Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction
Using the central extension of the Cuntz C*-algebra, we study the periodicity for corresponding fractals.
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2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509226170843136 |
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| author | Castillo, Alvarado F. L. Lawrynowicz, J. Suzuki, O. Кастіло, Альварадо Ф. Л. Лавринович, Й. Сузукі, О. |
| author_facet | Castillo, Alvarado F. L. Lawrynowicz, J. Suzuki, O. Кастіло, Альварадо Ф. Л. Лавринович, Й. Сузукі, О. |
| author_sort | Castillo, Alvarado F. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:45Z |
| description | Using the central extension of the Cuntz C*-algebra, we study the periodicity for corresponding fractals. |
| first_indexed | 2026-03-24T02:37:44Z |
| format | Article |
| fulltext |
UDC 517.5
J. Ławrynowicz (Inst. Phys., Univ. Łódź, Poland),
O. Suzuki (Coll. Human. and Sci., Nihon Univ., Japan),
F. L. Castillo Alvarado (Escuela Super. de Fı́s. Mat. Inst. Politecn. Nac., México)
BASIC PROPERTIES AND APPLICATIONS
OF GRADED FRACTAL BUNDLES RELATED
TO CLIFFORD STRUCTURES: AN INTRODUCTION*
ОСНОВНI ВЛАСТИВОСТI ТА ЗАСТОСУВАННЯ
СТУПIНЧАСТИХ ФРАКТАЛЬНИХ ЖМУТКIВ,
ЩО ПОВ’ЯЗАНI ЗI СТРУКТУРАМИ КЛIФФОРДА. ВСТУП
Using the central extension of the Cuntz C∗-algebra, we study the periodicity for corresponding fractals.
З допомогою центрального розширення C∗-алгебри Кунца вивчається перiодичнiсть для вiдповiд-
них фракталiв.
1. Introductory: dynamics of binary and ternary alloys. The idea of fractal modelling
of crystals comes back to Bethe [1] who observed its convenience when coming to first,
second, third nearest neighbours of an atom. Taking into account that it is a neighbour of
two or more other atoms, even in the case of one layer with a lattice formed by squares,
one naturally comes to the notion of cluster [2, 3]. It is then natural to cut the plane
of lattice correspondingly to the cluster involved and construct a Riemann surface or a
Bethe lattice — a fractal set of the branch type [4, 5]. The construction is parallel to that
related to the holomorphic function f(z) = exp z2 in C (Fig. 1). The example shows
already the importance of the corresponding group Γ of cover symmetry transformations
(Decktransformationengruppe), inoculation (of the branch corresponding to no. 1 on the
branch corresponding to no. 4), and gradation related to the points �, •, �, ◦.
The next important step was done by Kikuchi [6], who — within his theory of cooper-
ative phenomena — developed a method of approximation for order-disorder phenomena.
In this context, Sukiennicki, Wojtczak, Zasada, and Castillo Alvarado [7] investigated
an infinite thin film of anAB3 alloy. As examples we may take Ni3Fe or Cu3Au. They as-
sumed the (111) orientation of the alloy. Let z(j) denote the concentration of theA-atoms
in the layer j = 0, 1, . . . ; j = 0 corresponding to the surface. Then the concentration of
the B-atoms in that layer is 1 − z(j). If U is the energy of interaction of the system, T
— the absolute temperature, and g – the number of configurations possible, the entropy
S and the free energy F of the system are given by
S = k ln g and F = U − TS,
*Research of the first author partially supported by the State Committee for Scientific Re-
search grant PB1 P03A 03 27 (Sections 2 and 3 of the paper), and partially by the grant of the
University of Łódź no. 505/693 (Sections 1 and 4). Research of the third author supported by
CONACYT, Becario de COFAA-IPN.
c© J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 5 603
604 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO2 JULIAN ÃLAWRYNOWICZ, OSAMU SUZUKI, AND FRAY DE LANDA CASTILLO ALVARADO
Figure 1. Scheme showing the way of constructing the inoculated graded
fractal Ξ(f : c) of the branch type, where f(z) = exp z2 , with gradation
related to the points ¥ , • , ¤ , ◦ corresponding to the Riemann surface of Γ .
The next important step was done by Kikuchi [13], who – within his theory of cooperative
phenomena – developed a method of approximation for order-disorder phenomena.
In this context, Sukiennicki, Wojtczak, Zasada, and Castillo Alvarado [39] investigated
an infinite thin film of an AB3 alloy. As examples we may take Ni 3 Fe or Cu 3 Au. They
assumed the (111) orientation of the alloy. Let z(j) denote the concentration of the A -atoms
in the layer j = 0, 1, . . . ; j = 0 corresponding to the surface. Then the concentration of the
B -atoms in that layer is 1− z(j) . If U is the energy of interaction of the system, T – the
absolute temperature, and g – the number of configurations possible, the entropy S and the
free energy F of the system are given by
S = k ln g and F = U − TS,
respectively, where k is the Boltzmann constant, and the conditions for thermodynamic
equilibrium at a given temperature T are
(∂/∂τ) F |τ=τ(j) = 0, (∂/∂z) F |z=z(j) = 0 at each j, λ = const.
Thus
(1/T ) (∂/∂τ) U |τ=τ(j) = (∂/∂z) S|τ=τ(j).
The authors of [39] have calculated that
∂S
∂τ
=
3
16
k ln
(
z + 3
4
τ
) (
1− z + 1
4
τ
)
(
z − 1
4
τ
) (
1− z − 3
4
τ
) for z = z(j) and τ = τ(j),
and thus, finally,
16
3
kT (∂/∂τ) U =
z(1− z) + 1
4
(3− 2z) + 3
16
τ 2
z(1− z)− 1
4
(1 + 2z) + 3
16
τ 2
for z = z(j) and τ = τ(j).
Fig. 1. Scheme showing the way of constructing the inoculated graded fractal Ξ(f : c)
of the branch type, where f(z) = exp z2, with gradation related to the points �, •, �, ◦
corresponding to the Riemann surface of Γ.
respectively, where k is the Boltzmann constant, and the conditions for thermodynamic
equilibrium at a given temperature T are(
∂
∂τ
)
F
∣∣∣∣
τ=τ(j)
= 0,
(
∂
∂z
)
F
∣∣∣∣
z=z(j)
= 0 at each j, λ = const.
Thus
1
T
(
∂
∂τ
)
U
∣∣∣∣
τ=τ(j)
=
(
∂
∂z
)
S
∣∣∣∣
τ=τ(j)
.
The authors of [7] have calculated that
∂S
∂τ
=
3
16
k ln
(
z +
3
4
τ
)(
1− z +
1
4
τ
)
(
z − 1
4
τ
)(
1− z − 3
4
τ
) for z = z(j) and τ = τ(j),
and thus, finally,
16
3
kT
(
∂
∂τ
)
U =
z(1− z) +
1
4
(3− 2z) +
3
16
τ2
z(1− z)− 1
4
(1 + 2z) +
3
16
τ2
for z = z(j) and τ = τ(j).
Let us take the pseudometric
ds2 = c2dt2 − dx2
A − dy2
A − dz2
A − dx2
B − dy2
B − dz2
B − ησcσdσ
2 − ητ cτdτ
2, (1)
where ησ = 1 or −1, ητ = 1 or −1, cσ and cτ are positive constants, and σ is the
stochastic parameter (e.g., σ = S or σ = σ0, the short-range order parameter), and
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 5
BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES RELATED . . . 605
consider a Lorentz-like deformation of the nine-dimensional system of the degrees of
freedom: {(
xA, yA, zA, xB , yB , zB , t, σ, τ
)}
propagating itself as a wave-like perturbation. We assume, for the sake of simplicity, that
dxA = dx′A, dxB = dx′B , dyA = dy′A, dyB = dy′B , dt = dt′,
dzA =
(
1 + c2z
)−1/2
czdz, dzB =
(
1 + c2z
)−1/2
dz,
cz being a positive constant. Then, thanks to [8], in that deformation:
dz = ρz (dz′ + vz
σdσ
′ + vz
τdτ
′) ,
dσ = ρσ (vσ
z dz
′ + dσ′ + vσ
τ dτ
′) ,
dτ = ρτ (vτ
zdz
′ + vτ
σdσ
′ + dτ ′) ,
(2)
where ρz, ρσ, ρτ are positive constants and vz
σ, v
z
τ , v
σ
z , v
σ
τ , v
τ
z , v
τ
σ are real constants, we
already know the coefficients vσ
τ and vτ
σ. Indeed,
S = S (σ0, τ) , σ0 = σ0 (S(σ0, τ), τ) ;
(
∂
∂σ0
)
S ≈ 0,
(
∂
∂τ
)
S ≈ 0.
Similarly, in the case of ternary alloys (e.g., Cd SxTe1−x; cf. [9]), we have to consider
an additional order parameter and – in the simplest case — the deformation
dz = ρz (dz′ + vz
σdσ
′ + vz
τdτ
′ + vz
θdθ
′) ,
dσ = ρσ (vσ
z dz
′ + dσ′ + vσ
τ dτ
′ + vσ
θ dθ
′) ,
dτ = ρτ (vτ
zdz
′ + vτ
σdσ
′ + dτ ′ + vτ
θdθ
′) ,
dθ = ρθ
(
vθ
zdz
′ + vθ
σdσ
′ + vθ
τdτ
′ + dθ′
)
,
(3)
where also ρθ has to be a positive constant and vz
θ , v
σ
θ , v
τ
θ , v
θ
z , v
θ
σ, v
θ
τ — real constants.
Here we arrive at a thirteen-dimensional system of the degrees of freedom.
This setting, especially formulae (1) – (3) provide one of possible motivations for
studying fractals related to Clifford structures and their relationship with twistor-like
structures. At the end of our outline we sketch, as an application, a geometrical model of
the surface melting effect (five degrees of freedom). The research will be continued in the
second part of the paper.
2. From alloys to fractals related to Clifford structures. Given generators A1
1 =
= A1, A
1
2 = A2, . . . , A
1
2p−1 = Ap−1 of a Clifford algebra Cl2p−1(C), p = 2, 3, . . . , in
particular the generators
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606 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i 0
)
, σ3 =
(
1 0
0 −1
)
(4)
of the Pauli algebra [10, 11], consider the sequence
Aq+1
α = σ3 ⊗Aq
α ≡
(
Aq
α 0
0 −Aq
α
)
, α = 1, 2, . . . , 2p+ 2q − 3,
Aq+1
2p+2q−2 = σ1 ⊗ Ip,q ≡
(
0 Ip,q
Ip,q 0
)
, (5)
Aq+1
2p+2q−1 = −σ2 ⊗ Ip,q ≡
(
0 iIp,q
−iIp,q 0
)
,
of generators of Clifford algebras Cl2p+2q−1(C), q = 1, 2, . . . , and the sequence of the
corresponding systems of closed squares Qα
q+1 (of diameter 1, centred at the origin of
C, where Ip,q = I2p+q−2 , the unit matrix of order 2p+q−2) together with their 4p+q−3
subsquaresQα,j
q+1,k with sides parallel to the sides ofQα
q+1 for α ≤ 2p−1, and into 4p+q−2
analogous subsquares for α ≥ 2p. We endow the squares Qα
q with the gradating function
(colour) gα
q equal aqk
αj within Qαj
qk and 0 otherwise. We call (5) a basic construction [12].
It is convenient to start with q always from 1, i.e., to shift q for α ≥ 2p correspond-
ingly. This means that, in the case of the latter two generators in (5), we have to shift q by
p− 1. For
q = 5− p, p = 2, 3, 4,
the sequence (5) gives the expected fractal model for a binary alloy. For
q = 7− p, p = 2, 3, 4, 5, 6,
(5) gives the expected fractal model for a ternary alloy.
If, for fixed α in (5), we now consider
(Qα
n, (Q
αγ
nh)) , n = 2, 3, . . . ,
we obtain a graded (coloured) Clifford-type fractal Σα, α = 1, 2, . . . , of the flower type
[4, 5]. The fractal set of each Σα is a dense subset of the diagonal of the corresponding
squares, running
from
1
2
√
2
(−1 + i) to
1
2
√
2
(1− i);
namely, it consists of points whose distance from the begining of the diagonal is an in-
tegral multiple m of 1/2 to some power (a positive integer) times the length 1 of the
diagonal of the matrix represented by the unit square. The graded (coloured) Clifford-
type fractal bundle Σα, α = 1, 2, . . .) is well defined which can be proved [13] using the
Cuntz algebra O(4) [14], Kakutani dichotomy theorem [15] and properties of petals [13]
being suitable pairs of (ordered) quadruples of neighbouring, sufficiently small subsets of
Qα
q and of (ordered) quadruples of the corresponding matrix entries.
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BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES RELATED . . . 607
It appears that, independently which generator is taken into account in the basic con-
struction (5), we get precisely two kinds of periods (dependent on the algebra and gener-
ator), related to
m = 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, . . . (6)
and
m = 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, . . . (7)
respectively, independently of α in Σα. The result is precisely formulated and proved in
[13, 16, 17].
Clearly, (5) and (6) are not formal definitions of the sequences. Even if one writes
n = 1, 2, . . . , it is possible to imagine the sequence (1, 2, 4, 5, 7, 8, . . .). Formally, we
have to consider two double sequences:
(2m− 1)22p−1, m, p = 1, 2, . . . , (8)
and
(2m− 1)22p − 1, m, p = 1, 2, . . . . (9)
If we order the set of numbers (8) to make the sequence
(an), an < an+1, n = 1, 2, . . . , (10)
then an with n odd give a subsequence of (6), and an with n even give a subsequence of
(7). (Explicitly, 2, 8, 14, 22, 26, 32, . . . correspond to (6), and 6, 10, 18, 24, 30, 34, . . .
correspond to (7).) If we order the set of numbers (9) to make the sequence
(bn), bn < bn+1, n = 1, 2, . . . , (11)
then bn with n odd give the subsequence of (7), complementary to that previously given,
and bn with n even give the subsequence of (6), complementary to that previously given.
(Explicitly, 3, 15, 27, 43, 51, 63, . . . correspond to (7), and 11, 19, 35, 47, 59, 67, . . .
correspond to (6).)
It is natural to ask whether the sequences (6) and (7) really reflect some properties
of generators (4), in particular of σ1, σ2, σ3, or they are of more general, perhaps trivial
character like decomposition of an arbitrary integer into a linear combination of different
powers of 2. The role of 2 (more precisely, of 0, 1 and −1) is here replaced by 0, 1, −1,
i,−i, the entries constituting the Pauli matrices σ1, σ2, σ3 because of the crucial role of
the Pauli algebra and of the Cuntz algebra O(4)generated by four isometries. At present
we cannot answer this question in the form of a theorem, but we can see the following
three important facts.
1. For each Σα the corresponding fractal set is as described above, i.e., it is naturally
related to the sequences (6) and (7).
2. The fact that the Clifford product of two vectors in R3 has a symmetric part and an
antisymmetric part is trivial, but the fact that the symmetric part is the scalar product of
those vectors, whereas the antisymmetric part is the wedge product is quite important [18].
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608 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
3. It seems that the construction of Σα provides a successful quick search of inter-
esting geometrical objects involving Clifford algebras in the following sense (P. Jordan,
von Neumann, and Wigner [19]): Consider the family of algebras A with addition + and
multiplication ◦ such that
λA ∈ A for A ∈ A, λ ∈ R,
((A ◦A) ◦B) ◦A = (A ◦A) ◦ (B ◦A) for A,B ∈ A,
if (A ◦A) + (B ◦B) + (C ◦ C) + . . . = 0 for A,B,C ∈ A,
then A = B = C = . . . = 0.
The only irreducible such algebras are the following:
(i) the algebra of real numbers with A+B, λA and A ◦B defined in the usual way;
(ii) Cn, n = 3, 4, . . .; Cn being the algebra with the linear basis 1, s1, . . . , sn−1,where
A+B and λA are defined in the usual way, but A ◦B is defined by
1 ◦ 1 = 1, 1 ◦ sj = sj and sj ◦ sk = δjk ◦ 1, j, k = 1, . . . , n− 1,
and δjk denoting the Kronecker delta;
(iii) Hp
q , p = 1, 2, 4, 8, and q = 3, or p = 1, 2, 4 and q = 4, 5, . . .; Hp
q being the
algebra of Hermitian matrices of order q whose elements are real numbers for p = 1,
complex numbers for p = 2, quaternions for p = 4, and octonions for p = 8, A+B and
λA are defined in the usual way, but for ◦ we have
A ◦B =
1
2
(AB +BA), (12)
where AB represents the usual matrix multiplication.
This means that the subfamily of {Hp
q} including octonions (p = 8) is quite marginal.
In contrast those corresponding to complex numbers (p = 2) and quaternions (p = 4)
have an infinite number of members. As far as our graded fractal bundles are concerned,
Σ1 corresponds to H2
6, Σ2 corr. to H2
40, Σ3 corr. to H2
224, . . .
and
Σ1 corresponds to H4
3, Σ2 corr. to H4
20, Σ3 corr. to H4
112, . . . .
Generally, we have
Σα corr. to H2
22α−1(2α+1) and Σα corr. to H4
22α−2(2α+1).
Now, let us come back to the relatioship between the gradation and inoculation of
fractals [20, 21]. Let us recall [22, 23] that each fractal Σ of the flower type has its dual
Ξ of the branch type and vice versa. In the case of the Sierpiński gasket this is illustrated
by Fig. 2 a), b). We consider there two copies of Ξ : Ξ1 and Ξ2 in Fig. 2 b), c), differing
in gradation (colour) and inoculate Ξ1 of the first kind on its n-th embranchment by
a branch of Ξ2 (Fig. 2 d)). The second kind of inoculation appears when, together with
changing gradation at the embranchment we change the number of branches (Fig. 4 vs.
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BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES RELATED . . . 609
Fig. 3). If, for some n, the n-th branch of a fractal Ξ1 within the graded fractal bundle in
question is considered in the bundle together with the 1-st embranchment of a fractal Ξ2
within the fractal bundle in question, we say thet Ξ1 is inoculated of the third kind at its
n-th embranchment by Ξ2.BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES 7
a) d)
b) c)
b) and c)
Figure 2. – a) The Sierpiński gasket Σ1 (of the flower type) after the 2nd step
of construction. – b) The fractal Ξ1 , dual to Σ1 (of the branch type) after the
2nd step of construction. – c) The fractal Ξ2 differing from Ξ1 in gradation.
– b) and c) The bundle (Ξ1, Ξ2) . – d) A graded fractal obtained from (Ξ1, Ξ2)
by inoculation of the first kind.
Figure 3. The inoculated graded fractal Ξ of the branch type, corresponding
to the sequences (2.3) and (2.4) responsible for the types of periods related to
(2.2).
(v) If we order the set of numbers (2.5) to make the sequence (2.7), we realize that an
with n odd correspond to (2.3), and an with n even correspond to (2.4).
(vi) If we order the set of numbers (2.6) to make the sequence (2.8), we realize that bn
with n odd correspond to (2.4), and (bn) with n even correspond to (2.3).
Fig. 2. a) The Sierpiński gasket Σ1 (of the flower type) after the 2nd step of construction.
b) The fractal Ξ1, dual to Σ1 (of the branch type) after the 2nd step of construction.
c) The fractal Ξ2 differing from Ξ1 in gradation.
b) and c) The bundle (Ξ1, Ξ2).
d) A graded fractal obtained from (Ξ1, Ξ2) by inoculation of the first kind.
The corresponding definitions for fractals of the flower type are similar. For instance
in the case of inoculation of the third kind we just replace “embranchment” by “growing
step”. As far as the graded Clifford-type fractal bundle (Σα) is concerned, Σ1, Σ2, . . .
. . . , Σ2p−1 are inoculated of the third kind at their first growing step by Σ2p and Σ2p+1;
the fractals Σ2p and Σ2p+1 are inoculated of the third kind at their first growing step by
Σ2p+2 and Σ2p+3, also Σ1, Σ2, . . . , Σ2p−1 are inoculated of the third kind at their second
growing step by Σ2p+2 and Σ2p+3, etc.
Now, following [20] we define the graded fractal Ξ of the branch type, inoculated of
the first kind (Fig. 3), related to the sequences (6) and (7), in six steps:
(i) At the n-th embranchment we have the numbers 1, 2, . . . , 2n−1 in the growing
order. Then from 1 we get 1 and 2, from 2 we get 3 and 4; finally, from 2n−1 we get
2n − 1 and 2n.
(ii) Inoculation concerns the sequences (6) and (7), and we have to determine all the
related numbers preceding and following the inoculation.
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610 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES 7
a) d)
b) c)
b) and c)
Figure 2. – a)The Sierpiński gasket Σ1 (of the flower type) after the 2nd step
of construction. – b) The fractal Ξ1 , dual to Σ1 (of the branch type) after the
2nd step of construction. – c) The fractal Ξ2 differing from Ξ1 in gradation.
– b) and c) The bundle (Ξ1, Ξ2) . – d) A graded fractal obtained from (Ξ1, Ξ2)
by inoculation of the first kind.
Figure 3. The inoculated graded fractal Ξ of the branch type, corresponding
to the sequences (2.3) and (2.4) responsible for the types of periods related to
(2.2).
(v) If we order the set of numbers (2.5) to make the sequence (2.7), we realize that an
with n odd correspond to (2.3), and an with n even correspond to (2.4).
(vi) If we order the set of numbers (2.6) to make the sequence (2.8), we realize that bn
with n odd correspond to (2.4), and (bn) with n even correspond to (2.3).
Fig. 3. The inoculated graded fractal Ξ of the branch type, corresponding to the sequences (6) and (7)
responsible for the types of periods related to (5).
(iii) The numbers preceding the inoculation are of the form (8).
(iv) The numbers following the inoculation are of the form (9).
(v) If we order the set of numbers (8) to make the sequence (10), we realize that an
with n odd correspond to (6), and an with n even correspond to (7).
(vi) If we order the set of numbers (9) to make the sequence (11), we realize that bn
with n odd correspond to (7), and (bn) with n even correspond to (6).
The whole construction leading to the fractal Ξ is visualized on Fig. 3. The sixth
embranchment is drawn separately in the lower left and lower right parts of the figure.
We have the following Fractal Inoculation Theorem:
Theorem 1. The inoculated graded fractal Ξ of the branch type can be decom-
posed to the bundle of inoculated fractals Ξ1 and Ξ2 without gradation, where Ξ2 is
repeated infinitely many times. Here Ξ1 corresponds to (6) and Ξ2 corresponds to (7).
Precisely, the embranchments of Ξ1 are renumbered according to the scheme n 7→ n+1.
At the new first embranchment of Ξ1 this fractal is inoculated of the second kind by the
first copy Ξ1
2 of Ξ2. At the first embranchment of Ξ1
2 this fractal is inoculated of the
second kind by the second copy Ξ2
2 of Ξ2. At the first embranchment of Ξ2
2 this fractal
is inoculated of the second kind by the third copy Ξ3
2 of Ξ2, etc.
The whole construction is visualized on Fig. 4. A detailed proof will be given in Part
II of the paper.
3. Atomization theorem for fractals and Hurwitz twistor-like structures. The
fractal Ξ (shown on Fig. 3) has six types of embranchments:
, j ,
j jjj ,
j jj ,
j
,
jjj .
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8 JULIAN ÃLAWRYNOWICZ, OSAMU SUZUKI, AND FRAY DE LANDA CASTILLO ALVARADO
Figure 4. The bundle (Ξ1, Ξ
1
2, Ξ
2
2, . . .) of inoculated fractals without grada-
tion, corresponding to the sequences (2.3) and (2.4) responsible for the types
of periods related to (2.2).
The whole construction leading to the fractal Ξ is visualized on Fig. 3. The sixth em-
branchment is drawn separately in the lower left and lower right parts of the figure. We have
the following Fractal Inoculation Theorem:
Theorem 1. The inoculated graded fractal Ξ of the branch type can be decomposed to the
bundle of inoculated fractals Ξ1 and Ξ2 without gradation, where Ξ2 is repeated infinitely
many times. Here Ξ1 corresponds to (2.3) and Ξ2 corresponds to (2.4) . Precisely, the
embranchments of Ξ1 are renumbered according to the scheme n 7→ n + 1 . At the new first
embranchment of Ξ1 this fractal is inoculated of the second kind by the first copy Ξ1
2 of Ξ2 .
At the first embranchment of Ξ1
2 this fractal is inoculated of the second kind by the second
copy Ξ2
2 of Ξ2 . At the first embranchment of Ξ2
2 this fractal is inoculated of the second kind
by the third copy Ξ3
2 of Ξ2 , etc.
The whole construction is visualized on Fig. 4. A detailed proof will be given in Part II of
the paper.
3. Atomization theorem for fractals and Hurwitz twistor-like structures
The fractal Ξ (shown on Fig. 3) has six types of embranchments:
, j,
j j
j
j,
j j
j
,
j
,
j
j
j,
Here resp. grepresent members of (2.3) resp. (2.4). Only two latter structures correspond
to inoculation at the embranchment. They will be called gemmae of Ξ . Extending our
Atomization Theorem (on isometric embeddings) [26, 27] we are going to state the following
Atomization Theorem for Fractals [20]:
Theorem 2. Suppose that the pseudometric corresponding to the p -dimensional real vector
space appearing in the definition of an Hermitian Hurwitz pair of bidimension (p, n) has the
Fig. 4. The bundle (Ξ1, Ξ1
2, Ξ2
2, . . .) of inoculated fractals without gradation, corresponding
to the sequences (6) and (7) responsible for the types of periods related to (5).
Here resp. grepresent members of (6) resp. (7). Only two latter structures corre-
spond to inoculation at the embranchment. They will be called gemmae of Ξ. Extending
our Atomization Theorem (on isometric embeddings) [24, 25] we are going to state the
following Atomization Theorem for Fractals [21]:
Theorem 2. Suppose that the pseudometric corresponding to the p-dimensional
real vector space appearing in the definition of an Hermitian Hurwitz pair of bidimen-
sion (p, n) has the form
〈dx, dx〉 = dx2
1 − dx2
2 − . . .− dx2
5,
or dx2
1 + . . .+ dx2
3 − dx2
4 − dx2
5,
for p = 5,
(13)
respectively:
〈dx, dx〉 = dx2
1 − dx2
2 − . . .− dx2
9,
or dx2
1 + . . .+ dx2
5 − dx2
6 − . . .− dx2
9,
or dx2
1 + . . .+ dx2
3 − dx2
4 − . . .− dx2
9,
or dx2
1 + . . .+ dx2
7 − dx2
8 − dx2
9,
for p = 9,
(14)
respectively:
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612 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
〈dx, dx〉 = dx2
1 − dx2
2 − . . .− dx2
13,
or dx2
1 + . . .+ dx2
7 − dx2
8 − . . .− dx2
13,
or dx2
1 + . . .+ dx2
3 − dx2
4 − . . .− dx2
13,
or dx2
1 + . . .+ dx2
11 − dx2
12 − dx2
13,
or dx2
1 + . . .+ dx2
5 − dx2
6 − . . .− dx2
13,
or dx2
1 + . . .+ dx2
9 − dx2
10 − . . .− dx2
13,
for p = 13,
(15)
Then in each of these cases there exists a finite subfractal Ξ0 ⊂ Ξ, where all gemmae
can be obtained with the help of type-changing transformations listed in Table 1.
Table 1. Fractal atoms
No. Atom No. Atom
1 s1 :
( j00
)
⇐⇒
( j3
0
)
7 s41 :
( j35
27
)
⇐⇒
( j43
35
)
2 r1 :
( j3
0
)
⇐⇒
( i11
3
)
8 r31 :
( j43
35
)
⇐⇒
( j47
43
)
3 s31 :
( j11
3
)
⇐⇒
( j15
11
)
9 s21 :
( i47
43
)
⇐⇒
( j51
47
)
4 r21 :
( j15
11
)
⇐⇒
( j19
15
)
10 r32 :
( j51
47
)
⇐⇒
( j59
51
)
5 s32 :
( j19
15
)
⇐⇒
( j27
19
)
11 s33 :
( j59
51
)
⇐⇒
( j63
59
)
6 r41 :
( j27
19
)
⇐⇒
( j35
27
)
In the above table
(
j11
3
)
etc. stays for the pair of gemmae
21 22
11
6j and
j5 6
3
jj
2
with j6 and 2 taken away. The other symbols and numbers can correspondingly
be deduced from Fig. 5 illustrating the (constructive) proof. For the full transformations
listed in the table, the gemmae indicated have to be extended to finite sequences of petals
corresponding to these gemmae and related to the generators of the Clifford algebra con-
cerned. Under the type of gemmae we mean the class of abstraction of all the structures
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BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES RELATED . . . 61310 JULIAN ÃLAWRYNOWICZ, OSAMU SUZUKI, AND FRAY DE LANDA CASTILLO ALVARADO
Figure 5. Type-changing transformations (¤) ⇐⇒ (©) : (a) decomposed as
stigmas of the pistil, (b) compared with the stamens.
In the above table ( j
11
3
) etc. stays for the pair of gemmae
21 22
11
6j and
j5 6
3
2
j
j
with j6 and 2 taken away. The other symbols and numbers can correspondingly be deduced
from Fig. 5 illustrating the (constructive) proof. For the full transformations listed in the
table, the gemmae indicated have to be extended to finite sequences of petals corresponding
to these gemmae and related to the generators of the Clifford algebra concerned. Under the
type of gemmae we mean the class of abstraction of all the structures ( d) resp. (
d) . A
transformation of gemmae is type-changing if it sends ( d) to ( d) or vice versa. In analogy
to the fact that 0 ∈ C lies in the real and imaginary axis as well, we complete the both
sequences (2.3) and (2.4) by 0 at the beginning, so that we have objects 0 and i0 . This is
caused by the fact that we have no pairing for i3 [20], so we need this extension.
The table is illustrated by Fig. 5. The choice of coordinates is motivated by the notions
of pistil and stamens introduced and discussed in [29]. The choice of symbols r and s for
particular basic type-changing transformations, called fractal atoms, as well as the corre-
sponding lower indices seem natural. Except for r1 , the choice of upper indices refers to the
length of the corresponding vectors. The notation 5, 7, etc. informs on the end of the
sequence (
m/2p−1, m = 0, 1, . . . , 2p−1 − 1
)
for 2p− 1 = 5, 7, etc.
A detailed proof of Theorem 2 will be given in Part II of the paper.
The Hurwitz-twistor counterpart of the Penrose theorem in the semiglobal version states
a one-to-one correspondence of the space of holomorphic solutions of the above mentioned
spinor equations of spin 1
2
n with the one-dimensional Dolbeault cohomology group H1 de-
pending on O(n− 2) = O([e]n−2) , where [e] is the canonical effective divisor of P3(C) . On
Fig. 5. Type-changing transformations (�)⇐⇒ (©):
(a) decomposed as stigmas of the pistil, (b) compared with the stamens.
( f) resp.
( f)
. A transformation of gemmae is type-changing if it sends
( f) to
( f)
or vice versa. In analogy to the fact that 0 ∈ C lies in the real and imaginary axis as well,
we complete the both sequences (6) and (7) by 0 at the beginning, so that we have objects
0 and i0 . This is caused by the fact that we have no pairing for i3 [21], so we need this
extension.
The table is illustrated by Fig. 5. The choice of coordinates is motivated by the notions
of pistil and stamens introduced and discussed in [13]. The choice of symbols r and s
for particular basic type-changing transformations, called fractal atoms, as well as the
corresponding lower indices seem natural. Except for r1, the choice of upper indices
refers to the length of the corresponding vectors. The notation 5, 7, etc. informs on
the end of the sequence(
m/2p−1, m = 0, 1, . . . , 2p−1 − 1
)
for 2p− 1 = 5, 7, etc.
A detailed proof of Theorem 2 will be given in Part II of the paper.
The Hurwitz-twistor counterpart of the Penrose theorem in the semiglobal version
states a one-to-one correspondence of the space of holomorphic solutions of the above
mentioned spinor equations of spin
1
2
n with the one-dimensional Dolbeault cohomology
group H1 depending on O(n − 2) = O([e]n−2), where [e] is the canonical effective
divisor of P3(C). On the other side the analogous pseudotwistor counterpart of the Pen-
rose theorem states a one-to-one correspondence of the respective space of holomorphic
solutions with the group H1 depending on O(−αn − β), where α and β, β ≥ 2, are
some positive integers. Therefore, again, the both structures have to be linked by a proper
type-changing transformation.
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614 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
Let us analyze the situation more closely. In [26] we can find such dualities, expressed
as type-changing transformations, relating manifolds M with the following pseudomet-
rics.
Case I. dx2
1 + . . .+ dx2
3 − dx2
4 − dx2
5 and dx2
1 − dx2
2 − . . .− dx2
5 for p = 5.
In analogy to Penrose twistors [27] we arrive at the structure of Hurwitz twistors
(H for short) [28], determined by a system of
(
5
4
)
= 5 algebraic equations, and at their
anti-objects (aH), corresponding to the 5-dimensional Kałuża – Klein theory [29, 30].
Case II. dx2
1 − dx2
2 − . . .− dx2
9 and dx2
1 + . . .+ dx2
7 − dx2
8 + dx2
9 for p = 9.
Here we arrive at the structure of pseudotwistors (p) [24, 25, 31], determined by a
system of
(
9
4
)
= 126 algebraic equations, and at their anti-objects (ap).
Case III. dx2
1+ . . .+dx2
7−dx2
8− . . .−dx2
13 and dx2
1+ . . .+dx2
5−dx2
6− . . .−dx2
13
for p = 13.
Here we arrive at the structure of bitwistors (b) [24, 25, 31, 32], determined by a
system of
(
13
4
)
= 715 algebraic equations, and at their anti-objects (ab).
Case IV. dx2
1 + . . .+dx2
5−dx2
6− . . .−dx2
9 and dx2
1 + . . .+dx2
3−dx2
4− . . .−dx2
9
for p = 9.
Here we arrive at the structure of pseudobitwistors (pb) [31, 32], determined by a
system of 126 algebraic equations, and at their anti-objects (apb).
The other cases appearing in (15) were not discussed in [21] because of the (8,8)-
periodicity of the Clifford structure.
The above demands can be fulfilled with the help of Atomization Theorem for fractals.
Namely, we have, as a corollary to that theorem, the following Atomization Theorem for
Twistor-like Structures:
Theorem 3. Suppose that the pseudometric corresponding to the p-dimensional
real vector space appearing in the definition of an Hermitian Hurwitz pair of bidimen-
sion (p, n) has the form given in cases I – IV. Then in each of these cases there exists a
type-changing transformation of the form listed in Table 2.
Table 2. Basic type-changing transformations for Hurwitz twistor-like structures
No. Transformation No. Transformation
1 H ⇐⇒ ap 7 p ⇐⇒ apb
2 aH ⇐⇒ p 8 ap ⇐⇒ pb
3 H ⇐⇒ aH 9 p ⇐⇒ ap
4 p ⇐⇒ ab 10 H ⇐⇒ ab
5 ap ⇐⇒ b 11 aH ⇐⇒ b
6 b ⇐⇒ ab
The theorem may be regarded as a further contribution to the so-called double Cartan-
like triality of Hermitian Hurwitz pairs [18, 33].
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BASIC PROPERTIES AND APPLICATIONS OF GRADED FRACTAL BUNDLES RELATED . . . 615
4. Geometrical model of the surface melting effect. A good example of physical
application of Theorem 2 is provided by the geometrical model of the surface melting
effect [8, 34 – 36], related to the formulae (13).
If a very thin layer at the crystal surface is the solid-vapour interface whose appearance
defines the surface melting effects, it can be described by its tangent bundle with the arc-
length element dx2 + dy2. Let us embed each tangent space of the bundle in a three-
dimensional space-time with the arc-length element c2dt2 − dx2 − dy2, with the usual
meaning of c and t. We consider the further embedding in a four-dimensional space time
M4 with the arc-length element
ds2 := c2dt2 − dx2 − dy2 − ηdτ2 (16)
where η = 1, or its modification M∗
4 with arc-length element (16), where η = −1. Here
each atom at the surface is treated as a small oscillator and the stochastical character
of its behaviour is introduced by the stochastical dimension τ related to entropy, while
η = 1 or −1 corresponds to two different kinds of stochastic nature. This means that
η = 1 stands for the Kałuża – Klein-type of differential equations governing the motion.
Correspondingly, η = −1 stands for the temporal character of the stochasticity which
leads to the Penrose-type of differential equations governing the motion. Thereafter M0 =
= M4 or M∗
4 will be called the base space of our construction.
Fix now a point 0 in M0, consider a family of the analogues of M0, corresponding
to different further layers of the crystal, and take into account the uniquely determined
curve M# starting from 0, passing through all the layers in question, and normal to each
of them. Denote by M0 = M5 or M∗
5 the bundle of all those layers, with the arc-length
element
ds2 := c2dt2 − dx2 − dy2 − dz2 − η(z)dτ2, η(z) = 1 or − 1,
and this corresponds to the formulae (13) in Theorem 2 and Case I in Theorem 3.
The notation M5 or M∗
5 is justified by the theorem formulated in [37], Sect. 5.6, which
assures that: (i) an arbitrary system of particles governed by equations of the Kałuża –
Klein-type, completed by an even number of particles governed by equations of the Pen-
rose type, is again a system of particles governed by equations of the Kałuża – Klein-type,
(ii) an arbitrary system of particles governed by equations of the Kałuża – Klein-type,
completed by an odd number of particles governed by equations of the Penrose type, is a
system of particles governed by equations of the Penrose type.
In order to find the surface potential fs(ρs), where ρs stands for the value of the
order parameter ρ at the surface, we take into account the general idea that the inhomo-
geneous behaviour of thermodynamic parameters can be connected with a deformation
of the space-time in which a considered system is embedded. This idea introduced in
analogy to the considerations reported by Ruppeiner [38] in order to describe the ther-
modynamical curvature in terms of the pseudoriemannian geometry was preliminarily
applied to the model of the surface melting description [36]. In the present paper we
outline a self-consistent approach determining the surface energy characteristics.
Our purpose requires that the surface energy is proportional to the inverse of the space-
time curvature in the neighbourhood of the surface, i.e., in the region of inhomogeneity,
in analogy to the Ruppeiner’s hypothesis originally introduced for the thermodynamic
space. Our condition can be then expressed as follows:
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 5
616 J. ŁAWRYNOWICZ, O. SUZUKI, F. L. CASTILLO ALVARADO
Fs(T = Tm) = kBTmR2,
where Fs denotes the surface free energy at the phase transition solid-liquid temperature
Tm and R is the curvature describing a deformation of a very thin layer whose thickness
is given by
z̆ = b/R,
where b is a positive parameter determined in [8]. This is the self-consistent condition
connecting the physical deformation close to the surface with its geometrical description
in terms of the Clifford-algebraic structure.
From the methodological point of view the deformation can be considered fibre bun-
dles, which are very appropriate not only to global topological problems but also for local
problems of differential geometry and field theory. The concept of induced represen-
tations of Lie groups and algebras may be most easily explained using the language of
bundles as reported by Trautman [39].
The algebraical content of this staff is strictly related to the idea of one of us and
Rembieliński [29, 30] to consider two vector spaces, the so-called Hurwitz pair (S, V )
restricted by the Hurwitz-type condition:
〈a, a〉S〈x, y〉V = 〈a ◦ x, a ◦ y〉V , a ∈ S, x, y ∈ V,
where 〈a, a〉S and 〈x, y〉V are the corresponding pseudoscalar products and ◦ is the mul-
tiplication in the Clifford algebra involved.
In the same manner the methodological aspect can be extended to the case of surface
physics. In particular, the surface energy characteristics can be described in terms of such
structures where the behaviour of atoms within the crystallographic lattice is characterized
by a standard equation Hψ = E ◦ ψ where ψ is a spinor defined in a domain in S with
values in V ; S, V and ◦ being properly chosen. Practically, this choice is governed by the
so-called Clifford constant.
From the physical point of view we consider surface of a sample treated as an inho-
mogeneity of the space. We choose the simplest idealized situation which is sufficient to
be a proper example showing the usefullness of the proposed methodology.
The investigations in the field of surface phenomena, in particular, surface melting,
show that there are at least two different configurations which are observed at the surface.
One of them represents the surface melting where a liquid-like layer is formed at the sur-
face. The other configuration is of solid-like layer character, but its structure is different
from the structure inside a sample.
Two different configurations at the surface are described by means of the density
whose profile corresponds to the distribution in the superficial region and contributes to
the surface energy. A comparison between two surface energies shows that the phase
transition characterized by the change of the density can be expected.
Explicit calculations are left to Part II of the paper.
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Received 05.02.08
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| id | umjimathkievua-article-3179 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:44Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/69/1530ffd0099f486ebef038f5c9e8c969.pdf |
| spelling | umjimathkievua-article-31792020-03-18T19:47:45Z Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction Основні властивості та застосування ступінчастих фрактальних жмутків, що пов'язані зi структурами Кліффорда. Вступ Castillo, Alvarado F. L. Lawrynowicz, J. Suzuki, O. Кастіло, Альварадо Ф. Л. Лавринович, Й. Сузукі, О. Using the central extension of the Cuntz C*-algebra, we study the periodicity for corresponding fractals. З допомогою центрального розширення C*-алгебри Кунца вивчається періодичність для відповідних фракталів. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3179 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 603–618 Український математичний журнал; Том 60 № 5 (2008); 603–618 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3179/3105 https://umj.imath.kiev.ua/index.php/umj/article/view/3179/3106 Copyright (c) 2008 Castillo Alvarado F. L.; Lawrynowicz J.; Suzuki O. |
| spellingShingle | Castillo, Alvarado F. L. Lawrynowicz, J. Suzuki, O. Кастіло, Альварадо Ф. Л. Лавринович, Й. Сузукі, О. Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title | Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title_alt | Основні властивості та застосування ступінчастих фрактальних жмутків, що пов'язані зi структурами Кліффорда. Вступ |
| title_full | Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title_fullStr | Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title_full_unstemmed | Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title_short | Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction |
| title_sort | basic properties and applications of graded fractal bundles related to clifford structures: an introduction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3179 |
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