding of the requirements forsurvival.

Another type of society is the opposite extreme. The Individualist viewis that the good of the Individual must come before the state. The citizenuses Resources. The state is the instrument of the Individual to get his good.Again, good is subject to Individual desires and may not achieve survival. ThoseIndividuals who chose to be in conformity to the Vital Way will survive. Com-petition restricts the ability of the state to respond to other states.

A third view is a combination of the two extremes. A “well ordered and juststate” (whatever that is) serves the Individual. A civic–minded Individual serves



the whole. Often consumers that are not Individuals are allowed to remain amember of the society. This is a symptom of the causes of the collapse of thesociety.

The philosophy of anarchism requires Individuals or subgroups to cooperateto achieve survival without any coercive state or authority. Nature’s forces areto Compete and Change. Without a coercive authority the Competition canresult in violent destruction of Resources. The decentralization of Roman powerwas an anarchist’s dream. It led to violent, destructive conflict within Europethat slowed Europe’s growth (the dark age). Therefore, anarchy doesn’t resultin a more powerful Life Spirit.

The difficulty of all these views is that happiness or self–actualization isthe goal. Also, they tend to be treated as forms that can survive forever, inall circumstances. The force of Change precludes any one form from survivingforever. Also, a state’s survival depends on its pursuit of the Vital Ways and theabhorrence of the Opposite ideas. Any organization can do this. Consequently,history has seen each type of society conquer each other type.

A society’s Spirit has certain characteristics that can be described by 7 ac-tion concepts - Justice, Mercy, Truth, Love, Understanding, Hope, and Wisdom.Wisdom is approaching what has been defined as Godlike in its creation char-acteristic.

A group’s survival Life force depends on how its culture is structured in theseven action concepts of the Vital Way. The measure of one Life force structurehas little meaning. Meaning and surviving are established only in how a groupcan respond relative to the environment and other groups.

There is a problem in the use of these words as a shorthand way of dealingwith interrelationships in a society. The problem is that these words require re-definition. However, their common usage is the closest to the intended concept.Perhaps it is best to think of each part of the Vital Way as representing anorganizational concept with relationships to time, to Expanding, to Nurturingand to each other. The definition of each Vital Way differs from prior definitionsbecause they are survival oriented. This can lead to a better Wisdom in dealingwith the challenges facing us.

Another problem in using these words is that the application is a society, notthe mineral world. The analogy may be hard to follow. For instance, thinkingof a rock as having Truth will require a close attention to the definition. A rockhas a being in the sense that it has a past that led to its current characteristic.It has a present and a relationship to other rocks and Life forces. It has afuture and will Change with erosion. It can cause Change of its environment byvirtue of gravity. A rock has also achieved balance between its Nurturing andExpanding modes. This is why the rock has survived.

The seven candles of a society’s Life stand on a triangular base of Competi-tion, Change and Limited Resources whose point - Limited Resources - extendsinto another base of Energy and Spirit. The seven candles (stored energy) ofLife are Justice balanced with Mercy, Truth balanced with Love, Understandingbalanced with Hope, on a central column with Wisdom on its top.

A recurring pattern can be seen in the progression of increasing the Spirit


4 THE VITAL WAY TO LIFE 28

of Life. First is sameness. All items of an identifiable Life process are the same.Then another Spirit develops and adds to the many same items a means ofCoordinating them. The whole then becomes greater than the sum of the parts.

The history of increasing Spirit size such as from chemicals to bacteria to usshows that each level has a greater rate of Change than the previous level. Sexwas originated by the blue-green algae. Sexual selection is used in addition tonatural selection to accelerate Change. With the addition of Wisdom, Changecan be created with forethought rather than by reaction, only.

Each level of increasing Spirit size has a greater amount of Energy to beused by the Individual. Note also that each stage is stable in that they survivetoday. These earlier stages are needed by later stages. Elements do not losetheir identity to make compounds. Therefore, the problem of mankind’s sur-vival is one of having several types of national governments (e.g., totalitarian ordemocratic) to rule the earth.

4 The Vital Way to life

All Spirits must have some response to the 7 dimensions of action. A weakSpirit will have responses that do not aid survival and do no direct harm withoutCompetition. These responses are called passive opposites. Some of the possibleresponses are active in a sense that inhibits survival or causes self–destruction.These are called active opposites to survival. Mystical terminology would referto the active opposites as evil. The responses listed above as Justice, Mercy,Truth, Love, Understanding, Hope and Wisdom are the positive responses tosurvival and are called the Vital Ways.

Development of the life Spirit

The biological survival of the fittest mechanism is only the last resort of nature’sCompetition. For species at the top of a food chain with no natural enemies,survival of the fittest implies war, starvation, and destruction of limited Re-sources. When this answer is applied to groups, the groups war and starve.This entails a huge Disintegration of Resources. Some action to make the se-lection is required with less Disintegration. Using Justice is much more efficientand is, therefore, necessary for life.

Justice selects by comparing past actions to a set of rules. If the rules of anIndividual are compatible with the rules of the larger Individual, the Justice willaid survival of the Individual to the extent the larger Individual’s rules allowit to survive. Ultimately, the larger Individual is nature. Ultimately, all rulesmust be compatible with nature.

The application of Justice in all circumstances restricts the society fromlearning. To learn, errors will be made. If the error is allowed to reoccur, thena mistake is made and life is threatened. To allow learning requires an actionsuch as bankruptcy to recognize the error and not waste Resources because ofthe error. Mercy is necessary to balance Justice for life.



To decide whether a situation requires a just or merciful action requires thatthe total, true situation be known. Some action must be taken to discover thetotal, true situation. This action is the search for Truth. Truth is the relationof one event or thing to another. This action is data. Truth exists only in thepresent. For instance, that a person was at a given place yesterday is a Truthtoday.

The application of only Truth to a Mercy and Justice balancing requirementfails to recognize the requirement for an Individual to apply Resources. Someaction to balance Truth with the recognition of the exchange of Resources thatis occurring is necessary. This action requires an Individual expend or giveResources to another in exchange for some value. Love is a giving and a getting.Only in this way can life progress. Love is necessary to balance Truth.

Unilateral giving is beautiful but deadly. Note that getting without givingis allowed by nature. However, the Spirit between the taken-from-Individualand the getting-Individual is weak. The taken-from–Individual may soon disin-tegrate if other Individuals continue getting.

To decide whether a situation requires a Truth or Loving action requiresthat the future of any modification of the present situation be known. Someaction must be taken to discover the future possibilities. This action is thesearch for Understanding. Understanding is the ability to predict the unfoldingof events and future Truths. For instance, a present Truth may be that a paperis being held above the floor. Later, a new present has the Truth that thepaper is released. Understanding would predict the future Truth of the paper.If Understanding was poor and we wanted the paper to be in a wastebasket,we may be required to try several times. This is Resource wasting. Muchbetter is to be able to predict the outcome of releasing the paper from any givenposition so we may choose the more optimal position. Much better to haveUnderstanding.

An exchange of Resources for other Resources to aid our survival (Love)is far more productive (profitable) than war. Understanding helps us get theResources we need in exchange for the Resources expended or that anotherneeds. Taking Resources by destruction of another Individual is an allowedtransaction but leads to war. The war may not be profitable. If we fail to receiveadequate Resources on each Love transaction, we’ll soon be void of Resources.A Spirit that wars without receiving booty soon dies. Profit is a more efficientmeans of growth and survival.

The application of only Understanding to a Love and Truth balancing re-quirement fails to recognize the requirement for an Individual to have someunifying goal. Such a coordination of Resources produces a whole greater thanthe sum of its parts. Resources purchased could have a much greater effect ifthey are selected with some synergy with other Resources. This makes a givenResource more valuable to one Individual than another. Economically, this iscalled profit. The voluntary Love relationship involves least waste. Some ac-tion to balance Understanding with the recognition of the possibility of profitis necessary. Only in this way can life progress. Hope is necessary to balanceUnderstanding.



Some action must be taken to create the Hope and Understanding of thefuture. This action involves the application of forces. Wisdom is the ability tocreate Hope and Understanding for the unfolding of events and future Truths.

The mineral world forces are nuclear, electromagnetic, and gravity.For life to advance, some method to have the next stage in Spiritual devel-

opment is required. For instance, to have Hope, Wisdom is needed to balanceharsh Understanding. To have Understanding, Hope is needed to advance Love.Understanding is needed to balance harsh Truth and Love. To have Truth, Loveis needed to advance Mercy. To have Mercy, Truth is needed to balance harshJustice. To have Justice, Mercy is needed to advance Competition.

Organization of the concepts

Spirits have relationships that lead to survival: a time relationship and a bal-ance of Expanding and Nurturing actions. Other possible relationships (activeopposites and passive opposites) may hinder a Spirit’s survival.

7 dimensions shown in Table 2 are developed to describe a Spirit’s way tosurvive under the condition of limited Resources and the opposing responses ofChange and Competition.

Table 2: A Spirit’s Vital Way to survive

Orientation Great IdeaPAST Expanding JusticePAST Nurturing MercyPRESENT Expanding TruthPRESENT Nurturing LoveFUTURE Expanding UnderstandingFUTURE Nurturing HopeALL TIME balancing/coupling Wisdom

The orientation (e.g., past-Expanding) applies to all levels of survival. Thelabeling of Justice, Mercy, etc. is applied to groups and societies.

The dimensions are chosen in such a structure as to be orthogonal and acomplete description of a Spirit’s position and survival capability. “Orthogo-nal” means the organization of the concepts is such that the concepts are eachmutually exclusive. This aids analysis.

Vital Way

“Way” is used in the sense of “actions to pursue to accomplish a longer survivaltime”. “Vital” is the fundamental, necessary but not sufficient result of actionchoices made by an Individual. Vital is the adjective of way. Once the choiceof the actions of the Individual’s approach to the 7 dimensions is made, the



resulting survival or non-survival in the Individual’s environment is determined.This is a mixture of Newtonian determinism (the clockwork universe of deism)and free will.

When the genes of an Individual are selected, many of the later characterand actions of the Individual are determined. Species Change by modificationin the genetic structure. That is the way of biological Individuals.

The group is organized by choices of how the Individuals are related. Thischoice can include choice of the members of the group. These choices involveresponses to the 7 dimensions. These responses determine how vital the groupcan be in relation to other Individuals.

Because time passes and Change occurs, there is not a sense of perfect butonly a sense of better in the Competition of Individuals.

A traveler on a road is continually faced with choices of paths to follow. Thetraveler can perceive that not choosing an established path is difficult. Uponcoming to an intersection, he must choose one of several paths. Reading signsand maps may increase his Understanding of his choice. Once the choice is made,he is committed to follow that path and experience what that path holds.

Time

As noted before, the universe is in the process of becoming. Thus, a past,present, and future coordination is necessary for survival. The condition oflimited Resources and the responses of Change (of becoming) and Competitionare ever present and are timeless. Justice and Mercy are concepts that respondto past events in the balancing of Change and Competition. The ideas of Truthand Love are statements about the current condition so Justice and Mercy maybe balanced for survival. The ideas of Understanding and Hope are futureoriented. As the natural condition and responses have no time orientation (arealways present), Wisdom is concerned with all time. Wisdom is outside timeand is concerned with the application of forces to influence the flow of events ofnature from the past through time to the future.

The stage a Spirit is in can be likened to the growth of a person. For example,a Spirit with a well-balanced Justice and Mercy level but weak or imbalancedTruth and Love level is like a child. An adult Spirit has Wisdom to the extentit can cause Change and Competition to balance.

Expanding and Nurturing

The Nurturing processes and Expanding processes are the Spirit equivalent ofthe biological female and male. However, the tendency to think of Nurturing aspassive and Expanding as active is not accurate. Nurturing is as active a rolein a life process as it is in biology.

Competition, Mercy, Love, and Hope have Nurturing characteristics. Change,Justice, Truth, and Understanding have Expanding characteristics.



Table 3: Stages of development

Nurture Balance Expand StageWisdom adult

Hope UnderstandingLove Truth adolescentMercy JusticeCompetition Change child

Limited ResourcesSpirit Energy

Happiness

is the experience of living every moment with Hope, Love, and Mercy in theexisting environment in Competition with others.

Fulfillment

is the experience of living every moment with Understanding, Truth, and Justicein a Changing environment.

Happiness and fulfillment are opposites. To be happy, one does not need tohave Understanding and Truth of his situation. One must only accept. Thatfulfillment often brings unhappiness with little Hope and appreciation of Loveis no coincidence.

To survive, happiness and fulfillment must be balanced. The Great Spirit isthe balancing of happiness and fulfillment in the use of limited Resources. Thepursuit of happiness or fulfillment alone is a threat to survival.

Wisdom and limited Resources have neither a Nurturing nor an Expandingcharacteristic. As such, they have no opposites.

Active and passive opposites

Life uses energy. The second law of thermodynamics (the universe is runningdown and entropy is increasing) has been applied to life. Murphy’s law (anythingthat can go wrong, will) is another statement of the same concept. If a livingbeing’s approach to life is passive, it will run down and eventually cease.

Table 4 lists the active and passive opposites relationships. Living and con-tinuing to survive requires the constant acquisition of energy. Life is an ACTIVEprocess. Passive values and goals will not secure survival. Throughout humanhistory there have been many examples of people committing Resources to pas-sive goals. The building of beautiful palaces signaled the high point of MoorishSpain. The end was not long in coming. Solomon’s great expenditures in thepursuit of beauty were followed by the ruin of a great power in one generation.David was wise. Solomon was unwise.



The passive Nurturing ideals are Grace, Beauty, and Faith. Survival de-mands active Nurturing. Not the passivity of Grace, but the activity of Mercy.Not the passivity of Beauty, but the activity of Love. Not the passivity of Faith,but the activity of Hope.

The Jews created the word of God- of the balance of Justice and Mercy(substituting grace) through history. Christians were born to carry the wordof Truth and Love (Jesus substituted Love for beauty) through history. Thenext great Change will be the formation of a church to carry the word of Un-derstanding and Hope as opposed to faith.

If each of the 7 dimensions and 2 responses are “Vital Ways”, then theiropposites must be equally great. There are different types of opposites. One typemust serve to enhance survivability. Another type serves to inhibit survivability.

Nurturing and Expanding are active opposites required for survival. Eachof the 7 Vital Ways has passive opposites that allow an exhaustion of the lifeSpirit for the eventual conquering by another, more active life Spirit. There arealso opposites that are active in a manner that results in the self–disintegrationof life. For lack of a better term, these are called evil.

The more obvious is hate as the active opposite of Love. The telling orbelieving of lies (unTruths) is not the opposite of Truth. Many times a lie isnecessary for Love to help survival. The destructive opposite of Truth is betrayalor bearing false witness when a Justice/Mercy decision is to be made becausethis destroys the Justice - Mercy balance.

The mechanism of opposing forces is necessary for life. The balancing of VitalWay opposites (e.g., Justice and Mercy) is necessary for life. The avoidance ofpassive opposites and the fight against active opposites prolongs life. If VitalWay opposites do not exist, life will not survive. Sameness can be deadly becausediversity is required for Change and Competition. Therefore, the mechanism ofopposites must exist. It evolved as a necessary condition to survival.

The need for opposites and balance solved the greatest pollution problemof all time. The life process of Plants gave off oxygen. While sufficient carbondioxide existed, all was sameness (only plants). However, oxygen was a poisonto plants. Beings that converted oxygen to carbon dioxide were required. Thenew beings, animals, required a source of carbon and energy. The plants hadboth and were required to sacrifice their own bodies to the animals so the plantSpirit could live.

The conditions required for the continued existence of plant life may havebeen attempted many times by plants. The arrival of animals allowed the bal-ance to be achieved. The mechanism of life is a mechanism for achieving couplingand balance.

Predicting a Spirit’s survival potential

The ability of a Spirit to survive in Competition can be measured by classi-fying the way a Spirit responds to its environment in each of the dimensions.Also, each dimension of the Vital Way, passive opposite, and active oppositeare mutually exclusive and are all inclusive of survival–oriented responses. For



Table 4: Great Idea relationsHelp Inhibit Self-

Survival Survival destructiveGreat Great Passive ActiveIdea Opposite Opposite OppositeCompetition Change Sameness WelfareChange Competition Stagnation RegressionJustice Mercy Peace MurderMercy Justice Grace CrueltyTruth Love Illusion BetrayalLove Truth Beauty HateUnderstanding Hope Ignorance ObstinacyHope Understanding Faith Death wishWisdom Wisdom

instance, freedom as an ideal is not so important. However, tolerance seemsto allow probing for the Truth, rewards Understanding and Wisdom, and al-lows the believers in the passive and active opposites to be reduced in powerthrough Competition. The idea of freedom overlaps with concepts of libertyand authoritarianism.

For organization and description purposes each ideology can be described byits response to each of the Vital Ways and their opposites. The response is theactual response in practice and not the intended or theoretical response. Thetest of illusion or Truth is found in Understanding. For any given theoreticalTruth, a prediction about a future Truth according to a theory may be made.Note this requires a positive statement of a new existence or Truth. A statementabout a future disaster fails to qualify as a prediction. Disasters happen as aresult of some failure of Truth or Understanding. We are imperfect, so we canexpect disasters. Believing in false prophets results in unfounded faith andillusion. If the theory and Truth are correct, the prediction will come to pass.If not, the Truth and the Understanding are wrong. The surviving Spirit willmodify its Truth and Understanding.

Comparing the response of the Individual to the Vital Way can make anevaluation of the survival potential of an Individual. The survival potentialagainst Competing ideologies can predict which will survive. This system canalso be used to coordinate the ideology of a group so the survival of the groupcan be enhanced.

For instance, the ideology of a parliamentary government and democracyappeared to be the major export of European colonial powers to Africa. How-ever, as the colonies became more self–governing, many, non-democratic formsof government replaced the parliamentary ideal. The experience of the U. S. inexporting democracy and freedom ideology has also been less than successful.

Why? The survival model explanation is that the freedom ideal in practice



in the colony did not match the religious, economic and organizational (e.g.,tribal) ideals. Secondly, the export was not really freedom. The U.S. founddealing with strong, centralized government easier. The true export had manypassive and active opposites. Thus, gross imbalance was created in the colonythat could not use Resources, Compete, and Change as effectively as the older,tribal ideals. Note the reaction in the exporting power was to increase itssurvival potential because the colony, for a short time, gave more than it got.The exporter saw the transaction as a Love transaction. The colony, however,lost Resources and saw the transaction as Hate. That the people of mid–eastnations want to kill us is a natural reaction to the Hate toward them that weexported. Therefore, the relation between the exporter and the colony was nota Love transaction.

The Hate in the colony fostered rebellion. In time the colonies required thecontinued use of an internally oriented standing army from the homeland. Thecolonies were uneconomical to keep. The idea that the parliamentary system inthe colony was working is false. To maintain the pax Britannia and parliamen-tary government in the colonies, Britain had to pay too high a price or to extracttoo heavy a tax from the people. Britain was eventually bled of its Resourcesand its Competitiveness. The bleeding of Resources was more than economic.Many people migrated to other countries like the U. S. As time passed, the Lovetransaction became imbalanced in that the Truth was lost. People rejected theidea that they were predators. Therefore, grace and beauty as seen in Britainwere met with hate and war in the colonies. The real behavior of the passiveand active opposites caused the collapse of the empire. The same is true of paxRomania centuries ago and pax Americana of this century.

There is a Spirit of all human kind. However, so far in human development,this Spirit has oscillated between the active and passive opposites. Therefore,this Spirit is very weak. Nations continue to war. Much stronger is the Spiritsassociated with states, nations and families.

A Spirit must pursue the Vital Ways to survive and avoid the opposites.The pursuit of any of the opposite ideas will lead to a weakening of the Spirit.

Human growth in Spirit

Human growth in Spirit has been to recognize the need for survival morals, tomodify his Spirit and to fully appreciate the survival ideas. The historic pro-gression has been from Justice, to Mercy, to Truth, to Love, to Understanding.At each stage the future stages were misunderstood and defied the existing ra-tional views. Upon reaching Justice, man thought Grace, Beauty, and Faith(passive opposites) were needed. As human Spirit recognized Mercy and Truth,the quest for Beauty and Faith proved ruinous.

Surviving for longer periods is to approach Godliness. Thus, when Jesus, theherald of Love for western culture, says in the Book of John that he who wouldknow God (long term survival) must come by him (Jesus - Love) is correct.Moses (Justice) might have said the same.

When man had Justice and Mercy in balance, God was Mercy. Now man



has Truth and Love, God is Love. Next man will have Understanding and Hope,God will be Hope. The God concept is all of those and Wisdom, too.

Through human history, people have tended to think of themselves as be-ing unique and an exception to nature’s rules. Thus, they create morals andpronounced these morals to have higher meaning. Seeking a new balancingmechanism is better than thinking we have found an exception to nature’s rules.

Certainly, human opportunity is to form ever–larger groups with ever–greaterinfluence on his environment. We must form a supernation that has the militarypower and that allows the nations to Compete without war.

What goes wrong?

The universe as we know it is not one Individual life Spirit. We cannot predictthe future indefinitely. Therefore, much expansion is needed along the VitalWay. If the Vital Way is the way, why are there still active and passive oppositesin life? Misapplication of the great Ideas causes the active and passive opposites.Misapplication at a higher (more future oriented) level causes the lower (morepast oriented) level to have active and passive opposites.

The pursuit of the Vital Way includes balancing the Nurturing and Expand-ing methods, prohibiting the active opposites and allowing the passive oppositesto cease. If this is not done, various types of Resource wasting events occur whichresults in disintegration. The Spirit either destroys itself or is made prone todestruction through the process of Competition. This is the evil that we sense.

A surviving Spirit must take action to restrain the active and passive op-posites. The model shows the causes of the existence of active and passiveopposites are found on the next more future level. Ultimately, Wisdom mustact to restrain the active and passive opposites and achieve balance.

First Wrong Way

Allowing the continued existence of a passive opposite will result in promotingthe passive opposites of the next lower level. For instance, not knowing andnot attempting to increase Understanding is Ignorance. Allowing ignorance tocontinue will foster a need for Illusion. Illusion creates Peace and Grace. Peaceand Grace in a society creates Stagnation and Sameness that soon looses a waror disintegrates. Another example is that the quest for Faith fosters lies andIllusion.

Second Wrong Way

A Nurturing imbalance is pursuing a Nurturing method and not the correspond-ing Expanding method. A Nurturing imbalance will result in promoting boththe passive opposites in the next lower level. For example, a Spirit’s placingmore emphasis on Hope than Understanding will place the Individuals of theSpirit in a situation were they will respond with beauty and illusion rather thanLove and Truth. For example, the reaction of citizens of a society placing more



emphasis on Love than Truth in sentencing criminals results in the resignationby the citizens that nothing can be done. Soon crime will pay and, therefore,increase. Therefore, Love without Truth will result in a quest for grace andpeace. The Spirit of the state will decay.

The expansion of western civilization often overran societies that were pur-suing peace, grace, illusion, beauty, ignorance, and faith.

Third Wrong Way

Pursuing an active opposite will result in promoting the active opposites in thenext lower level. For example, the reaction to lack of Understanding (inaccurateforecasting) and obstinately refusing to correct the models is to place great strainon the Individuals of a Spirit. The Individuals then must resort to betrayalin their interaction to other Individuals in the Spirit lest their own Spirit bethreatened.

Fourth Wrong Way

An Expanding imbalance is pursuing an Expanding method and not the corre-sponding Nurturing method. An Expanding imbalance will result in promotingboth the active opposites in the next lower level.

Throughout history there have been many who have placed heave empha-sis on the Expanding way of Justice without Mercy, Truth without Love andUnderstanding without Hope. These often engage not only in war but murder.Standing against such a great conqueror is often destructive. The conquerorends in self–destruction or by causing another to destroy them (a form of selfdestruction).

The degree of impact differs among the Wrong Ways

A significant part of Wisdom is to choose the WrongWay it will tolerate. Makingno choice is really choosing the Fourth Wrong Way.

For instance, if a society is suffering unjust, reduced effectiveness of someof its member (murder) in the past level, then the cause and correction canbe found on the present level. Love may be misapplied or betrayal not beingrestrained. If the society fails to act sufficiently to reduce bearing false witness(betrayal), murder results.

If a society finds too often to have illusions rather than Truth, the cause ishaving too much Hope and not enough Understanding, of having inappropriatefaith and allowing ignorance to persist.

There are limits to the Wisdom a Spirit possesses. Therefore, ignorance andfaith exists. Therefore, some acceptance of the passive opposites exists. TheFirst Wrong Way must exist. This is felt by a society as the ever–present eviland as a foreboding that Disintegration is near. The goal is to develop a Spiritstronger than other Spirits. A Spirit need only be better not perfect.


5 DISCUSSION AND CONCLUSION 38

The error toward Expanding ways creates a high risk and rapid self–destruction.The error toward Nurturing ways allows others to conquer. The tendency forsocieties to error toward Nurturing ways at least assures longer survival if thereare no immediate wars or Competitive threat.

The Nurture imbalance is worse than allowing only one passive opposite.Striving for a balance is within a Spirit’s control. This Second Wrong Way isunnecessary and can be corrected by a Spirit. Nurture imbalance is worse for aSpirit than the First Wrong Way.

Pursuing an active opposite causes more destructive responses within a Spiritthan passive opposites. These responses reduce the Spirit’s ability to Compete.Therefore, a Competing Spirit need not advance life to conquer. The Competitorneed only wait. The Spirit must take action by prohibiting the Third WrongWay. The Third Wrong Way is worse for a Spirit than the First or SecondWrong Way.

Expanding imbalance is much worse than dealing with only one active op-posite. Expanding imbalance is difficult to prohibit by laws. The Spirit willsoon disintegrate. A Spirit must exercise care lest another Spirit in Expandingimbalance drags it to disintegration.

The misapplication of Expanding ways has a much more severe result thanthe misapplication of Nurture ways. The risk is much higher. Therefore, soci-eties have tended to imbalance in favor of the Nurturing ways. This is especiallytrue when the Competing risk (war) appears remote.

The meek shall indeed inherit the earth. The meek are those that pursuethe Passive Opposite (Peace, Grace, Illusion, Beauty, Ignorance, and Faith).The meek seek to survive from misapplication, often the imbalance form, ofNurturing ways. The meek shall not have the Great Spirit embodied in theVital Ways.

Previous attempts to eliminate war have failed. This means the Understand-ing and Truths of those movements are faulty. Many of these attempts wereand are grounded in the Second and Third Wrong Ways. Therefore, they willcontinue to fail.

Preventing war means the Competition and Change processes must be en-couraged to continue in a manner that does not destroy Resources. The Second,Third, and Fourth Wrong Ways must be inhibited. The First Wrong Way willalways be with us in ever–smaller degrees. The Second, Third, and FourthWrong Ways can be eliminated by organizing our Spirit to allow Competitionwithout war.

5 Discussion and conclusion

Cycles exist in nature. The cycle of plenty to drought has caused many asociety to weaken and collapse. Science has allowed such blows to be softened.For example, the invention of fertilizer in the late 18th century allowed thecold cycle in the early 19th century to have a significantly smaller impact. Weare in a crisis caused by the federal government deciding the one way to fund


5 DISCUSSION AND CONCLUSION 39

a collective society. It decides the morals for health, welfare, ecology, workerrelations, etc. If we were only ignorant of the optimum morals, we would haveonly the First Wrong Way where we need only be better than our competitors.

However, we are following models that have been proven false. Proven falseby the collapse of the USSR. Proven false in that the predictions failed to ma-terialize such as with the bailouts. Even worse, models have proposed thatpredicted today’s economic and internal political reality by Rand, by MiltonFriedman4, and by others. Neither discusses the military or criminal realities.The federal government is not only ignorant of nature’s way as shown by theever–worsening condition, it is pursuing the active opposite of the Third andFourth Wrong Ways. The result is our wars, wasted resources, and soon disastersuch as happened to the USSR.

We must not be Ignorant of a method to deal with today’s problems. Rand’soutcome may yet be avoided. Nature has used natural selection. We could leavethe decision on all those morals to states, and then allow the states to competefor survival. The federal government must not interfere if some state seems tobe failing. The pure objectivist solution leads to the Tragedy of the Commonsbetween states. Some adjudication of state’s interests is needed at the federallevel. Adjudication does not imply regulation. Federal regulation did seem towork for a few decades after FDR. Then stagnation developed in the ’90s (SecondWrong Way) followed by the looming disaster. The difficulty of decentralizationis the ability of people to allow other moral systems to succeed while theirs fail- Tragedy of Tolerance.

Choose to survive. Change your morals to be in harmony with nature.

4Capitalism and freedom, 1962, Free to choose, 1980


Teacher security and counter-terrorism in DOW as the direction of scientific

research department

Baisheva MI

North-Eastern Federal University, Yakutsk, Russia

AbstractThe problem of security and counter-terrorism in the DOW. Authors settle extreme urgency of the problem and

pay attention to its ethnic and cultural conditioning.

Pedagogical support for security and counter-terrorism in preschools special attention

should be paid to the training of teachers at the university. Students in high school have to

acquire the knowledge, skills and abilities to ensure the life safety of small children in

preschools. They should be competent and willing to take adequate measures in emergencies:

natural (earthquakes, floods, hurricanes, cold), socio-biological (diseases, epidemics), man-

made (emergency or disaster-related technology), social (criminal, family, domestic, religious,

etc.).

Department of Pre-school education is the winner of the Grant AVTSP on "Pedagogical

security and counter-terrorism in preschool educational institutions" (deadlines research:

beginning 01/01/2012 end 31/12/2014). The goal of our research is to determine the theoretical

and methodological bases of the problem, develop a holistic concept and system of work in

teacher security, counter-terrorism in the mental terrorism and in extreme situations in

preschool institutions.

At present, the research department is actively searching for interactive teaching

technologies that form in children is not only a basic presentation skills for life safety, but also

valuable attitude to safe living in the North. The technology developed in working with

children provides heuristic, the problem method, a variety of forms of joint activities with

them.

For the training of professionals and educators in preschool education, we have

developed an innovative educational and methodical complex "Teacher ensure life safety and

counter-terrorism in preschool educational institutions." The main objective of the course is to

develop in future teachers the safety culture of life and to prepare them for preparedness in

providing professionally competent defense and protection of children of preschool age in cases

of emergencies. In the learning process of students insist on the promotion of a culture of


personal, environmental, security, social and legal protection, safety in emergencies. Healthy

way of life, knowledge of norms and rules of behavior in extreme or emergency situations, the

development of moral and psychological stability is the core professional ethics of a preschool

teacher in the name of conservation, saving the lives of young children.

Today, terrorism in all its manifestations has become one of the most dangerous

problems that mankind entered the XXI century. In modern society, one of the global threats of

Russia today is mental terrorism. Subject to attack this threat is both individual and mass

consciousness, the protection of which requires the company timely and deliberate

consolidation not only of intellectual effort and timely large-scale events to protect the inner

spiritual world of every citizen and the whole Russian society as a whole. Particularly

vulnerable to these children without stable outlook and perceived moral principles.

Because the preparation of teachers to ensure a safe environment in the DOW, able to

work with children on a highly professional level to create a culture of safe life for ethnic and

cultural traditions is one of the most important tasks of the university.

In a study of AI Sadretdinova emphasized that today the children and their teachers are

the "one-fifth of the population, ac considering their families - more than half the country's

population. This is what the place and the role of the security of educational institutions in the

national security of Russia "[1, p.3]. Because each teacher in an emergency should be prepared

to take urgent and adequate measures to save the lives of children.

Safety features of life in the North due to the climatic and ethnic and cultural

peculiarities of the land. Harsh climatic conditions require sustainable production in children

life skills. Formation of human security, which has the best climate for the health, stamina,

dexterity and stress psyche, is the key challenge for educational institutions. In this regard, the

experience of the people on the safety of life, to develop a set of preventive measures in

conditions of extreme continental climate, of course, require substantial research and scientific

and methodological grounds.

For the safety of human life must be aware of the value and meaning of their existence.

Absolutely true statement ND Nikandrova that "... they find themselves in an emergency

situation, people often survive if moved by high ideals, sees the meaning of life" [2, p.44]. In

this regard, the traditional culture of life and human survival, accumulated over centuries

people in the harsh conditions of the North is an invaluable source.


The main idea of the essence of traditional culture and education of the Sakha people is

the formation of building up a person. Awareness, support harmony macro-and microcosm, the

understanding of basic values and meanings of life, man's rise to the "spiritual-I" are the main

goals of education of the perfect man. The philosophical doctrine of the Sakha (scientists Aiyy)

highlights the multidimensional and value the essence of man, its great potential to understand

the world and support harmony in reality.

Professor of Chuvashia TN Petrov said that "in the current conditions of reforming the

system of education in Russia, the revival and implementation, and the national-ethnic

potential, especially important to immersion in historical research and teaching of national

spiritual culture, is of fundamental substantive basis for the formation of young generation" [3,

.3]. AB Pankin comes from the fact that for thousands of years to form original ethno-cultural

space, in which the means, methods, and technology education of children have not lost their

significance today. Professor argues that culture has always serves the needs of the individual,

and above all its three basic needs: basic (need for food, meet the physical needs), derivatives

(in the division of labor, protection, social control), integrity (the psychological needs of

security, social harmony, life purpose, etc.) [4, p.50].

Each historical period in its essence fills safe living human society. Especially vital is

the adoption of measures to ensure safe living teacher in educational institutions: the

justification of the theoretical and methodological approaches to the problem, the development

of its conceptual framework, the creation of an integrated system of security and counter-

terrorism, etc.

However, analysis of scientific papers on the issues of security and fighting terrorism in

educational institutions shows that in teaching science has not yet developed an integrated

scientific concept and developed a system of work safety conditions in preschool institutions.

We believe that the pedagogical life safety of children in pre-school educational

institutions must take into account the climatic features of the North, worked out over centuries

and accumulated ethnic and cultural heritage of indigenous peoples, their value for nature,

harmonization of reality. In this respect, the expected result is forecasted as education and

personal development of preschool children take responsibility for secure livelihoods and

acquire basic skills to predict the risk and its consequences, the skills to properly evaluate their

ability to make informed decisions of safe behavior in extreme situations.


Noting the importance of ensuring security in the mental terrorism, anti-terrorist

protection of children in preschool institutions, emphasize that there is a need to address the

legal, organizational, scientific and methodological bases of the problem.

References

Sadretdinova AI Instructional design of the educational environment a culture of life safety in

preschoolers: Author. dis. . Candidate. ped. Science. - Yekaterinburg: CORNER, 2009. - 64.

Nikandrov ND Russia: socialization and education at the turn of the millennium. - M.:

Pedagogical Society of Russia, 2000. - 303 p.

Baisheva MI , A. Grigoriev Etnopedagogicheskie views of the Sakha people: on a material

Olonkho. - Novosibirsk: Nauka. - 2008. - 167 p.

AB Pankin Formation of ethnic and cultural identity: studies. allowance. - M.: Publishing MPSI,

Voronezh: Publisher NGO "MODEK", 2006. - 280

Continue in Russian

Педагогическое обеспечение безопасности личности детей в условиях

ментального терроризма

Baisheva MI

North-Eastern Federal University, Yakutsk, Russia

Терроризм в любых формах своего проявления превратился в одну из самых

опасных проблем, с которыми человечество вошло в XXI столетие.

Существуют различные концептуальные подходы к определению понятия

«терроризм». В широком смысле терроризм определяется как способ управления

людьми через их устрашение. В узком смысле данное понятие сужается до набора

отдельных террористических актов, что обедняет его суть, лишая целого рода базовых

характеристик.

Одним из глобальных угроз современного общества является ментальный

терроризм. В этом контексте следует выяснить само понятие «менталитет». Термин

менталитет (ментальность) - латинского происхождения (лат. mens - ум, мышление,

образ мыслей). Терроризм, согласно Толковому словарю Владимира Даля, — это

«устращивание, устрашенье смертными казнями, убийствами и всеми ужасами

неистовства».


Национальный менталитет формируется на протяжении длительного времени,

охватывающего практически всю историю данного народа. Основополагающие духовно-

нравственные устои народа складываются в зависимости от его традиций, культуры,

социальных структур, внешней, природной среды обитания. В свою очередь сама

ментальность выступает как порождающее сознание, задавая определенные образцы

мышления и поведения личностей, социальных групп и народа в целом. Таким образом,

национальный менталитет - определенный способ видения мира и типичных образцов

социального действия, регулирующих поведение народа на протяжении длительного

времени.

Цель ментального терроризма — уничтожить духовный стержень общества, на

котором держится нравственность, политика, экономика и обороноспособность страны.

Основной задачей атак на ментальном уровне является разрушение позитивных

ментальных установок (убеждения и ценности личности), внедрение и закрепление в

сознании индивидума стереотипных образов мышления, ориентированных на

игнорирование принципа разумной достаточности и осознания образов здравого смысла,

которые и обусловливают соответствующее этим ментальным установкам, поведение.

Объектом атаки этой угрозы является как индивидуальное, так и массовое сознание,

защита которого требует своевременной и осознанной консолидации не только

интеллектуальных усилий, но и своевременного проведения масштабных мероприятий,

направленных на защиту внутреннего духовного мира каждого гражданина и всего

общества в целом.

Особенно беззащитна от разрушительных воздействий ментального терроризма

молодежь, дети, не имеющие устойчивого мировоззрения и осознанных моральных

принципов. Под внешним воздействием возникает внутренняя благоприятная среда для

распространения заданных мыслеобразов и ментальный геноцид начинает работать

изнутри, разрушая культурную общность ее собственными негативными ресурсами.

Ментальный терроризм - многогранный феномен и обладает чрезвычайно

сложной структурой, его различные формы переплетаются и часто смыкаются между

собой. Ущерб, нанесенный ментальным терроризмом латентен и устойчив.


В этом контексте особо следует подчеркнуть о последствиях ментального

терроризма как ментальная дестабилизация. К сожаленью, ментальная дестабилизация

как процесс не стала еще объектом достаточного исследовательского внимания.

Ментальная дестабилизация проявляется в росте социального напряжения и эскалации

этнической розни многонационального государства, в искажении роли и места стран в

мировой истории, уничижении её значимых достижений в науке и культуре,

нивелировании вековых духовно-нравственных традиций. Последствия ментальной

дестабилизации - ущерб духовности, морали и нравственности общества, коррупция,

криминал, разного рода и характера насилия, правонарушения. Если мы не решим, как

обуздать ментальный терроризм, мы потеряем нашу молодежь и с нею будущее страны.

Противостоять ментальному терроризму можно только на духовном, ментальном

уровне. Современная идеология противодействия терроризму должна основываться не

только на мерах по предотвращению террористической угрозы, но и противостоянию

граждан терроризму. В этом плане можно сказать, что вся государственная политика и

идеология должны носить контртерористический характер, предупреждая и разрешая те

конфликты, эскалация которых может привести к терроризму как средства политической

борьбы.

Таким образом, противодействие терроризму становится одной из важнейших

функций современного демократического государства, условием сохранения его базовых

характеристик, в том числе, таких как приверженность основополагающим принципам

правового государства, призванного защищать права и свободы человека, гражданина.

При разработке и принятии новых мер безопасности противодействия терроризму

следует руководствоваться не только мерами конкретных антитеррористических мер

безопасности, но определить роль и место основополагающих духовно-нравственных

идей и системы воспитания и образования молодежи, детей. Эти меры безопасности

должны осуществляться в рамках целостной и стабильной стратегии противодействия.

Успешное противодействие терроризму в большой степени зависит от кардинальной

перестройки сознания, когда для стран и народов станут главными не корпоративные,

националистические или узкорелигиозные интересы, а общечеловеческие, гуманитарные

ценности.


Современный мир, рождающийся в ходе современных трансформационных

процессов, не един. Дело не только в наличии различных политических факторов, но и в

колоссальном разнообразии культурных, национальных идентичностей, включенных в

процесс создания целостного мира и вынужденных искать способы сохранения своей

самобытности.

Современный ментальный терроризм, как сложный политический феномен, меняя

свои формы, средства и масштабы насилия, но, не меняя свою античеловеческую

сущность, представляет собой научную и практическую проблему, которая не потеряет

своей актуальности еще многие десятилетия и будет требовать пристального внимания

со стороны экспертов разных отраслей наук, в том числе педагогики и психологии.

Ориентация сознания молодого поколения на высокие духовно-нравственные

ценности и идеалы является необходимым условием: сохранения духовной целостности

этноса, нации; стабильного развития государства и адекватной реализации

противостояния ментальному терроризму.

Традиции во все времена признавались хранителем лучших духовно-

нравственных, этнонациональных ценностей народа. В этом отношении историю надо

помнить, чтить, с ней не надо бороться.

Последние десятилетия XX века показали, что в критические моменты

исторического развития традиционные ценности обретают особый смысл, придавая

человеку дух защищенности и становясь опорой для дальнейшего продвижения по

жизни. Ценнейшим качеством этих этнических регуляторов социальной жизни является

их способность к самообновлению, обогащению и совершенствованию.

В этом процессе исключительно важную роль играют критерии морали и

ценностные ориентации, трактовка деструктивных явлений терроризма в рамках

духовно-нравственных предпочтений.

Основными источниками этноценностных ориентаций традиционной культуры

безопасности, регламентирующих ментальное поведение народов, являются нормы и

ценности вероисповедания, обычаи, традиции.


Понятие «ценности» и «ценностные ориентации» имеют сложную природу,

определяющуюся единством объективных и субъективных факторов. Предпосылкой

формирования ценностных ориентаций субъекта является система социальных

ценностей. Ценностное сознание индивида определяется и развивается в процессе

социализации, когда им усваиваются ценности в обществе. В тоже время в процессе

усвоения тех или иных объективно существующих ценностей, в сознании индивида

формируются определенные оценки как внешней социальной действительности, так и

самого себя, как субъекта. Ориентация сознания молодого поколения на высокие

духовно-нравственные ценности и идеалы способствует стабилизации процессов его

социализации, является необходимым условием: сохранения духовной целостности

этноса, нации; стабильного развития государства и адекватной реализации

противостояния.

Актуализация духовно-нравственной традиции в сфере культуры и образования

должна осуществляться на уровне личности и общества, на уровне сознания и

поведения. Современная система образования органически объединяет четыре

взаимосвязанных уровня культуры: региональный, национальный, российский, мировой.

Для воссоздания сакральных основ духовно-нравственной традиции институт

образования, выполняющий сегодня функцию культурной трансмиссии, обращается к

идеалам народной педагогики, этническим и надэтническим традициям.

В контексте изучения духовно-нравственных ценностей в условиях ментального

террора следует обратить внимание на выявление нравственной позиции субъектов,

которая предстает как единство знаний, убеждений, поведения. Важным здесь явилось

выявление: 1) нравственной информированности, этических знаний; 2) нравственных

убеждений, включающих суждения, оценку каких-либо нравственных явлений и

поведенческий компонент (предполагаемые активность, поведение в соответствующей

ситуации); 3) нравственного поступка (поведения), неразрывно связанного с мотивами.

Подчеркивается также необходимость анализа условий, объективных и субъективных

факторов, определяющих характер формирования и содержания духовно-нравственных

ценностей у молодежи и детей.


Ценностное сознание обладает уникальной силой сплочения, выступает как акт

духовной близости в противостоянии и противодействию ментальному терроризму.

Формирование этнокультурной и общероссийской идентичности, поиск оптимальных

путей их сочетаемости определяет суть педагогического процесса духовного

становления личности в условиях ментального терроризма.

Исходной функцией духовно-нравственной традиции является обеспечение

преемственности и связности. Связывая настоящее с прошлым и будущим,

преемственность и связность обусловливают устойчивость целого. С одной стороны,

каждое новое поколение никогда не начинает с пустого места; оно осваивает духовно-

нравственные ценности, которые накоплены предшествующими поколениями.

Рассмотрение проблемы «этнонациональные ценности и социализация личности» в

историческом контексте позволило сделать вывод, что важнейшие духовно-

нравственные ценности, составляющие фундамент духовной культуры, сохраняют свою

значимость и для современной молодежи. К числу таких ценностей относятся: доброта,

справедливость, милосердие, сострадание, простота, открытость, честность,

толерантность.

Приобщение детей с самых малых лет к культуре безопасной жизнедеятельности,

безопасности личности, знание основополагающих умений и навыков безопасности

определяются сегодня реальными потребностями общества. Трагические события,

которые произошли и происходят в современном обществе, говорят о необходимости

формирования культуры безопасной жизнедеятельности детей с дошкольного возраста и

подготовки педагогов, обеспечивающих безопасную среду в дошкольных

образовательных учреждениях и школах.

В исследовании А.И. Садретдиновой подчеркивается, что сегодня дети и их

педагоги составляют «пятую часть населения, a c учетом членов их семей - более

половины населения страны. Именно этим определяется место и роль обеспечения

безопасности образовательных учреждений в системе национальной безопасности

России» [1, с.3]. Потому каждый должен быть готовым не только к принятию

экстренных и адекватных мер по спасению жизни детей, но и к формированию культуры

безопасности личности.


Для безопасности жизнедеятельности человек должен осознавать ценности и

смыслы своего существования. Абсолютно верно утверждение Н.Д. Никандрова, что

«…оказываясь в экстремальной ситуации, человек чаще выживает, если движим

высокими идеями, видит смысл жизни» [2, с.44]. В этом плане традиционная культура

жизнедеятельности и выживания человека, веками накопленная народами в суровых

условиях Севера является бесценным источником.

Общеизвестно, что особенность безопасной жизнедеятельности на Севере

обусловлена природно-климатическими и этнокультурными особенностями края.

Суровые природно-климатические условия требуют выработки у детей устойчивых

навыков безопасной жизнедеятельности. Формирование безопасной личности,

обладающей оптимальным для данного климата здоровьем, выносливостью, сноровкой и

стрессоустойчивой психикой, является важнейшей задачей учреждений образования. В

этом отношении богатый опыт народа по безопасности жизнедеятельности,

выработанный комплекс профилактических мер в условиях резко континентального

климата, безусловно, требуют содержательного изучения и научно-методического

обоснования.

Главной идеей и сущностью традиционной культуры воспитания у народов

Севера является формирование созидающего человека. Осознание, поддержка гармонии

макро- и микрокосмов, понимание ценностно-смысловых основ жизни, восхождение

человека в «духовное-Я» являются главными целями воспитания созидающего человека.

Профессор из Чувашии Т.Н. Петрова утверждает, что «в современных условиях

реформирования всей системы российского образования, возрождения и реализации, при

этом национально-этнического потенциала, особую актуальность приобретает

погружение исследователей в историко-педагогическую национальную духовную

культуру, представляющую фундаментальную содержательную основу формирования

молодого поколения» [3, с.3]. А.Б. Панькин исходит из того, что на протяжении

тысячелетий формировалось оригинальное этнокультурное пространство, в котором

средства, методы и технология воспитания детей не утратили своего значения и сегодня.

Профессор утверждает, что культура всегда служит нуждам индивида, и прежде всего

трем его основным потребностям: базовым (необходимости в пище, удовлетворении


физических потребностей), производным (в разделении труда, в защите, в социальном

контроле), интегративным (потребностям психологической безопасности, социальной

гармонии, цели жизни и т.д.) [4, с.50].

Каждая историческая эпоха по своему наполняет суть безопасной

жизнедеятельности человека, общества. Сегодня особенно актуальным становится

принятие комплекса мер по педагогическому обеспечению безопасной

жизнедеятельности в учреждениях образования: обоснование теоретико-

методологических подходов к проблеме, разработка её концептуальных основ, создание

целостной системы безопасности и противодействия терроризму и т.д.

Формирование основ безопасности у детей старшего дошкольного возраста

рассматривается в исследованиях Н.Н. Авдеевой, К.Ю. Белой, Г.К. Зайцева, В.Н.

Зимониной, О.Л. Князевой, А.И. Садретдиновой, Т.С. Сантаевой, О.П. Синельниковой,

Р.Б. Стеркиной, Л.Г. Татарниковой, Л.Ф. Тихомировой, Т.Г. Хромцовой и др.

Тем не менее, анализ научных трудов по проблемам обеспечения безопасности и

противостояния терроризму в учреждениях образования показывает, что в

педагогической науке пока еще не сложилась целостная научная концепция и не

разработана система работы по обеспечению безопасной жизнедеятельности в

дошкольных образовательных учреждениях.

Отсюда и цель нашего исследования: определение теоретико-методологических

основ проблемы, разработка целостной концепции и системы работы по

педагогическому обеспечению безопасности личности в условиях ментального

терроризма.

Мы считаем, что в педагогическом обеспечении безопасной жизнедеятельности

детей в дошкольных образовательных учреждениях должны учитываться климатические

особенности Севера, веками выработанное и накопленное этнокультурное наследие

аборигенных народов, их ценностное отношение к природе, гармонизации окружающей

действительности.

Для педагогического обеспечения безопасности личности и противодействия

терроризму в дошкольных образовательных учреждениях особое внимание должно быть

обращено на подготовку педагогов в вузе. Студенты в вузе должны овладевать


знаниями, умениями и навыками для обеспечения безопасности жизнедеятельности

маленьких детей в дошкольных образовательных учреждениях. Они должны быть

компетентными и готовыми для принятия адекватных мер при чрезвычайных ситуациях:

природных (землетрясения, наводнения, ураганы, холода); социально-биологических

(болезни, эпидемии); техногенных (аварии или катастрофы, связанные с техникой);

социальных (криминальные, семейно-бытовые, религиозные и т.д.) и для формирования

культуры безопасности личности в условиях ментального терроризма.

Для подготовки квалифицированных специалистов в нашем вузе разрабатываются

учебно-методические комплексы, программы, главной задачей которых является

формирование у будущих педагогов культуры безопасности личности и подготовка

их к обеспечению готовности в оказании профессионально грамотной защиты и охраны

жизни детей, обращается особое внимание на воспитание культуры личной,

экологической безопасности, социальной и правовой защищенности. Ведение здорового

образа жизни, знание норм и правил поведения в экстремальных и чрезвычайных

ситуациях, выработка морально-психологической устойчивости, дисциплины является

стержнем профессиональной этики педагога ДОУво имя охраны, спасения жизни

маленьких детей.

Отмечая актуальность обеспечения безопасности и антитеррористической

защищенности детей в дошкольных образовательных учреждениях, подчеркиваем, что

назрела необходимость комплексного решения правовых, организационных, научно-

методических основ проблемы.

Литература

1. Садретдинова А.И. Педагогическое проектирование образовательной среды

формирования культуры безопасности жизнедеятельности у дошкольников: автореф. дис. . канд.

пед. наук. - Екатеринбург: УГЛУ, 2009. – 64 с.

2. Никандров Н.Д. Россия: социализация и воспитание на рубеже тысячелетий. – М. :

Педагогическое общество России, 2000. – 303 с.

3. Баишева М.И. , Григорьева А.А. Этнопедагогические воззрения народа саха: на

материале олонхо. – Новосибирск: Наука. – 2008. – 167 с.

4. Панькин А.Б. Формирование этнокультурной личности: Учеб. пособие. – М.: Издательство

МПСИ; Воронеж: Издательство НПО «МОДЭК», 2006. – 280 с.


Polycultural education of students

based on the pedagogical projection

Natalia Filatova, PhD

Pedagogical Institute, North-Eastern Federal University,

the Republic of Sakha (Yakutia), Russia

Abstract

A component of the global challenge of the future is an objective necessity of unity, convergence,

spiritual integration of human communities. Polycultural education and training are relevant to the multi-

national, multi-religious Russia. In these circumstances, there is a social need in the formation of polycultural

competence of personality, combining focus on ethno-cultural moral values, tolerance and capacity for

intercultural dialogue.

We consider polycultural competence as a complex integrative quality of a future school-leaver,

reflecting his/her knowledge in the content, tools and modes of interaction with the world of culture, which is

realized in his/her ability to navigate freely in a polycultural world, to understand its value and meaning,

embodying them in decent samples of civilized behavior in the process of positive interaction with people of

different cultures (nations, races, beliefs, social groups).

We determined the most effective phases, structure and pedagogical conditions of formation of students’

polycultural competence.

Essential factors to the effective work on the formation of polycultural individual are polycultural

competence of the teacher, his/her personality, professionalism and ability to shift from teaching as a passive,

reproductive learning on learning as a process of active and productive knowledge of reality.

Success of the process of the formation of students’ polycultural competence is provided by a set of

pedagogical conditions: enhancing of project and research capacity of the educational process, modeling

polycultural educational environment, the use of active learning methods of intercultural education based on

competence, synergetic and practice-oriented approaches.

Memoir

Relevance of polycultural personality development is caused by dynamic changes in

the life of modern society, polycultural information, economic and legal space. Polycultural

education and training are relevant for the multi-national, multi-religious Russia, a country

with many different cultures and problems in intercultural relations. In these circumstances,

there is a social need in the formation of polycultural competence of personality, combining

focus on ethno-cultural moral values, tolerance and capacity for intercultural dialogue.

Accordingly, the educational system should provide the individual the opportunity to self-

identify as a representative of a culture and tradition, ensure the integration of the individual

in the modern world civilization.


We consider polycultural competence as a complex integrative quality of a future

school-leaver, reflecting his/her knowledge in the content, tools and modes of interaction with

the world of culture, which is realized in his/her ability to navigate freely in a polycultural

world, to understand its value and meaning, embodying them in decent samples of civilized

behavior in the process of positive interaction with people of different cultures (nations, races,

beliefs, social groups).

We consider that the most effective phases of a polycultural competence of students

are:

- Propaedeutic: saturation of the polycultural component of general education and core

subjects in order to create demand for polycultural learning, and positive motivation for

polycultural interaction through mainstreaming of social and cultural functions of language,

increasing of cultural knowledge and providing an idea of the different peoples and their

cultures;

- Developing: activation of the process of formation of an integrated polycultural

picture in the minds of students, diving into the elements of native culture, access to the

multicultural environment and global cultural space through active forms of intercultural

learning such as discussions, debates, role-playing and simulation, interactive cross-cultural

training;

- Project-Research: gaining experience of polycultural interaction in the process of

working on research projects at various levels.

In the formation of a polycultural competence it is important to follow age-appropriate

form of ethnic identity:

- National education, understood as the formation of ethnic identity: inculcation of love

and respect for his/her people, proud of his/her cultural and historical achievements,

displaying of activity and independence in the understanding and awareness of the specificity

of his/her culture (the direction "I am a representative of my nation");

- Knowledge of the features of polycultural Russia, familiarizing students with the

ethnic environment, the formation of attitudes towards representatives of neighboring nations,

being proactive in understanding and appreciation of cultural knowledge, developing the

ability to exercise situational flexibility and to find ways to cross-cultural interaction (the

direction "I am Russian");


- Imparting knowledge about ethnic identity of distant peoples and the formation of

emotional and positive attitude to the ethnic diversity of the planet, the general cultural

knowledge of specific cultural values and awareness of the laws of development of world

cultures, their uniqueness (the direction "I am a world citizen").

The structure of the polycultural competence of students includes:

- Cognitive component - the formation of the system of polycultural knowledge, acting

as an orienting basis of individual’s activity in a multicultural society;

- Motivation and values component is the current system of motives and values

formation: the motives, values, interests, needs, polycultural qualities, governing daily life and

activities of the individual in a multicultural society;

- Activity component ensures the formation of the polycultural skills, respect for social

norms and rules of behavior in a multicultural society, the experience of positive interaction

with people of different cultures.

We made changes in the content of education by saturating a polycultural component

of general education and core subjects in order to create the demand for polycultural learning,

and positive motivation to interaction with different cultures. The most potential have the

lessons of history, social studies, world culture, foreign languages, Russian language and

literature, Yakut language and literature, geography, physical education, labor training, music,

and art.

Elective courses of multicultural focus: "Conflictology", "Psychology of

Communication", "Intercultural Communication", "Russian folk culture and Orthodoxy".

Outdoor activity is organized in seven centers: "Health", "Family", "Leisure", "Intelligence",

"Teenager", "Center of Friendship and Peace", “Aesthetic centre” are introduced in the

variable part of the curriculum.

An essential factor for effective work on the formation of polycultural individual is

polycultural competence of a teacher, his/her personality, his/her professionalism and ability

to shift from teaching as a passive, reproductive learning on learning as a process of active and

productive knowledge of reality. Following L.N. Berezhnova, I.L. Nabokov, V. Shcheglov,

we consider polycultural competence of a teacher as a set and the ability to function in a

polycultural society, the knowledge of the problems of the society, the understanding of the

mechanisms of its development, social activism and implementation of ethnopedagogically


directed projects, the ability to efficiency in the selection of priority values in a polycultural

society and the ability of a tolerant interaction in a polycultural society, to an adequate relation

to him/herself and to other people from a position of respect for human rights; taking

responsibility for decision of ethnopedagogical problems in the conditions of multi-ethnic

composition of students, the ability to implement him/herself as a representative of his/her

own culture.

Polycultural education of students based on pedagogical projection is realized through:

- Modeling of the educational environment on the principles of recognition, acceptance

and understanding of members of other social and cultural (ethnic, religious, subcultural,

social) groups;

- The focus on the development of the individuality of each student;

- The integration of educational and training process, based on competent, synergetic,

practice-oriented approaches;

- The organization of project and research activities of students within pedagogical

projects;

- The use of active forms of intercultural education.

The management structure was established in order to achieve success in the formation

of a polycultural competence of students on the basis of pedagogical projection. The council

rules the process of projection and implementation of polycultural education. Its responsibility

includes: the definition of social order for educational results, the definition of the goals and

objectives of polycultural education, motivation and commitment of the teaching staff to the

goals and objectives; specification of polycultural competence and defining a set of its

constituent skills, the development of educational programs and educational projects, training

of teachers to new technologies, assessment and appraisal of processes and results, and

analysis of consumers’ satisfaction.

The group of tutors passed advancing training towards the others so as to transmit their

knowledge and experience to other teachers, the group also performs instructional role in

relation to their colleagues.

Modeling pairs (dyads) are formed and used in the project activity, when it is

necessary to create an experience of action, attitude, emotional response in the paired


professional (interpersonal) interaction (Deputy Director - teacher, teacher - teacher of

additional education, etc.) for the participants of different professional (social, age) categories.

Under the project team it is understood the main group of people (experts) directly

involved in the educational project. Functions that allow to ensure the implementation of the

project are distributed among them. These functions include research, training, appraisal,

document preparation, coordination, technical and social development, methodological

support, administration.

The pilot group was established, which performs experimental actions in the logic of

the projects on polycultural education. A variety of training and working groups are formed,

performing the various functions according to the purpose of the project activity and the stage

of its implementation.

Training group of psychologists is formed to propaedeutics and overcome any

difficulties associated with the development of the mode and certain procedures of

experimentation. Training group’s job focuses on the diagnosis and correction of any personal

manifestations; removal of internal barriers to productive activities, the creation of a favorable

psychological and emotional atmosphere.

The process of forming a polycultural competence of students is more effective on the

basis of pedagogical projection, which provides an active, independent and initiative position

of students in mastering this competence, forming not only polycultural skills, but also

polycultural competence - skill directly associated with the experience of its application in

practical activities aimed at developing cognitive interest of students, developing general

educational skills (research, reflective, self-assessment, etc.), which implements the principle

of connection training with life.

Teachers worked out individual, group, network pedagogical projects in order to model

a polycultural learning environment on the following directions, "I am a representative of my

nation," "I am Russian," "I am a world citizen".

Projects in the primary school are: "Games from around the world", "Child of the

country “Olonkho", "Khomus - the music of friendship", "Info-culture", "Every little helps"

for applied arts in common with the Palace of Childhood, "The future of the country

“Olonkho" together with the Children Dance Sport Club "Sparkle" and the Club of vocal

singing "Skylark", "Every nation is an artist." These projects are carried polycultural and


primary school education by means of applied, artistic and musical arts, by means of folklore,

games, informational culture, etc.

Projects in the secondary school "Sunday School", Mega-project of networking "Arctic

idea - creativity and perspective", "Dialogue of Cultures", "Young Journalist", "Media

Center", “Communication without borders” implement polycultural education by means of

distance learning, dual education, practice-oriented and research projects, etc.

All educational projects exercise their functions in a single educational space, which

includes cultural studies (spiritual culture, cultural identity) component, institutional (family,

schools, universities, colleges, Palace of Children, the House of Peoples' Friendship, Center of

technical creativity, libraries, museums, theaters , cinemas, Physics and Mathematics Forum

"Lensky krai", Children's TV and Radio Academy "Polar Star", the media, etc.), social-

communicative (rules, regulations, intercultural communication and behavior), technological

(human and technical capacity) components.

Teachers, forming polycultural competence of schoolchildren, face the following

problems: to help students to overcome their own stereotypes of social reality; to use the

school environment in the education; to use of active learning methods, psychological

trainings, simulating typical to adult world relationships that allow students to experience the

complexity and inconsistency, encouraging the inclusion of students in a discussion of various

social problems; to establish a trusting relationship with the student, to show his/her

relationship to the students as an adults; to maintain their freedom of valuable self-

determination, to show the tolerance to the youth subculture.

Other conditions of formation of polycultural competence are organizing students’

meetings with other cultures in a specially prepared pedagogical environment, joint cultural

events, camping trips; meetings of students with other cultures can be modeled by a teacher in

special game situations where the children take on the role of different cultures’

representatives and come into provided intercultural dialogue; problematization of students’

relations to representatives of other cultures; the organization of problem discussion and

debate on the interpersonal, intercultural communication; organization of the reflection by

students of their attitude towards representatives of other cultures.

Thus, the relevance of preparation of graduates for life and work in a changing,

polycultural society requires teachers’ serious attention to ethnocultural and polycultural


educational component. Adding ethnocultural knowledge to the educational system,

realization of its potential and focus on the formation of Russian national identity, a modern

tolerant polycultural awareness to the development of interethnic and interfaith relations, so

important to multicultural Russia, develop a common strategic policy of the state to meet the

cultural needs of all the people of the country.

Success of the process of the formation of students’ polycultural competence is

provided by a set of pedagogical conditions: enhancing of project and research capacity of the

educational process, modeling polycultural educational environment, the use of active

learning methods of intercultural education based on competence, synergetic, practice-

oriented approaches.

The personality of the teacher, his/her professionalism contribute to the formation of

polycultural competence of students, as they train and educate not only on the intellectual

(knowledge transfer), but also on the affective (emotional connection) and behavioral levels.

References

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Publishing Center "Academy", Moscow, 2008. – 240 p.

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3. Davydov, Y.S. The concept of polycultural education in higher education of the Russian

Federation / Y.S. Davydov, L.L. Suprunova. - Pyatigorsk, 2003. - 49 p.

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Publishing Center "Academy", Moscow, 2007. – 288 p.

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O.V. Akulova. - CARO, St. Petersburg, 2005. – 272 p.

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2005. – 84 p.


Professional training of teachers-bachelors in the conditions

of modernization of research activity of higher education

Lena Maximova, Ph.D., Associate Professor

Department of Preschool Education

Pedagogical Institute

The North-Eastern federal university the Republic of Sakha (Yakutia), Russia

Priority orientation of education to modernization, improving its efficiency and quality

in the prevailing socio-economic conditions require new approaches to training of specialists

with higher education. There is a need to develop a Concept of modernization of the research

training system of bachelors in the conditions of transition to tier Higher Vocational

Education (HVE), considering a regional perspective of the North-East of Russia in the

North-Eastern Federal University (NEFU) named after M.K. Ammosov, which is caused by

the following circumstances:

- Modernization of the Russian education system, public order for the quality of higher

education in accordance with international educational standards;

- Gradual transition to tier higher education, the introduction of the Federal State

Educational Standard of HVE of the third generation, based on the competent development of

students;

- Creation HVE "North-Eastern Federal University named after M.K. Ammosov" as a

leading research and methodological center in the region, designed to innovate and manage

the integration of science, education and industry.

- The introduction of a set of measures to ensure the development of Russian science-

based universities, research institutions and advanced research laboratories and schools.

In the Concept are formulated theoretical and methodological approaches, objectives,

goals, direction, resource provision and modernization of the basic mechanisms of research

training of higher education in the field of study “Teacher Education” 050100 (qualification


(degree) "Bachelor") through the development of scientific and research activities of the

university.

The Concept of modernization of the research training of bachelors in transition to tier

HVE is worked out with the regional characteristics of the North-East of Russia in accordance

with the principles of educational policy in Russia.

In the North-East of Russia the improvement of the system of research training is the

issue of special importance, defined by the set of factors, associated with the most important

geo-strategic interests of the country, natural and economic potential of the north-eastern

region - the main resource for innovative ways of developing of Russian economy.

Contemporary theory and practice of higher education are faced with a contradictory

situation. On the one hand, it is awareness of the need to improve the efficiency and quality of

higher education, on the other hand, it is found a low level of productivity and quality of

work, education, in particular, the lack of coverage of adult higher education, a structural

imbalance between the purpose of higher education, its content and direction of preparation,

labor market demand and innovation; growing demands of modern enterprises to the level of

professional competence, the relevance of the regional market innovation and training, the

long-term regional needs.

The analysis showed that the basic ideas of modernization of preparing students for

research in transition to tier higher vocational education should advocate following ideas:

- The unity of the educational and research activities;

- Scientific support for research activities;

- Continuity of development of students’ research activity through various forms of

organization of educational activities;

- The creation of innovative educational environment of the university, which

integrates the main factors of personal development of students.

Considering a regional perspective of the North-East of Russia it is required to

establish a system of research for bachelors, oriented to the needs of innovative-based

economy, its priority sectors in the long term, science, technology, education, culture and

social development, region-specific, spatial labor market, the distribution of labor in it. As its

mission the North-Eastern Federal University has identified nurturing competitive

professionals, carrying out researches, innovative and technological development for the


formation of sustainable economic and social developed circumpolar region, ensuring a high

quality of life, the preservation and development of the culture of the peoples of the North-

East of Russia.

Consequently, the purpose of bachelors’ research training should be to ensure the

progressive development of the region through the organization of level of professional

education, aimed at the research and innovative activities, appropriating to the international

and European standards.

This goal encompasses the following interrelated objectives:

- Development and implementation of bachelors’ educational programs considering a

regional perspective and needs of the social and productive capacity;

- A systematic methodological monitoring to develop the system of monitoring,

analysis and prediction, which is aimed at improving the quality of the educational process;

- Development and testing of pedagogical innovation, interactive technologies, active

learning and training methods and their introduction to the work of the teaching staff of the

university;

- Organizing and conducting workshops with students in the scientific and cultural

centers, youth parliaments of the public assembly, in enterprises, institutions, organizations,

etc.;

- Systematic training and retraining of the teaching staff;

- Forecasting and simulation training options with innovative thinking in the changing

conditions of the region and the country;

- Analysis of feedback from companies and institutions on the graduates and research,

to help correct the improvement of students’ research training;

- Development by students of social projects in various sectors of the real economy for

the city administration, specific agencies, etc.;

- Fundamental and applied research in high school (together with research and

manufacturing facilities), including on educational innovation;

- The use of educational technologies for students to select courses and build

individual learning paths;

- The creation of innovative educational environment and logistics, realizing the major

characteristics of the innovative higher education institution.


Competence development paradigm of higher education is defined as a strategic goal

in education. Modernization of the vocational training system is expected by increasing the

competence of bachelor's and master's degrees. Consequently, the federal state educational

standards of the third generation are aimed at developing competences of students. Therefore,

modernization of the system of Bachelors’ research training in transition to tier HVE should

be implemented in the framework of the competence approach, allowing to increase the

quality of research work and to strengthen the professionalization of students.

The introduction of the competence approach, implemented in the Russian educational

system is dictated by several factors. Firstly, by the trends in the world and the European

community for the integration and globalization of the economy and educational systems.

Russia signing the Bologna process requires radical structural modernization of the entire

system as a general education and higher education. Secondly, the need to improve the quality

of education. In this connection there is a paradigm shift of education, the transition from

"knowledge, skills and abilities paradigm" to the "paradigm of competence”, that entails the

change in the content of education, the development of new approaches to the construction of

the educational process. Thirdly, the rapid update of the professions and professional

activities affect the improvement of the educational process for the formation of university

graduates ready to meet new professional challenges, that arise during the labor process.

Modern high school graduate should possess polyfunctional competencies that allow him to

quickly adapt to the working environment, be prepared for constant updating of his level of

competence, learning throughout life. Innovative development of economy and production

requires a creative approach to the profession, maintenance of professional activity.

Professional research competencies by students are formed in the frame of the

competence approach, that are part of the professional competence of the prospective

employee.

Professional research competences (PRC) allow to do professional work in the field of

scientific research. PRC is a characteristic of personality, reflected in the willingness and

ability to carry out research activity.

Set of professional applications provides formation of following competencies for

bachelors:


- To develop skills to complex analysis of changing socio-economic processes, solving

professional problems through the generation of innovation;

- Development of skills to the organization and management of project modalities with

the unconventional approach, including practical projects of the educational system’s

development;

- The development of competences: analytical approaches to combining, searching,

processing and analysis of mixed information to generate innovation;

- Development of professional competencies that are in demand by employers in the

modern target and individual training;

- The development of competence for the research and the practical use of the results

of basic and applied researches.

The strategy of development of the bachelors’ professional training in the North-

Eastern Federal University in the conditions of modernization of the research activity involves

the following main areas:

- Providing the integration of pedagogical theory and practice through the

development and implementation of major educational programs to meet the needs of

educational institutions of different types and species of different educational services in

accordance with the principles of openness and diversification of education;

- Mastering by the students basic skills of research and innovative activities through

their inclusion in the appropriate practice;

- Co-operation and partnership with social institutions in order to search for their

applications, and in search of a fundamental subject to the needs of the region;

- The transformation of the university into the center of communications of business,

society and the state on the problems of scientific and technological forecasting, the exchange

of advanced knowledge, research training, addressing issues of effective innovation in the

educational system of the North-East of the country.

- Interdisciplinary humanitarian researches through the development of

interdisciplinary research teams based on institutes and departments of the NEFU, as well as

non-university research institutions, social and cultural institutions of the region, country and

other countries.


- The expansion of ties with the leading Russian and foreign universities to enable

bachelors to realize the right to pursue academic mobility within and outside the country.

To improve the quality of training and the promotion of employment of graduates it is

planned to use opportunities of social partnership with other educational institutions and

employers. In this connection it is necessary to create a university consortium of social

partners to establish cooperation with the university of manufacturing organizations, attracting

students to the participation in the innovative project and research work with them, which will

allow employers to objectively assess the quality of the graduates.

The necessary conditions for improving the efficiency and quality of research, the

efficient use of existing intellectual, material and manufacturing resources are created in the

North-Eastern Federal University.

The university developed their own traditions and accumulated extensive experience

of training of scientific personnel for various industries, including the use of modern

information technologies to integrate education, research and innovation in order to provide a

new level of professional training.

In the transition to the tier higher professional education it is necessary to ensure the

occupational mobility of bachelors. Considering the current realities of the information

society, the future of the employee must possess information technologies, be able to process

and transmit information and knowledge.

The implementation of information approach in the training of bachelors will allow:

- To create a holistic educational environment;

- To apply information and communication technologies for improving the quality of

students’ training;

- to use a structural representation of information for collection, processing, storage

and transmission;

- to accumulate the hardware and software for information providing of bachelors’

research training.

This conceptual approach is particularly relevant in regard to the computerization of

education and training systems, the saturation of information products, tools and technologies.

At the same time it allows to provide bachelors’ research training with necessary resources.


To ensure the quality of bachelors’ research training it is necessary to use resource-

based approach that promotes the integrity of the coverage of all agencies involved in this

work, areas of research in the university, the necessary resources in order to strengthen the

scientific potential.

The following principles dominate during the implementation of the resource

approach:

- The principle of conceptual focus, involving the development of conceptual

approaches and theoretical foundations of change;

- The principle of systems, providing a connection together of all the elements and

components of the system;

- The principle of invariability, involving the establishment of common positions and

approaches for a wide class of objects; •

- The principle of variability, providing the possibility of accounting and the use of

specific features of the object and the transformed situation;

- The principle of the organizational and resource provision, including organizational

reasoning of all the actions that convert an object, targeted of investment areas and programs,

which are expected to best effect;

- The principle of openness, which ensure an interaction of all the subjects of the

educational process;

- The principle of integrity and complexity, involving the union of all the structures

and trends.

For the effective management of the formation of professional research skills of

students it is important to use a variety of resources: financial, economic, logistical,

regulatory, organizational, personnel, scientific, information ones.

Financial and economic support will be implemented on the Development Program of

the university/

Fundraising - through:

• Participation in the federal and regional grant programs;

• Providing students and professionals of additional educational services;

• Conducting training courses, training and re-training on the methodology of research

and the organization of research activities.


Logistical support includes:

• Purchase of equipment for the implementation of exchange programs and research.

Purchase of equipment and technology to equip new databases and upgrade logistics facilities

training and students’ and graduate students’ work experience.

• Establishment of the Institute of Interdisciplinary Humanities Research of NEFU,

coordinating research activities of humanitarian departments of the University in order to

create a common space of research in the humanitarian work of the university. Establishment

within the institutions of the research laboratories and scientific and educational centers in

priority humanitarian directions. Development of integrated research projects and

interdisciplinary educational programs.

Regulatory support includes the development of regulations:

• The scientific student communities of teachers-researchers;

• The e-University;

• The resource centers.

Organizational maintenance involves the creation of:

• Association of the strategic partners and the consortium of the social partners of

NEFU;

• Resource centers, training and counseling offices and representative offices of NEFU

in the cities and districts of the region;

• University Council of educators-researchers-mentors;

• The organization of small innovative enterprises, educational and industrial

complexes, grounds.

Staffing includes:

• Personnel training, holding interdisciplinary methodology;

• Creating sustainable inter-university and international scientific relations of NEFU;

• Organizing training courses for teachers on pedagogical support of personal and

professional development of students.

Scientific and methodological support includes the development of:

- Programs of pedagogical support of professional self-determination, personal and

professional development of bachelors.


- Curriculum, textbooks, electronic teaching methods courses based on modular

system.

Information provision provides:

- The creation of a common information space for the education of the North-East of

Russia;

- Software and technical support for the operation of scientific schools, e-university;

- The development of electronic portfolios, case studies on the formation of research

skills of students, e-learning resources;

- Development of software for conducting research;

- The creation of e-science libraries to provide free access to the learning content;

- Software support of bachelors’ research training.

For successful implementation of the modernization of bachelors’ research training it

is necessary to perform the following steps:

1. improvement of methodological, scientific, educational, administrative system of

the educational process, adequate needs of employers, the formation of research competence

in students;

2. creation of scientific communities of teachers and researchers and students to

generate pedagogical innovations that shape students' innovative thinking and learning

through the development and implementation of the results of innovative projects in their own

teaching activities;

3. wide dissemination of advanced information technology in the projecting and

implementation of training plans and programs for bachelors

4. functioning of the system for monitoring and forecasting of innovation in education;

5. effective use of scientific and educational-methodical information support of the

educational process;

6. improvements in technology of research, innovative activity of teachers of all levels

of education;

7. creating subsystems of students preparation for research: preparation for research in

educational activities, extracurricular activities and the inclusion of students in the research

and production, research and innovation;


8. implementation of continuity of purpose and content, methods and means of

preparation for the research of future professionals;

9. development of scientific and methodological support to the preparation of students’

research;

10. implementation of the university joint research projects, joint research laboratories

for scientific and educational activities, the creation of institutional structures that facilitate

the development of activities in the field of science and innovation.

Implementation mechanisms are aimed at enhancing the practice of scientific research

and experimental, project works, implementation of the results of joint research into practice

of educational institutions of all kinds and types, of different forms of ownership, having or

shaping the modern innovative environment, or in research institutions.

In general, the concept of modernization of bachelors’ research training in NEFU till

2019 will improve the quality of training of highly qualified personnel in the North-Eastern

Federal University named after M.K. Ammosov, as in the leading innovative research,

educational and cultural center of the North-East of Russia.


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