DOI: 10.2478/auom-2014-0044
An. Şt. Univ. Ovidius Constanţa
Vol. 22(2),2014, 239–251
Extremal problems on the generalized
(n, d)-equiangular system of points
Abstract
The paper of Lavrent’ev [1] was the beginning of geometrical theory
of functions of the complex variable. He solved a problem on the product
of conformal radiuses of two non-overlapping domains. In many papers
(see [2] – [13]) the Lavrent’ev’s result are generalized. In this paper are
obtained the new results of this direction.
A. Targonskii
Let N, R and C be the
S sets of natural, real and complex numbers respectively. We define C := C {∞} and R+ := (0, ∞).
Let n, m, d ∈ N such that m = nd. Consider the set of natural numbers
{mk }nk=1 such that
n
X
mk = m.
(1)
k=1
The following system of points
An,d := ak,p ∈ C : k = 1, n, p = 1, mk ,
are called the generalized (n, d)-equiangular system of points on the rays, if
the condition (1) is fulfilled and if for all k = 1, n, p = 1, mk the following
relations are true:
0 < |ak,1 | < . . . < |ak,mk | < ∞;
arg ak,1 = arg ak,2 = . . . = arg ak,mk =
2π
n (k
− 1).
(2)
Key Words: inner radius of domain, quadratic differential, piecewise-separating transformation, the Green function, radial systems of points, logarithmic capacity, variational
formula.
2010 Mathematics Subject Classification: Primary 30C70, 30C75.
Received: June, 2013.
Accepted: August, 2013.
239
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
240
An arbitrary generalized (n, d)-equiangular system of points (with variable
n
number of points on the rays An,d ) formed the set of domains {Pk }k=1 , where
2π
2π
Pk := w ∈ C\{0} :
(k − 1) < arg w <
k , k = 1, n.
n
n
For an arbitrary generalized (n, d)-equiangular system of points (with the
variable amount of points on the rays An,d ) we consider the following ”managing” functional
mk n Y
n2 Y
|ak,p | ,
µ (An,d ) :=
χ ak,p k=1 p=1
−1
where χ(t) = 12 t + t
.
Let {B0 , Bk,p } , {Bk,p , B∞ } be the arbitrary non-overlapping domains such
that
0 ∈ B0 , ak,p ∈ Bk,p , ∞ ∈ B∞ , B0 , Bk,p , B∞ ⊂ C,
k = 1, n, p = 1, mk .
Let D ⊂ C be an arbitrary open set and w = a ∈ D. Then D(a) is
a connected component of D which contain the point a. For an arbitrary
system of points An,d and for open set D, An,d ⊂ DTwe define Dk (al,p ) as
the connected component of the set for which D(al,p ) Pk contain the point
al,p for k = 1, n, l = k, k + 1, p = 1, ml , where mn+1 = m1 , an+1,p := a1,p .
We have thatTDk (0) (respectively Dk (∞))
T define the connected component of
the set D(0) Pk (respectively D(∞) Pk ) which contain the point w = 0
(respectively w = ∞).
An open set D with {0}∪An,d ⊂ D satisfies the non-overlapping conditions
with respect to the system of points {0} ∪ An,d if satisfies the condition:
h
i[h
i
\
\
Dk (as,p ) Dk (al,q )
Dk (0) Dk (al,q ) = ∅,
(3)
k = 1, n,
l, s = k, k + 1,
p = 1, ms ,
q = 1, ml
for all corners Pk .
An open set D with {∞} ∪ An,d ⊂ D satisfies the non-overlapping conditions with respect to the system of points {∞} ∪ An,d if satisfies the condition:
h
i[h
i
\
\
Dk (as,p ) Dk (al,q )
Dk (∞) Dk (al,q ) = ∅,
(4)
k = 1, n,
for all corners Pk .
l, s = k, k + 1,
p = 1, ms ,
q = 1, ml
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
241
The system of domains {B0 ∪ Bk,p } ({B∞ ∪ Bk,p }), k = 1, n, p = 1, mk , are
n m
S
Sk
defined as system of partially non-overlapping domains if D :=
Bk,p ∪
k=1 p=1
B0 (D :=
n m
S
Sk
Bk,p ∪ B∞ ) is an open set and if it satisfies the condition (3)
k=1 p=1
(condition (4)).
The definition of inner radius r(B; a) of domain B ⊂ C with respect to a
point a ∈ B can be found in the papers [4 – 6].
(1)
For an arbitrary n, d ∈ N, n ≥ 2 we denote by An,d the generalized (n, d)equiangular system of points which is formed by poles of the quadratic differential Q1 (w)dw2 , where
wn−2 (1 + wn )2d−1
2
Q1 (w)dw2 := − 2 dw .
n
n
2d+1
2d+1
(1 − iw 2 )
− (1 + iw 2 )
(5)
(2)
We denote by An,d the generalized (n, d)-equiangular system of points which
is formed by poles of the quadratic differential Q2 (w)dw2 , where
wn−2 (1 + wn )2d−1
2
Q2 (w)dw2 := 2 dw .
n
n
2d+1
2d+1
(1 − iw 2 )
+ (1 + iw 2 )
(6)
(1)
We remark that the condition (2) is satisfied for the system of points An,d ,
(2)
An,d when mk = d, k = 1, n. This statement easy follows from the general
theory of quadratic differentials [16].
In this paper we investigate the following problem.
Problem. Let n, m, d ∈ N, m = nd, n ≥ 2. We intend to find a maximum
of the functional In and to describe all its extremals, if
In := r
n2
4
(D, 0) ·
mk
n Y
Y
r(D, ak,p ),
k=1 p=1
where An,d = {ak,p } is an arbitrary generalized (n, d)-equiangular system of
points satisfying relation (2) and D is an arbitrary open set satisfying condition
(3).
We remark that this problem is more general with respect to the conditions
which are considered in [8 – 13].
Theorem 1. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(1)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies the condition (1); an arbitrary
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
242
open set D, An,d ⊂ D ⊂ C satisfies the non-overlapping conditions with respect
to the system of points An,d . Then we have the inequality
r
n2
4
mk
n Y
Y
(D, 0) ·
r(D, ak,p ) ≤
k=1 p=1
8
2m + n
The equality sign holds, if the open set D =
m ·
n m
S
Sk
2n
2m + n
n2
· µ.
Bk,s , where Bk,s is the
k=1 s=1
system of circular domains of the quadratic differential (5).
Corollary 1. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(1)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies the condition (1). Let also {Bk,p },
ak,p ∈ Bk,p ⊂ C be an arbitrary set of non-overlapping domains. Then we have
the inequality
r
n2
4
(B0 , 0) ·
mk
n Y
Y
r(Bk,p , ak,p ) ≤
k=1 p=1
8
2m + n
m ·
2n
2m + n
n2
· µ.
The equality sign holds, if the points ak,p and domains Bk,p are the poles and
the circular domains of the quadratic differential (5).
Corollary 2. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(1)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies the condition (1). Let also {Bk,p },
ak,p ∈ Bk,p ⊂ C be an arbitrary set of partially non-overlapping domains.
Then the inequality of Corollary 1 is true.
Theorem 2. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(2)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies condition (1); an arbitrary open
set D, An,d ⊂ D ⊂ C satisfies the non-overlapping conditions with respect to
the system of points An,d . Then we have the inequality
r
n2
4
(D, ∞) ·
mk
n Y
Y
k=1 p=1
r(D, ak,p ) ≤
8
2m + n
The equality sign holds, if the open set D =
m ·
n m
S
Sk
2n
2m + n
n2
· µ.
Bk,s , where Bk,s is the
k=1 s=1
system of circular domains of the quadratic differential (6).
Corollary 3. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(2)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies condition (1). Let also {Bk,p },
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
243
ak,p ∈ Bk,p ⊂ C be an arbitrary set of non-overlapping domains. Then we
have the inequality
m n2
mk
n Y
Y
n2
2n
8
·
r 4 (B∞ , ∞) ·
· µ.
r(Bk,p , ak,p ) ≤
2m + n
2m + n
p=1
k=1
The equality sign is holds, if the points ak,p and domains Bk,p are the poles
and the circular domains of the quadratic differential (6).
Corollary 4. Let n, m, d ∈ N, m = nd, n ≥ 2 and let An,d = {ak,p },
(2)
µ(An,d ) = µ(An,d ) be an arbitrary generalized (n, d)-equiangular system of
points; the set of numbers {mk }nk=1 satisfies the condition (1). Let also {Bk,p },
ak,p ∈ Bk,p ⊂ C be an arbitrary set of partially non-overlapping domains.
Then the inequality of Corollary 3 is true.
Proof of Theorem 1. We note that from the non-overlapping condition
follows that cap C\D > 0 and the set D with respect to a point a ∈ D
possesses the Green generalized function gD (z, a), which has the form



gD(a) (z, a), z ∈ D(a),
gD (z, a) := 0, z ∈ C\D(a),


 lim gD(a) (ζ, a), ζ ∈ D(a), z ∈ ∂D(a),
ζ→z
where gD(a) (z, a) is the Green function of the domain D(a) with respect to a
point a ∈ D(a).
Further, we will use the methods of the paper [8]. Consider the sets E0 =
C\D; U t = {w ∈ C : |w| 6 t} , E(ak,p , t) = {w ∈ C : |w − ak,p | 6 t}, k =
1, n, p = 1, mk , n > 2, n, mk ∈ N, t ∈ R+ . For a rather small t > 0, we
consider the condenser
C (t, D, An,d ) = E0 , U t , E1 ,
where E1 =
n m
S
Sk
E(ak,p , t). The capacity of the condenser C (t, D, An,d ) is
k=1 p=1
defined as
Z Z
capC (t, D, An,d ) = inf
0 2
(Gx ) + (G0y )2 dxdy
(see [5]), where an infimum takes in C over all Lipschitzian functions G =
G(z), such that G = 0, G = 1, G = n2 .
E0
E1
Ut
The module of the condenser C is defined as
−1
|C| = [capC]
.
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
244
From the Theorem 1 from [8] we get
|C (t, D, An,d ) | =
1
·
2π
n2
4
1
1
· log + M (D, An,d ) + o(1),
t
+m
t → 0, (7)
where
M (D, An,d ) =
+
1
·
2π
mk
n X
X
1
n2
4
2 ·
+m
h n2
4
log r(D, ak,p ) +
k=1 p=1
log r(D, 0) +
mk
n X
X
gD (0, ak,p )+
k=1 p=1
i
gD (ak,p , aq,s ) .
X
(8)
(k,p)6=(q,s)
The function
n
zk (w) = (−1)k i · w 2 ,
k = 1, n realizes univalent and conformal transformations of domain Pk on the
right half-plane Rez > 0.
Therefore function
1 − zk (w)
(9)
ζk (w) :=
1 + zk (w)
is a univalent and conformal mapping of the domain Pk on the unit circle
U = {z : |z| ≤ 1}, k = 1, n.
Obviously, we have ζk (0) = 1, k = 1, n.
(1)
(2)
(2)
(2)
Let ωk,p := ζk (ak,p ), ωk−1,p := ζk−1 (ak,p ), an+1,p := a1,p , ω0,p := ωn,p ,
ζ0 := ζn (k = 1,n, p = 1, mk ). For any domain ∆ ∈ C, we define (∆)∗ :=
w ∈ C : w1 ∈ ∆ .
From the formula (9) from [7], we obtain the following asymptotic expressions
−1
2
n2 · χ ak,p |ak,p |
· |w − ak,p |,
ζk (w) − ζk (ak,p ) ∼
n
w → ak,p , w ∈ P k .
−1
n2 2
|ak,p |
· χ ak,p · |w − ak,p |,
ζk−1 (w) − ζk−1 (ak,p ) ∼
n
w → ak,p ,
w ∈ P k−1 ,
k = 1, n, p = 1, mk .
(10)
The coefficients of piece-dividing transformation at the point w = 0 are
defined by the following asymptotic equalities
n
0
(11)
ζk (w) − 1 ∼ 2|w| 2 , w → 0, w ∈ P k , k = 1, n.
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
245
(1)
Let
Ωk,p
be
a
connected
component
T S
T ∗
(1)
(2)
ζk D P k
ζk D P k
containing the point ωk,p and let Ωk−1,p be a
S
∗
T
T
containing the
ζk−1 D P k−1
connected component ζk−1 D P k−1
(2)
(2)
(2)
point ωk−1,p , k = 1, n, p = 1, mk , P 0 := P n , Ω0,p := Ωn,p . It is clear that
(s)
in generally Ωk,p are multiconnected domains, k = 1, n, p = 1, mk , s = 1, 2.
(2)
(1)
Pair of the domains Ωk−1,p and Ωk,p is the result of piece-dividing transformation of the open set D with respect to the family {Pk−1 , Pk }, {ζk−1 , ζk }
(0)
at the point ak,p , k = 1, n, p = 1, mk . Let Ωk be a connected component
T S
T ∗
ζk D P k
containing the point 1, k = 1, n. The family of
ζ D Pk
n k on
(0)
the domains Ωk
is the result of piece-dividing transformation of the
k=1
n
n
open set D with respect to the family {Pk }k=1 and the functions {ζk }k=1 at
the point w = 0, k = 1, n.
In the following, we consider the condensers
(k)
(k)
(k)
Ck (t, D, An,d ) = E0 , U t , E1
,
where
\ [ h \ i∗
Es(k) = ζk Es P k
ζk Es P k
,
\ [ n \ o∗
(k)
U t = zk U t Pk
,
zk U t Pk
n
k = 1, n, s = 0, 1, {Pk }k=1 is a system of corners corresponding to a system
of points An,d ; the set [A]∗ is a set which is symmetrical to the set A with
respect a unit circle |w| = 1. From this, it follows that for dividing transforn
n
mation with respect to {Pk }k=1 and {ζk }k=1 for the condenser C (t, D, An,d )
n
corresponds the set of condensers {Ck (t, D, An,d )}k=1 . The last condensers
are symmetrical with respect to {z : |z| = 1}. According to the paper [8], we
obtain
n
1X
capC (t, D, An,d ) >
capCk (t, D, An,d ) .
(12)
2
k=1
Therefore we obtain
|C (t, D, An,d ) | 6 2
n
X
!−1
|Ck (t, D, An,d ) |−1
.
(13)
k=1
The formula (7) gives a module of asymptotic C (t, D, An,d ) when t → 0
and M (D, An,d ) is a module of the set D with respect to An,d . Using the
formulae (10), (11), and the fact that the set D satisfies a non-overlapping
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
246
conditions with respect to the system of points 0 ∪ An,d , we have the following
asymptotic representations for the condensers Ck (t, D, An,d ), k = 1, n:
|Ck (t, D, An,d ) | =
=
2π
n
2
1
1
log + Mk (D, An,d ) + o(1),
t
+ mk + mk+1
t → 0,
mn+1 := m1 ,
(14)
where

Mk (D, An,d ) =
1
2π
n
2
+ mk + mk+1
2 · log
(0)
r Ωk , 1
2
+

(1)
(1)
(2)
(2)
mk+1
r Ωk,p , ωk,p
r Ωk,t , ωk,t
X

+
log h
log h
i−1 +
i−1  ,
n
n
2
2
2
2
p=1
t=1
|ak,p |
|ak+1,t |
n · χ |ak,p |
n · χ |ak+1,t |
mk
X
and k = 1, n.
Using (14), we get
|Ck (t, D, An,d )|
2π
=
n
2
+ mk + mk+1
×
log 1t
!−1
+ mk + mk+1
1
× 1+
Mk (D, An,d ) + o
=
log 1t
log 1t
2π n2 + mk + mk+1
=
−
log 1t
!2
2 !
2π n2 + mk + mk+1
1
Mk (D, An,d ) + o
, t → 0.
log 1t
log 1t
2π
−
−1
n
2
Using the equality
n
P
(15)
mk = m and the condition (15), we have
k=1
n
X
k=1
−
2π
log 1t
−1
|Ck (t, D, An,d )|
2π
=
n2
2
+ 2m
log
1
t
−
2 X
2 !
n 2
n
1
·
+ mk + mk+1 Mk (D, An,d ) + o
, t → 0.
2
log 1t
k=1
(16)
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
247
In turn, from the relation (16), we obtain the following asymptotic representation
!−1
n
X
log 1t
1
2π
−1
1 − n2
·
=
|Ck (t, D, An,d )|
1 ×
n2
log
2π 2 + 2m
t
2 + 2m
k=1
×
n X
n
k=1
+
2
+ mk + mk+1
2
Mk (D, An,d ) + o
1
log 1t
!−1
=
log
2π
n 2
X
n
·
+ mk + mk+1 Mk (D, An,d ) + o(1),
2
2
+ 2m
k=1
1
n2
2
n2
2
1
t
+
+ 2m
t → 0. (17)
From the inequalities (12) and (13), using (7) and (17), we obtain
1
n2
4
2π
6
+m
log
1
+ M (D, An,d ) + o(1) 6
t
n 2
X
n
·
+ mk + mk+1 Mk (D, An,d )+o(1).
2
n2
2
n
2
2π 4
k=1
2 + 2m
(18)
From (18) when t → 0, we get
1
2
1
log +
t
+m
n 2
X
n
·
+
m
+
m
Mk (D, An,d ) .
k
k+1
2
2
+ 2m
k=1
2
M (D, An,d ) 6
n2
2
(19)
The formulae (8), (14) and (19) imply the following expression
1
·
2π
+
mk
n X
X
1
n2
4
+m
2 ·
h n2
4
(k,p)6=(q,s)

(0)
r Ωk , 1
mk
n X
X
gD (0, ak,p )+
k=1 p=1
X
log r(D, ak,p ) +
k=1 p=1
log r(D, 0) +
i
1
gD (ak,p , aq,s ) ≤
·
4π
1
n2
2
2 ×
+m
(1)
(1)
r
Ω
,
ω
k,p
k,p

×
+
log h
log
i−1 +
n
2
2
2
p=1
k=1
·
χ
|a
|
|a
|
k,p
k,p
n

(2)
(2)
mk+1
r Ωk,t , ωk,t
X

log h
k = 1, n.
+
i−1  ,
n
2
2
t=1
·
χ
|a
|
|a
|
k+1,t
k+1,t
n
n
X
mk
X
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
SYSTEM OF POINT
248
Therefore, we have
r
n2
4
mk
n Y
Y
(D, 0) ·
m
2
·
· µ (An,d ) ×
n
−n
2
r(D, ak,p ) ≤ 2
k=1 p=1
×
n
Y
(
mk k+1
Y
mY
(0)
(1)
(1)
(2)
(2)
r Ωk , 1 ·
r Ωk,p , ωk,p ·
r Ωk,t , ωk,t
p=1
k=1
) 12
.
(20)
t=1
From results of the paper [6, 8, 9], we have the following inequalities
r
mk k+1
Y
mY
(1)
(1)
(2)
(2)
·
r Ωk,p , ωk,p ·
r Ωk,t , ωk,t ≤
(0)
Ωk , 1
p=1
t=1
mk +mk+1 +1
Y
≤
2π
(s−1)
i m +m
k
k+1 +1
,
r G(k)
,
e
s
(21)
s=1
(k)
where Gs
is a system of circular domains of the quadratic differential
mk +mk+1 −1
Q (ζk ) dζk2 = − ζk
m +m
+1
ζk k k+1
−1
2
2 · dζk .
Using the inequalities (20), (21), we obtain
r
n2
4
(D, 0) ·
mk
n Y
Y
−n
2
r(D, ak,p ) ≤ 2
k=1 p=1
×
n
Y
m
2
·
· µ (An,d ) ×
n
(mk +mk+1 +1
)1
2
Y
2π
i m +m
(s−1)
k
k+1 +1
r G(k)
.
s ,e
(22)
s=1
k=1
Now consider the family of functions
p
2π
ξk = n ζk · ei n (k−1) ,
k = 1, n,
(k)
which transform the unit circle to a sector with size 2π
n . Then the domains Gs ,
(k)
k = 1, n, s = 1, mk + mk+1 + 1 will be transformed to the domain
Σs and the
i
2π
(s−1)
i 2π
s−1
+k−1
points e mk +mk+1 +1
will be transformed into e n mk +mk+1 +1
.
By union all sectors we obtain the unit circle containing (2m + n) non(k)
overlapping domains Σs , k = 1, n, s = 1, mk + mk+1 + 1. Then
s−1
2π
2π
(k) i mk +mk+1 +1 (s−1)
(k) i n mk +mk+1 +1 +k−1
r Gs , e
≤ n · r Σs , e
.
(23)
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
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249
Using the inequalities (22), (23), we have
r
n2
4
(D, 0) ·
mk
n Y
Y
r(D, ak,p ) ≤ 2m ·
n n2
2
k=1 p=1
· µ (An,d ) ×
1
(
×
) 2
n mk +m
k+1 +1
Y
Y
s−1
2π
i
+k−1
n
mk +mk+1 +1
.
r Σ(k)
s ,e
(24)
s=1
k=1
Using the results of the paper [6, 8, 9], we obtain the following inequality
n mk +m
2m+n
k+1 +1
Y
Y
Y
s−1
i 2π
n
mk +mk+1 +1 +k−1
≤
r Σ(k)
,
e
r (Bt , bt )(25)
s
k=1
=
s=1
4
2m + n
t=1
2m+n
.
The sign of equality is obtained when the domains Bt and the points bt are
the circular domains and the poles of the quadratic differential
Q (ξ) dξ 2 = −
ξ 2m+n−2
(ξ 2m+n − 1)
2
· dξ 2 .
(26)
Finally, from the inequalities (25), (24), we obtain
r
n2
4
(D, 0) ·
mk
n Y
Y
k=1 p=1
r(D, ak,p ) ≤
8
2m + n
m ·
2n
2m + n
n2
· µ (An,d ) . (27)
The statement of the theorem follows directly from the inequality (27)
and from the quadratic differential (26), in which we must make a necessary
exchange of variables. The theorem is proved.
Proof of the Theorem 2 is similar to the proof of the Theorem 1.
Acknowledgement. The author is grateful to Prof. A. K. Bakhtin for formulation of the problem and useful discussions.
EXTREMAL PROBLEMS ON THE GENERALIZED (n, d)-EQUIANGULAR
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250
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Andrey L. TARGONSKII,
Department of Mathematical Analysis,
Zhytomyr State University,
Velyka Berdychivs’ka, 40, 10008 Zhytomyr, Ukraine.
Email: targonsk@mail.ru, targonsk@zu.edu.ua.
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SYSTEM OF POINT
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