Critical phenomena and critical dimensions in anisotropic nonlinear systems
The model that allows one to generalize the notions of the multicritical and Lifshitz points is considered. The model under consideration includes the higher powers and derivatives of order parameters. Critical phenomena in such systems were studied. We assess the lower and upper critical dimensions...
Збережено в:
| Дата: | 2012 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Institute of Electrophysics and Radiation Technologies NAS of Ukraine
2012
|
| Назва видання: | Вопросы атомной науки и техники |
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/107155 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Critical phenomena and critical dimensions in anisotropic nonlinear systems / A.V. Babich, S.V. Berezovsky, L.N. Kitcenko, V.F. Klepikov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 268-272. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-107155 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1071552025-02-09T21:25:35Z Critical phenomena and critical dimensions in anisotropic nonlinear systems Критические явления и критические размерности анизотропных нелинейных систем Критичні явища і критичні розмірності анізотропних нелінійних систем Babich, A.V. Berezovsky, S.V. Kitcenko, L.N. Klepikov, V.F. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases The model that allows one to generalize the notions of the multicritical and Lifshitz points is considered. The model under consideration includes the higher powers and derivatives of order parameters. Critical phenomena in such systems were studied. We assess the lower and upper critical dimensions of these systems. These calculation enable us to find the fluctuation region where the mean field theory description does not work. Предложена модель, позволяющая обобщить понятия точек Лифшица и мультикритических точек. Предложенная модель учитывает в термодинамическом потенциале как высшие градиенты параметров порядка, так и высшие нелинейности. Рассчитаны верхняя и нижняя критические размерности для такой модели. Полученные результаты позволяют определить флуктуационную область, в которой приближение среднего поля не работает. Запропоновано модель, яка дозволяє узагальнити поняття точок Ліфшица и мультикритичних точок. Модель, що запропоновано, враховує в термодинамічному потенціалі як вищи градієнти параметрів порядку, так і вищи нелінійності. Розраховано верхню і нижчу критичні розмірності для такої моделі. Здобуті результати дозволяють визначити флуктуаційну область, в якій наближення середнього поля не дійсно. 2012 Article Critical phenomena and critical dimensions in anisotropic nonlinear systems / A.V. Babich, S.V. Berezovsky, L.N. Kitcenko, V.F. Klepikov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 268-272. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 64.60.Kw, 64.60-i https://nasplib.isofts.kiev.ua/handle/123456789/107155 en Вопросы атомной науки и техники application/pdf Institute of Electrophysics and Radiation Technologies NAS of Ukraine |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| spellingShingle |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Babich, A.V. Berezovsky, S.V. Kitcenko, L.N. Klepikov, V.F. Critical phenomena and critical dimensions in anisotropic nonlinear systems Вопросы атомной науки и техники |
| description |
The model that allows one to generalize the notions of the multicritical and Lifshitz points is considered. The model under consideration includes the higher powers and derivatives of order parameters. Critical phenomena in such systems were studied. We assess the lower and upper critical dimensions of these systems. These calculation enable us to find the fluctuation region where the mean field theory description does not work. |
| format |
Article |
| author |
Babich, A.V. Berezovsky, S.V. Kitcenko, L.N. Klepikov, V.F. |
| author_facet |
Babich, A.V. Berezovsky, S.V. Kitcenko, L.N. Klepikov, V.F. |
| author_sort |
Babich, A.V. |
| title |
Critical phenomena and critical dimensions in anisotropic nonlinear systems |
| title_short |
Critical phenomena and critical dimensions in anisotropic nonlinear systems |
| title_full |
Critical phenomena and critical dimensions in anisotropic nonlinear systems |
| title_fullStr |
Critical phenomena and critical dimensions in anisotropic nonlinear systems |
| title_full_unstemmed |
Critical phenomena and critical dimensions in anisotropic nonlinear systems |
| title_sort |
critical phenomena and critical dimensions in anisotropic nonlinear systems |
| publisher |
Institute of Electrophysics and Radiation Technologies NAS of Ukraine |
| publishDate |
2012 |
| topic_facet |
Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/107155 |
| citation_txt |
Critical phenomena and critical dimensions in anisotropic nonlinear systems / A.V. Babich, S.V. Berezovsky, L.N. Kitcenko, V.F. Klepikov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 268-272. — Бібліогр.: 11 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT babichav criticalphenomenaandcriticaldimensionsinanisotropicnonlinearsystems AT berezovskysv criticalphenomenaandcriticaldimensionsinanisotropicnonlinearsystems AT kitcenkoln criticalphenomenaandcriticaldimensionsinanisotropicnonlinearsystems AT klepikovvf criticalphenomenaandcriticaldimensionsinanisotropicnonlinearsystems AT babichav kritičeskieâvleniâikritičeskierazmernostianizotropnyhnelineinyhsistem AT berezovskysv kritičeskieâvleniâikritičeskierazmernostianizotropnyhnelineinyhsistem AT kitcenkoln kritičeskieâvleniâikritičeskierazmernostianizotropnyhnelineinyhsistem AT klepikovvf kritičeskieâvleniâikritičeskierazmernostianizotropnyhnelineinyhsistem AT babichav kritičníâviŝaíkritičnírozmírnostíanízotropnihnelíníinihsistem AT berezovskysv kritičníâviŝaíkritičnírozmírnostíanízotropnihnelíníinihsistem AT kitcenkoln kritičníâviŝaíkritičnírozmírnostíanízotropnihnelíníinihsistem AT klepikovvf kritičníâviŝaíkritičnírozmírnostíanízotropnihnelíníinihsistem |
| first_indexed |
2025-11-30T23:03:28Z |
| last_indexed |
2025-11-30T23:03:28Z |
| _version_ |
1850258290809765888 |
| fulltext |
CRITICAL PHENOMENA AND CRITICAL DIMENSIONS IN
ANISOTROPIC NONLINEAR SYSTEMS
A.V. Babich∗, S.V. Berezovsky, L.N. Kitcenko, V.F. Klepikov
Institute of Electrophysics and Radiation Technologies NAS of Ukraine, 61108, Kharkov, Ukraine
(Received October 30, 2011)
The model that allows one to generalize the notions of the multicritical and Lifshitz points is considered. The model
under consideration includes the higher powers and derivatives of order parameters. Critical phenomena in such
systems were studied. We assess the lower and upper critical dimensions of these systems. These calculation enable
us to find the fluctuation region where the mean field theory description does not work.
PACS: 64.60.Kw, 64.60-i
1. INTRODUCTION
Second order phase transitions (PT) are ones of the
most intensively investigated phenomena in theoret-
ical physics. A lot of different kinds of PT have
been observed and investigated in various physical
systems [1–3]. Initially the basics objects of PT the-
ory application were various condensed matter sys-
tems. However such fundamentals of PT theory
as the spontaneous symmetry breaking, the group-
theoretical approach allow one to use predictions of
the theory of PT in the fields of physics which from
the first view have nothing in common with above
mentioned condensed matter systems.
One of the most important properties of the sys-
tems in vicinity of critical point is a strong increase
of fluctuations influence. The influence of the fluc-
tuations strongly depends on spatial dimension. In
modern theory of critical phenomena the space di-
mensionality “d” is usually considered as continuous
value [4,5]. It appears in thermodynamic relations as
one of the parameters of the system. As mentioned
above a critical behavior of the system strongly de-
pends on it.
One of the effects of this dependence is an ex-
istence of the 2 critical (or borderline) dimensions.
Lower critical dimension determines the range of the
existence ordering states: there are no PT at nonzero
temperature if the space dimensionality is less than
lower critical dimension, or in other words at lower
critical dimension Goldstone bosons start to interact
strongly [6]. Upper critical dimension determines a
range of the mean field based theories applicability.
So if we plot an axis (Fig. 1) of dimensionality then
the critical dimensions divide it into 3 regions (dl and
du are the upper and lower critical dimensions).
The critical dimensions are important not only
as “borders”. They are necessary elements for cal-
culations of critical indexes in the fluctuation region
by various methods based on the renormalization
group. In condensed matter theory a consideration
of systems with d > 3 is just a trick that allows one
to calculate the critical indexes by the renormgroup
methods. But in such fields as particle physics, gen-
eral relativity and cosmology it is necessary to use
models with higher space dimensionalities [7, 8].
dl du1 2 3
d
Fig. 1. 1. No ordering. 2. Fluctuation region. PT
are possible, but the mean field approximation is
invalid. 3. The fluctuations are damped. The mean
field approximation is valid
2. MULTICRITICAL AND LIFSHITZ
POINTS
In the simplest model of PT an expansion of thermo-
dynamic potential (TP) looks as follows:
Φ = Φ0 + aϕ2 + bϕ4. (1)
Here lower and upper critical dimensions equal 2 and
4 correspondingly. In more complicated models crit-
ical dimensions depend on the model parameters.
There are a few ways to generalize model (1).
The first one is to take into account terms with
higher powers of the order parameters in the ex-
pansion of the TP. It leads one to the notion
of the multicritical point. Multicritical points of
various types have been observed in many physi-
cal systems. The most known example of a sys-
tem with the tricritical point (TCP) is the mixture
of helium isotopes He3-He4. There are TCPs on
phase diagrams of some ferroelectrics for example
in KH2PO4 (Fig. 2), antiferromagnetics. Upper CD
of systems with TCP is equal to 3, so correspond-
ing models are renormalazed in usual physical space.
∗Corresponding author E-mail address: avbabich@yahoo.com
268 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 268-272.
Fig. 2. Schematic phase diagram for systam with
TCP. β > 0 – second order PT, β < 0 – first order
PT
Another way is to take into account an inhomo-
geneity of the order parameters. It leads one to the
notion of the Lifshitz point (LP). The theoretical in-
vestigations of PT in systems with anisotropic mod-
ulations of the order parameters near the Lifshitz
points began in the 1970th and have continued up
to date. Initially the main fields of application of
these theories were systems with anisotropic mag-
netic ordering. But recently systems with Lifshitz
points have increasingly arisen interest in such fields
of physics as particle physics, quantum gravity and
cosmology [7,8]. Lifshitz points arise in systems with
modulated structures of OP. There are 3 phases co-
exist in LP: initial ordering phase, homogeneous dis-
ordered phase and modulated phase (Fig. 3). Modu-
lated structures of different types have been observed
in a lot of magnetic and ferroelectric crystals (MnP,
NaNO2, K2SeO4 etc.).
Fig. 3. Schematic phase diagram for system with
Lifshitz point. 1 – ordered state, 2 – state with
modulated structures, disordered state
This paper is about CDs of systems with mutual
multicritical and Lifshitz behavior. The simplest ex-
ample of such point is tricritical Lifshitz point (Fig.4).
Point of that kind has been studied in ferroelectrics
of type Sn2P2(SexS1−x)6. Corresponding TP:
Φ = Φ0 +
α
2
ϕ2 +
β
4
ϕ4 +
γ
6
ϕ6 +
δ
2
ϕ′2 +
g
4
ϕ′′2. (2)
We introduce a model that allows one to study a
system with a new type of critical point. The gra-
dients of OP and the highest power of OP in such
model may take arbitrary values.
Fig. 4. Phase diagram for Sn2P2(SexS1−x)6 in the
vicinity of TCLP
In the vicinity of the critical point under consid-
eration the effective Hamiltonian can be written as
follows:
H =
∫
dmxid
d−mxc
{
r
2
η2 +
γ
2
(
Δ
1
2
i η
)2
+
+
δ
2
(
Δ
1
2
c η
)2
+
β
2
(
Δ
p
2
i η
)2
+ uη
N+1
}
, (3)
where η is a one-component order parameter, d is the
space dimension and r, γ, δ, β, u are the model para-
meters. We assume that the space can be divided into
two subspaces of dimensions m and d−m denoted by
i and c respectively. There are wave modulation vec-
tors in the first subspace and none in the second one.
Let us assume d and m to be continuous variables
and d > m. Δc and Δi are the Laplacian opera-
tors available in the corresponding subspaces. In this
case the operators Δl are defined as Δl = Δ
(
Δl−1
)
.
For non-integer values of l and m the corresponding
operators are determined using the inverse Fourier
transformation. In the critical point r = γ = 0, and
the other parameters in (3) are positive quantities.
Using the Fourier transformation :
η (x) =
∫
ddq · η (q) eiqx, (4)
we rewrite the Hamiltonian (3) in momentum space:
H =
1
2
∫
ddqν (q) η (q) η (−q)+
+ u
∫
ddq1 · ... · ddqN+1 (5)
× δ (q1 + ... + qN+1) (η (q1) ...η (qN+1)) .
Here ν (q) = r+γq2
i +βq2p
i +δq2
c , qi and qc are the ab-
solute values of the wave vector−→q being in the sectors
i and c respectively, and q2
i =
m∑
α=1
q2
α, q 2
c =
d∑
α=m+1
q2
α.
The main parameters which determine the criti-
cal behavior of the system with the Hamiltonian (3)
269
are the order of higher gradients p, the dimension of
the subspace of modulation, and the order of nonlin-
earity of the model.
3. THE CRITICAL DIMENSIONS OF THE
ANISOTROPIC SYSTEMS
The lower critical dimension dl is defined by the fol-
lowing condition: there are no ordering states in
space with d < dl under condition of nonzero tem-
perature. From the thermodynamic point of view it
means that fluctuation contribution to entropy is a
divergent function of temperature. The fluctuation
contribution to entropy looks as follows:
Sfl = sτσ(d), (6)
here τ = (T − Tc)/Tc is the reduced temperature,
σ(d) is a function of space dimensionality and does
not depend on T. We are interested in the critical
behavior of Sfl, so:
lim
τ→0
Sfl =
{
0 σ(d) < 0,
∞ σ(d) > 0.
(7)
One can see that under condition Tc �= 0 the fluctu-
ation contribution to entropy is a divergent function
if σ(d) > 0 otherwise it goes to zero. Thus one can
find the lower critical dimension from the following
condition:
σ(dl) = 0. (8)
In order to calculate σ(dl) let us find how the fluc-
tuation contribution to entropy depends on the tem-
perature. By definition:
Sfl(τ) =
∂Gfl
∂τ
, (9)
here Gfl is a fluctuation contribution to the thermo-
dynamic potential, in model (3) it looks as:
Gfl = A
∫
dmqi · dd−mqc ln
β q2p
i + δ q2
c + α τ
πT
, (10)
where A is a parameter that does not depend on tem-
perature. From (9) and (10):
Sfl = αA
∫
dmqi · dd−mqc
(
β q2p
i + δ q2
c + α τ
)−1
,
(11)
after some manipulations:
Sfl = τ−1αA
∫
dmqi · dd−mq2
c
{
β ·
(
qi · τ− 1
2p
)2p
+
+γ ·
(
qc · τ− 1
2
)2
+ α
}−2
. (12)
After the change of variables κi = qi · τ− 1
2p , κc =
qc · τ− 1
2 the final expression for Sfl takes the form:
Sfl = τσ(d) · I (κi, κc) , (13)
here σ(d) = 2 − (m/2p + (d − m)/2) and the func-
tion I (κi, κc) does not depend on t. And finally for
σ(d):
σ(d) =
m
2p
+
d − m
2
− 1, (14)
From (8) and (14):
dl = m
(
1 − 1
p
)
+ 2. (15)
There are several ways to calculate the upper crit-
ical dimension. First is similar to way we calculated
dl - comparing the fluctuation contribution to a heat
capcity:
CV = T
∂2Φ
∂T 2
= Φ = B
2 (N + 1)
(N − 1)2
· t−αl , (16)
where αl = 2 − N+1
N−1 .
With fluctuation correction to the heat capacity:
C1,fl = t−αfl · I (κi, κc) , (17)
where αfl = 2 −
(
m
2p + d−m
2
)
and the function
I (κi, κc) does not depend on t.
Second is to find du from the stability condition
of the fixed point of corresponding renomgroup trans-
formation:
ν′(q) = z2a−mb−(d−m)
×
(
r + δ
q2
c
b2
+ γ
q2
i
a2
+ β
q2p
i
a2p
+ ...
)
, (18)
u′(q) = zN+1a−Nb−N(d−m) · (u+...) . (19)
Here we take into account that the scale parameters
changing the scale for the momenta are independent
in the sectors i and c. We denote them a and b re-
spectively. The scale parameter for u is denoted by z.
Both of those ways lead us to the following expression
for du [9]:
du = m
(
1 − 1
p
)
+ 2
N + 1
N − 1
. (20)
Also one can find du from the condition of scale varia-
tion invariance of the model (3) under transformation
with generator:
X =
N − 1
2
∂
∂xc
+
N − 1
2p
∂
∂xi
− ∂
∂ϕ
. (21)
4. DISCUSSION
The spatial distribution of an order parameter is de-
fined by the solution of the variational equation for
the functional of the free energy. This equation is a
nonlinear differential equation (NDE). In our model
the order of corresponding equation is 2p. At present
there are no universal methods for solving the NDE,
therefore of fundamental importance are the meth-
ods enabling to simplify the NDE analysis. One of
the most effective ways for the NDE analysis is the
270
analysis of its symmetries. Investigation in the group
NDE structure allows one to determine the behavior
of solutions without defining their explicit form, to
find different NDE invariants, in particular, to con-
struct the conservation laws. Knowing the symmetry
groups of the assumed NDE enables also to construct
important classes of particular solutions. Every addi-
tional NDE-admitted Lie group allows reducing the
equation order by one. The variational symmetries
play here a very important role. These symmetries
belong to the variational equation, as well as, to the
functional, for which the given equation is variational.
The presence of the variational symmetry in the NDE
makes it possible to reduce its order by 2. The im-
portance of the variational symmetries is due to the
conservation laws related to them. In paper [9] was
shown that model (3) is invariant under scale varia-
tional transformations with generators:
X̂ =
N − 1
2
∂
∂xc
+
N − 1
2p
∂
∂xi
− ∂
∂ϕ
. (22)
Let us find the range of the fluctuation region:
Δd ≡ du −dl =
4
N − 1
, and lim
N→∞
(du −dl) = 0. (23)
So, the fluctuation region decreases as a function
of power of nonlinearity (see Fig. 5). This fact is
physically reasonable. Strong coupling supresses the
fluctuations. As it is expected, the lower critical di-
mension of any systems is not less then 2.
N
1
3
2
Fig. 5. Dependence of Δd on the order of nonlin-
earity of the model
Let us compare the obtained results with those
known previously. As mentioned above, in the case
of the simple critical point (N = 3, p = 1, m = 0)
the lower and upper CDs are equal to 2 and 4 respec-
tively. The CDs of the system with m-axial Lifshitz
point (N = 3, p = 2) depend on m linearly (some
special cases in 3 and 4 dimensional spaces will be
discussed later) . An interesting case is the model
with an isotropic Lifshitz-like point with derivatives
of order up to p (Lifshitz point of order p). The varia-
tional equations in this model coincide with Multidi-
mensional Polywave Equation that is invariant under
conformal transformations.
Let us consider what types of anisotropic system
in 3 and 4 dimensional spaces are possible accord-
ing to the condition d > dl. There are 2 types of
anisotropic systems in three-dimensional space:
1) usual critical point (without anisotropy) dl =
2, and
2) 1-axial Lifshitz point dl = 2, 5. In other cases
dl ≥ 3.
In the case of 4-dimensional space:
1) usual critical point (without anisotropy) dl = 2,
and
2) 1-, 2-, 3-axial Lifshitz point of order 2 dl =
2.5, 3, 3.5.
3) 1-, 2-axial Lifshits point of order p > 2, dl =
3 − 1/p, 4 − 1/p. In other cases dl ≥ 4.
In general cases: for some initial order p of the
Lifshitz point, the ordering in the space with dimen-
sionality d > 2p with any kind of anisotropy is pos-
sible. In case of d ≤ 2p, there are various situations
and this case demands additional investigation.
The obtained results are correct for classical PTs.
As we know, in the theory of quantum PTs the ef-
fective dimension of a system in the vicinity of the
quantum critical point is higher than a dimension of
space. So it is apparent that there are more possi-
ble types of PTs in a quantum case. In particular,
our results do not contradict a possibility of quantum
PTs in 2-dimensional systems.
References
1. J.C. Toledano and P. Toledano. The Landau
Theory of Phase Transitions. Application to
Structural, Incommensurate, Magnetic and Liq-
uid Crystal Systems. World Scientific Publishing
Company, 1987, 461 p.
2. A.D. Bruce and R.A. Cowley. Structural Phase
Transitions. 1981, London: Taylor & Francis
Ltd., 302 p.
3. A.I. Olemskoi, V.F. Klepikov. The theory of
spatiotemporal patterns in nonequilibrium sys-
tems // Physics Reports. 2000, v. 338, p. 571-677.
4. A.Z. Patashinskii and V.L. Pokrovskii. Flucta-
tion Theory of Phase Transition. 1979, New
York: “Pergamon”, p. 524.
5. Shang-keng Ma. Modern Theory of Critical Phe-
nomena. 1976, Perseus Books, U.S., p. 299.
6. A.M. Polyakov. Gauge Fields and Strings.
Switzerland: “Harwood Academic Publishers”,
1987, p. 327.
7. B.E. Meierovich. Vector order parameter in gen-
eral relativity: Covariant equations // Phys. Rev.
2010, v. D82, 024004.
8. K.A. Bronnikov, S.G. Rubin, and
I.V. Svadkovsky. Multidimensional world,
inflation, and modern acceleration // Phys. Rev.
2010, v. D81, 084010.
271
9. A.V. Babich, S.V. Berezovsky, V.F. Klepikov.
Spatial modulation of order parameters and crit-
ical dimensions // Int. J. Mod. Phys. 2008,
v. B22, p. 851-857.
10. A.V. Babich, L.N. Kitcenko, V.F. Klepikov.
Critical dimensions of systems with joint multi-
critical and lifshitz-point-like behavior // Mod.
Phys. Lett. 2011, v. B25, p. 1839-1845.
11. W. Fushchych, W. Shtelen, N. Serov Symme-
try Analysis and Exact Solutions of Equations
of nonlinear Mathematical Physics. 1993, Dor-
drecht: Kluwer Academic Publishers, 306 p.
����������� �
���� � ����������� ��
��������
���
�������� ��
������� ������
���� �����
��� ����
������ ���� ������� ���� ��������
���������
������
�������� � ��������
������ ����� ������ �
���������������� ������
���������� �
����� ���� � �� � ���
����
������
������ �� � � � ���� !� �����
�
��"
���
����� � � � � � ���� ������#������ $ ����� � ������� � ������ ����������� � �
�������
��� � ��#
������ �������� � ������� �
�������� �
�������� ������ ������� ��� ���� � ����"
��#
���������� ������!�
��� �� � ��� ���
�������� � ��� � �������� ��
��������
���
�������� ��
������� ������
���� �����
��� ����
������� ���� ������� ���� ��������
%
��
���� ��
������ �� �������& �� ! ������
������ ����� �'���� �
�������������� ������
(������ �� �
��
���� ��� �� ����& � ���
����
����
�
�����' �' �� ���� !� �'&���
�
���'�
������� � � ' ���� ���'�'#����'� $��� ��� �� ������ ' ����� �������' � �
'�����' ��� � ��)
����'�
%�����' ������� �� ���������� ���� ���� ������ �'#�� ��� ���� � ��'# � �������� ��������!�
���
�� �'#����
*+*
|