On the Dirichlet problem for generalized Cauchy-Riemann equations
Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the Dirichlet problem to generalized Cauchy-Riemann equations ▽v = B▽u in the real plane R² that describe flows of fluids in anisotropic and inhomogeneous media, where B is a 2 × 2 matrix valu...
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Gutlyanskii, V.Yu. Ryazanov, V.I. Salimov, A.R. Salimov, R.R. 2025-09-12T14:19:55Z 2025 On the Dirichlet problem for generalized Cauchy-Riemann equations / V.Yu. Gutlyanskii, V.I. Ryazanov, A.R. Salimov, R.R. Salimov // Доповіді Національної академії наук України. — 2025. — № 2. — С. 3-10. — Бібліогр.: 15 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/206498 517.5 https://doi.org/10.15407/dopovidi2025.02.003 Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the Dirichlet problem to generalized Cauchy-Riemann equations ▽v = B▽u in the real plane R² that describe flows of fluids in anisotropic and inhomogeneous media, where B is a 2 × 2 matrix valued coefficient and the gradients ▽u and ▽v are interpreted as vector columns. Moreover, we clarify the relationships of the latter to the A-harmonic equation div (A▽u) = 0 with matrix valued coefficients A that is one of the main equations of the potential theory, namely, of the hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media in the plane. The survey includes a series of effective integral criteria for existence of regular solutions of the Dirichlet problem with continuous data in arbitrary bounded simple connected domains to generalized Cauchy-Riemann equations with matrix coefficients in the case of anisotropic and inhomogeneous media. Стаття містить огляд наслідків теорії рівнянь Бельтрамі з комплексного аналізу для задачі Діріхле до узагальненого рівняння Коші—Рімана ▽v = B▽u на дійсній площині R², що описує потоки рідини в анізотропних та неоднорідних середовищах, де коефіцієнт B представлено у вигляді 2 × 2 матриці, а градієнти ▽u та ▽v інтерпретуються як вектор-стовпці. Крім того, з’ясовується зв’язок цього рівняння з A-гармонічним рівнянням div(A▽u) = 0 з матричними коефіцієнтами A, яке є одним із головних рівнянь теорії потенціалу, а саме гідромеханіки (механіки рідин) в анізотропних та неоднорідних середовищах на площині. Огляд включає низку ефективних інтегральних критеріїв існування регулярних розв’язків задачі Діріхле з неперервними даними в довільних обмежених однозв’язних областях для узагальнених рівнянь Коші—Рімана з матричними коефіцієнтами в умовах анізотропних та неоднорідних середовищ. The first 2 authors are partially supported by the project “Mathematical modeling of complex dynamical systems and processes caused by the state security”, No. 0123U100853, of the National Academy of Sciences of Ukraine and by a grant from the Simons Foundation PD-Ukraine-00010584. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика On the Dirichlet problem for generalized Cauchy-Riemann equations Про задачу Діріхле для узагальнених рівнянь Коші—Рімана Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the Dirichlet problem for generalized Cauchy-Riemann equations |
| spellingShingle |
On the Dirichlet problem for generalized Cauchy-Riemann equations Gutlyanskii, V.Yu. Ryazanov, V.I. Salimov, A.R. Salimov, R.R. Математика |
| title_short |
On the Dirichlet problem for generalized Cauchy-Riemann equations |
| title_full |
On the Dirichlet problem for generalized Cauchy-Riemann equations |
| title_fullStr |
On the Dirichlet problem for generalized Cauchy-Riemann equations |
| title_full_unstemmed |
On the Dirichlet problem for generalized Cauchy-Riemann equations |
| title_sort |
on the dirichlet problem for generalized cauchy-riemann equations |
| author |
Gutlyanskii, V.Yu. Ryazanov, V.I. Salimov, A.R. Salimov, R.R. |
| author_facet |
Gutlyanskii, V.Yu. Ryazanov, V.I. Salimov, A.R. Salimov, R.R. |
| topic |
Математика |
| topic_facet |
Математика |
| publishDate |
2025 |
| language |
English |
| container_title |
Доповіді НАН України |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| format |
Article |
| title_alt |
Про задачу Діріхле для узагальнених рівнянь Коші—Рімана |
| description |
Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the Dirichlet problem to generalized Cauchy-Riemann equations ▽v = B▽u in the real plane R² that describe flows of fluids in anisotropic and inhomogeneous media, where B is a 2 × 2 matrix valued coefficient and the gradients ▽u and ▽v are interpreted as vector columns. Moreover, we clarify the relationships of the latter to the A-harmonic equation div (A▽u) = 0 with matrix valued coefficients A that is one of the main equations of the potential theory, namely, of the hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media in the plane. The survey includes a series of effective integral criteria for existence of regular solutions of the Dirichlet problem with continuous data in arbitrary bounded simple connected domains to generalized Cauchy-Riemann equations with matrix coefficients in the case of anisotropic and inhomogeneous media.
Стаття містить огляд наслідків теорії рівнянь Бельтрамі з комплексного аналізу для задачі Діріхле до узагальненого рівняння Коші—Рімана ▽v = B▽u на дійсній площині R², що описує потоки рідини в анізотропних та неоднорідних середовищах, де коефіцієнт B представлено у вигляді 2 × 2 матриці, а градієнти ▽u та ▽v інтерпретуються як вектор-стовпці. Крім того, з’ясовується зв’язок цього рівняння з A-гармонічним рівнянням div(A▽u) = 0 з матричними коефіцієнтами A, яке є одним із головних рівнянь теорії потенціалу, а саме гідромеханіки (механіки рідин) в анізотропних та неоднорідних середовищах на площині. Огляд включає низку ефективних інтегральних критеріїв існування регулярних розв’язків задачі Діріхле з неперервними даними в довільних обмежених однозв’язних областях для узагальнених рівнянь Коші—Рімана з матричними коефіцієнтами в умовах анізотропних та неоднорідних середовищ.
|
| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/206498 |
| citation_txt |
On the Dirichlet problem for generalized Cauchy-Riemann equations / V.Yu. Gutlyanskii, V.I. Ryazanov, A.R. Salimov, R.R. Salimov // Доповіді Національної академії наук України. — 2025. — № 2. — С. 3-10. — Бібліогр.: 15 назв. — англ. |
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2025-11-25T21:31:31Z |
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3
ОПОВІДІ
НАЦІОНАЛЬНОЇ
АКАДЕМІЇ НАУК
УКРАЇНИ
МАТЕМАТИКА
MATEMATICS
ISSN 1025-6415. Dopov. nac. akad. nauk Ukr. 2025. No 2: 3—10
C i t a t i o n: Gutlyanskiĭ V.Ya., Ryazanov V.I., Salimov A.R., Salimov R.R. On the Dirichlet problem for generalized Cau-
chy—Riemann equations. Dopov. nac. akad. nauk Ukr. 2025. No 2. P. 3—10. https://doi.org/10.15407/dopovidi2025.02.003
© Publisher PH «Akademperiodyka» of the NAS of Ukraine, 2025. Th is is an open access article under the CC BY-NC-
ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)
https://doi.org/10.15407/dopovidi2025.02.003
UDC 517.5
V.Ya. Gutlyanskiĭ1, 2, https://orcid.org/0000-0002-8691-4617
V.I. Ryazanov 1, 2, https://orcid.org/0000-0002-4503-4939
A.R. Salimov1, https://orcid.org/0009-0000-1021-7527
R.R. Salimov2, https://orcid.org/0000-0001-9395-3334
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine
2 Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine
E-mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com, salimov.artem@gmail.com,
ruslan.salimov1@gmail.com
On the Dirichlet problem
for generalized Cauchy-Riemann equations
Presented by Academician of the NAS of Ukraine I.I. Skrypnik
Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the
Dirichlet problem to generalized Cauchy-Riemann equations v B u in the real plane R2 that describe fl ows of
fl uids in anisotropic and inhomogeneous media, where B is a 2 × 2 matrix valued coeffi cient and the gradients u and
v are interpreted as vector columns. Moreover, we clarify the relationships of the latter to the A-harmonic equation
( ) 0div A u with matrix valued coeffi cients A that is one of the main equations of the potential theory, namely, of
the hydromechanics (fl uid mechanics) in anisotropic and inhomogeneous media in the plane. Th e survey includes a
series of eff ective integral criteria for existence of regular solutions of the Dirichlet problem with continuous data in
arbitrary bounded simple connected domains to generalized Cauchy-Riemann equations with matrix coeffi cients in
the case of anisotropic and inhomogeneous media.
Keywords: Cauchy-Riemann system, generalized Cauchy-Riemann equations, Dirichlet problem, Beltrami and A-har-
monic equations.
1. Introduction. As it is well-known, the characteristic property of an analytic function f u iv
in the complex plane C is that its real and imaginary parts satisfy the Cauchy-Riemann system
, , .u v u v z x iy
x y y x
(1)
Euler was the fi rst who has found the connection of the system (1) with the analytic functions.
4 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 2
V.Ya. Gutlyanskiĭ, V.I. Ryazanov, A.R. Salimov, R.R. Salimov
A physical interpretation of (1), going back to Riemann works on function theory, is that u
represents a potential function of the incompressible fl uid steady fl ow in homogeneous isotropic
media and v is its stream function.
Th is system can be written as the one equation in the matrix form
,v H u (2)
where v and u denotes the gradient of v and u , correspondingly, interpreted as vector-col-
umns in 2R , and 2 2:H R R is the so-called Hodge operator represented as the 2 2 matrix
0 –1
,
1 0
H
(3)
which carries out the counterclockwise rotation of vector columns by the angle 2 in 2R .
Th us, (2) shows that streamlines and equipotential lines of the fl uid fl ow are mutually orthog-
onal. Note also that H is an analog of the imaginary unit in the space 2 2M of all 2 2 matrices
with real entries because
2 –H I , (4)
where I is the unit 2 2 matrix.
Here we consider the generalized Cauchy-Riemann equations of the form
v B u (5)
with the matrix valued coeffi cients 2 2:B D M that describe fl ows in anisotropic and inho-
mogeneous media and, on the basis of the well-developed theory of the Beltrami equations, see
e.g. monographs [1]—[6] and articles [7]—[9], we give the corresponding consequences for the
Dirichlet problem with continuous data to these equations.
Moreover, let us clarify the relationships of the equations (5) and the A -harmonic equation
2( ( ) ( )) 0, : ( , ) ,div A Z grad u Z Z x y R (6)
with matrix valued coeffi cients 2 2:A D M that is one of the main equations of hydromechan-
ics in anisotropic and inhomogeneous media.
For this purpose, recall that the Hodge operator H transforms curl-free fi elds into diver-
gence-free fi elds and vice versa. Th us, if 1, 1
locu W is a solution of (6) in the sense of distributions,
then the fi eld V HA u is curl-free and, consequently, V v for some 1, 1
locv W and the pair
( , )u v is a solution of the equation (6) in the sense of distributions with
:B H A . (7)
Vice versa, if u and 1, 1
locv W satisfi es (5) in the sense of distributions, then u satisfi es (6) also
in the sense of distributions with
–1:A H B H B (8)
because the curl of any gradient fi eld is zero in the sense of distributions.
5ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 2
On the Dirichlet problem for generalized Cauchy—Riemann equations
Let us denote by 2 2B space of all 2 2 matrices with real entries,
11 12
21 22
,
b b
B
b b
(9)
with det 1B , antisymmetric with respect to its auxiliary diagonal, i.e., with 22 11–b b , and
with the ellipticity condition 1 , where
12 21 11
B
12 21
2
: .
2
b b ib
b b
(10)
Note that, under the above conditions det 1B and 22 11–b b , the ellipticity condition 1
is equivalent to the condition 21 12b b and, furthermore, to the conditions 12 0b and 21 0b .
We will call the quantity (10) a complex characteristic of (5). Criteria of solvability of the
equation (5) will be formulated in terms of its dilatation quotient
B
B
1
: .
1B
K
(11)
Let us consider the Dirichlet problem for the generalized Cauchy-Riemann equations (5)
consisting in fi nding its solutions ( , )u v with prescribed continuous data : D R of potential
u at the boundary
lim ( ) ( )
z
u z
D (12)
in arbitrary bounded simply connected domains D in 2R .
Given a simply connected domain D in 2R , we say that a pair ( , )u v of continuous functions
:u D R and :v D R in the class 1, 1
locW is a regular solution of the Dirichlet problem (12) for
the generalized Cauchy-Riemann equation (5) in D if ( , )u v satisfi es (5) a.e. in D and, moreover,
0u and 0v a.e. in D , and the correspondence ( , ) ( , )x y u v is a discrete and open
mapping of D into 2R . Recall that a mapping of a domain D in 2R into 2R is called discrete if
the preimage of each point in 2R consists of isolated points in D and open if the mapping maps
every open set in D onto an open set in 2R .
2. Criteria in terms of BMO, FMO and VMO. Hereaft er ( )dL Z corresponds to the Lebesgue
measure in 2R with the notation 2: ( , )Z x y R .
Recall fi rst of all that a real-valued function in a domain D of 2R is called of bounded
mean oscillation in D , abbr. ( )BMO D , if
B*
B
1sup ( ) ( ) ,
BB
Z dL Z (13)
where 1 ( )locL D , the supremum is taken over all discs B in D and
B
B
1: ( ) ( ).
B
Z dL Z
6 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 2
V.Ya. Gutlyanskiĭ, V.I. Ryazanov, A.R. Salimov, R.R. Salimov
We also write (D)BMO if * *( )BMO D for some extension * of the function
into a domain *D containing D .
Th e class BMO was introduced by John and Nirenberg (1961) in the paper [10] and soon
became an important concept in harmonic analysis, partial diff erential equations and related ar-
eas, see e.g. monographs [11] and [12].
Further we always assume by defi nition that 1
B
K outside D .
Th eorem 1. Let D be a bounded simply connected domain in 2R and let 2 2:B D B be a meas-
urable function. Suppose also that
B
K has a dominant 2Q:R [1, ) in the class (D)BMO .
Th en the generalized Cauchy-Riemann equation (5) has a regular solution ( , )u v of the Dirichlet
problem (12) in D for each continuous inconstant boundary date : D R .
A function in BMO is said to have vanishing mean oscillation, abbr. ( )VMO D , if the
supremum in (13) taken over all balls B in D with B converges to 0 as 0 . VMO has
been introduced by Sarason in [13]. Th ere are a number of papers devoted to the study of PDEs
with coeffi cients of the class VMO . Note that 1,2( ) ( )W D VMO D , see e.g. [14].
Corollary 1. In particular, the conclusion of Th eorem 1 on existence of a regular solution for the
Dirichlet problem (12) to the generalized Cauchy-Riemann equation (5) holds if the dominant Q of
B
K belongs to the class 1, 2( )W D .
Following [15], we say that a locally integrable function :D R has fi nite mean oscilla-
tion at a point 0Z D , abbr. 0( )FMO Z , if
0
00 0 B(Z , )
1lim ( ) ( ) ( ) ,
B(Z , )
Z Z dL Z
(14)
where
0
0
0 B(Z , )
1( ) ( ) ( ) ,
B(Z , )
Z Z dL Z
(15)
is the mean integral value of the function ( )Z over disk 2
0 0B(Z , ): { : }.Z R Z Z
Th eorem 2. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a meas-
urable function in D . Suppose also that
0Z( ) ( )
B
K Z Q Z a.e. in
0ZU for every point 0 DZ , a
neighbourhood
0ZU of 0Z and a function
0 0Z Z( ) : [0, ]Q Z U in the class 0( )FMO Z . Th en the
generalized Cauchy-Riemann equation (5) has a regular solution ( , )u v of the Dirichlet problem (12)
in D for each continuous inconstant boundary date : D R .
Corollary 2. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function in D . Suppose also that
0
00 0 B(Z , )
1lim ( ) ( ) D
B(Z , ) È
K Z dL Z Z
(16)
Th en the conclusion of Th eorem 2 holds.
Corollary 3. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function. Suppose also that
0Z( ) ( )
B
K Z Q Z a.e. in D with a function Q of the class
(D)FMO . Th en the conclusion holds.
B
K
7ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 2
On the Dirichlet problem for generalized Cauchy—Riemann equations
3. Criteria of the Calderon-Zygmund type.
Th eorem 3. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function. Suppose also that
0 0
2
02
0| |
( )
( ) ([log(1 )] )
| |È
Z Z
dL Z
K Z o Z D
Z Z
(17)
as 0 for some 0 0( ) 0Z . Th en the generalized Cauchy-Riemann equation (5) has a regular
solution ( , )u v of the Dirichlet problem (12) in D for each continuous inconstant boundary date
: D R .
Remark 1. We are also able here to replace (17) by
0 0
2
02
| |
0
0
( ) ( )
([log log(1 )] )
1| | log
_
È
Z Z
K Z dL Z
o Z D
Z Z
Z Z
(18)
In general, we are able to give here the whole scale of the corresponding conditions in terms of
iterated logarithms, i.e., in terms of functions of the form 1 log1 log log1 log log1t t t t .
4. Criteria of the Lehto type. Further 0( , )
B
k Z r denotes the integral mean of ( )
B
K Z over
the circle 2
0 0( , ) : { :| | }S Z r Z R Z Z r .
Th eorem 4. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function with 1( )
B
K L D . Suppose also that, for some 0 0( ) 0Z ,
0
00 ( , )
B
dr
rk Z r
0Z D . (19)
Th en the generalized Cauchy-Riemann equation (5) has a regular solution ( , )u v of the Dirichlet
problem (12) in D for each continuous inconstant boundary date : D R .
Corollary 4. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a mea-
surable function in D with 1( )
B
K L D . Suppose also that
0
1( , ) log
B
k Z O
as 0 0Z D . (20)
Th en the conclusion of Th eorem 4 holds.
Remark 2. In particular, the conclusion of Th eorem 4 holds if
0
1( ) log
B
K Z O
Z Z
as 0Z Z 0Z D . (21)
Moreover, the condition (20) can be replaced by the whole series of more weak conditions
0
1 1 1( , ) log log log log log
B
k Z O
as 0 0Z D . (22)
B
K
B
K
8 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 2
V.Ya. Gutlyanskiĭ, V.I. Ryazanov, A.R. Salimov, R.R. Salimov
5. Criteria of the Orlicz type.
Th eorem 5. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function. Suppose also that
( ( )) ( ) .
B
D
K Z dL Z (23)
for a convex non-decreasing function :[0, ] [0, ] such that, for some 0 ,
2log ( ) .dtt
t
(24)
Th en the generalized Cauchy-Riemann equation (5) has a regular solution ( , )u v of the
Dirichlet problem (12) in D for each continuous inconstant boundary date : D R .
Remark 3. Note that the condition (24) is not only suffi cient but also necessary to have regular
solutions ( , )u v of the Dirichlet problem (12) in D to the generalized Cauchy-Riemann equations
(5) with the integral constraints (23) for all continuous inconstant date : D R .
Corollary 5. Let D be a bounded simply connected domain in 2R and 2 2:B D B be a
measurable function. Suppose that, for 0 ,
exp[ ( )] ( ) .
B
D
K Z dL Z (25)
Th en the conclusion of Th eorem 5 holds.
Th e corresponding survey of consequences on the Dirichlet problem to generalized Cauchy-
Riemann equations with sources from the theory of the Beltrami equations will be published
elsewhere.
Acknowledgments. Th e fi rst 2 authors are partially supported by the project “Mathematical mo-
deling of complex dynamical systems and processes caused by the state security”, No. 0123U100853,
of the National Academy of Sciences of Ukraine and by a grant from the Simons Foundation PD-
Ukraine-00010584.
9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2025. № 2
On the Dirichlet problem for generalized Cauchy—Riemann equations
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1. Ahlfors, L. (1966). Lectures on Quasiconformal Mappings. New York: Van Nostrand.
2. Astala, K., Iwaniec, T. & Martin, G. J. (2009). Elliptic differential equations and quasiconformal mappings in the
plane. Princeton Math., Ser. 48. Princeton: Princeton Univ. Press.
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Received 18.01.2025
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2025. No 2
V.Ya. Gutlyanskiĭ, V.I. Ryazanov, A.R. Salimov, R.R. Salimov
В.Я. Гутлянський1, 2, https://orcid.org/0000-0002-8691-4617
В.І. Рязанов1, 2, https://orcid.org/0000-0002-4503-4939
А.Р. Салімов1, https://orcid.org/0009-0000-1021-7527
Р.Р. Салімов2, https://orcid.org/0000-0001-9395-3334
1 Інститут прикладної математики і механіки НАН України, Слов’янськ, Україна
2 Інститут математики НАН України, Київ, Україна
E-mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com,
salimov.artem@gmail.com, ruslan.salimov1@gmail.com
ПРО ЗАДАЧУ ДІРІХЛЕ ДЛЯ УЗАГАЛЬНЕНИХ РІВНЯНЬ КОШІ—РІМАНА
Стаття містить огляд наслідків теорії рівнянь Бельтрамі з комплексного аналізу для задачі Діріхле до уза-
гальненого рівняння Коші—Рімана v B u на дійсній площині R2, що описує потоки рідини в
анізотропних та неоднорідних середовищах, де коефіцієнт B представлено у вигляді 2 × 2 матриці, а
градієнти u та v інтерпретуються як вектор-стовпці. Крім того, з’ясовується зв’язок цього рівняння з
A-гармонічним рівнянням ( ) 0div A u з матричними коефіцієнтами A, яке є одним із головних рівнянь
теорії потенціалу, а саме гідромеханіки (механіки рідин) в анізотропних та неоднорідних середовищах на
площині. Огляд включає низку ефективних інтегральних критеріїв існування регулярних розв’язків задачі
Діріхле з неперервними даними в довільних обмежених однозв’язних областях для узагальнених рівнянь
Коші—Рімана з матричними коефіцієнтами в умовах анізотропних та неоднорідних середовищ.
Ключові слова: система Коші—Рімана, узагальнені рівняння Коші—Рімана, задача Діріхле, рівняння Бель-
трамі, A-гармонічні рівняння.
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