Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development
In the work activity of the city system, described by generalized Lorentz model, is considered. It is assumed the existence of external influence on the land rate change in the city that is described by diffusive term and ergodic Markov processes. With the made assumptions, asymptotic dissipativity...
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| Zitieren: | Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development / A.V. Kinash, U.T. Khimka // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 78-86. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1338892018-06-09T03:05:12Z Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development Kinash, A.V. Khimka, U.T. In the work activity of the city system, described by generalized Lorentz model, is considered. It is assumed the existence of external influence on the land rate change in the city that is described by diffusive term and ergodic Markov processes. With the made assumptions, asymptotic dissipativity conditions of the initial system are set. Stationary points of the model of city economical activity are studied based on the results obtained for asymptotic dissipativity of generalized Lorentz model. У роботі розглянуто діяльність системи міста, що описується узагальненою моделлю Лоренца. Припускається наявність зовнішнього впливу на зміну величини земельної ренти в місті, який описується дифузійним доданком та ергодичним марковським процесом. За зроблених припущень, встановлено умови асимптотичної дисипативності вихідної системи. Досліджено стаціонарні точки моделі економічної діяльності міста на основі результатів, отриманих для асимптотичної дисипативності узагальненої моделі Лоренца. 2016 Article Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development / A.V. Kinash, U.T. Khimka // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 78-86. — Бібліогр.: 4 назв. — англ. 2308-5878 http://dspace.nbuv.gov.ua/handle/123456789/133889 519.21+62 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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In the work activity of the city system, described by generalized Lorentz model, is considered. It is assumed the existence of external influence on the land rate change in the city that is described by diffusive term and ergodic Markov processes. With the made assumptions, asymptotic dissipativity conditions of the initial system are set. Stationary points of the model of city economical activity are studied based on the results obtained for asymptotic dissipativity of generalized Lorentz model. |
| format |
Article |
| author |
Kinash, A.V. Khimka, U.T. |
| spellingShingle |
Kinash, A.V. Khimka, U.T. Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| author_facet |
Kinash, A.V. Khimka, U.T. |
| author_sort |
Kinash, A.V. |
| title |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development |
| title_short |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development |
| title_full |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development |
| title_fullStr |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development |
| title_full_unstemmed |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development |
| title_sort |
asymptotic dissipativity of lorenz model with markov switching for the city development |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/133889 |
| citation_txt |
Asymptotic Dissipativity of Lorenz model with Markov Switching for the City Development / A.V. Kinash, U.T. Khimka // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 13. — С. 78-86. — Бібліогр.: 4 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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AT kinashav asymptoticdissipativityoflorenzmodelwithmarkovswitchingforthecitydevelopment AT khimkaut asymptoticdissipativityoflorenzmodelwithmarkovswitchingforthecitydevelopment |
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2025-07-09T19:48:49Z |
| last_indexed |
2025-07-09T19:48:49Z |
| _version_ |
1837200086149365760 |
| fulltext |
Математичне та комп’ютерне моделювання
78
UDC 519.21+62
A. V. Kinash, postgraduate student,
U. T. Khimka, Ph. D.
National University Lviv Polytechnic, Lviv
ASYMPTOTIC DISSIPATIVITY OF LORENZ MODEL
WITH MARKOV SWITCHING FOR THE CITY DEVELOPMENT
In the work activity of the city system, described by general-
ized Lorentz model, is considered. It is assumed the existence of
external influence on the land rate change in the city that is de-
scribed by diffusive term and ergodic Markov processes. With the
made assumptions, asymptotic dissipativity conditions of the initial
system are set. Stationary points of the model of city economical
activity are studied based on the results obtained for asymptotic
dissipativity of generalized Lorentz model.
Key words: Lorenz model, city system, asymptotic dissipativ-
ity, stationary point.
Introduction. Lorentz system was first formulated in 1963 by
E. Lorenz in considering the convective fluid motion heated from the
bottom. Lorentz system is a system of three first order differential
equations, which has the form
= ,
= ,
= ,
dX
Y X
dt
dY
rX Y XZ
dt
dZ
XY bZ
dt
where ( )X t describes the intensity of convective motion, ( )Y t —
temperature difference between ascending and descending liquid streams,
and ( )Z t — vertical change of temperature. Lorentz model is widely used
in meteorology, economics, physics and others.
In particular, in economic theory Lorenz system can be used to study
the distribution function of demand for products [1], and economic
dynamics of the city system. In this paper the generalized system of
Lorentz that describes the economic system of the city in the presence of
external disturbances, which have diffusive nature. Namely, asymptotic
dissipativity of such a model is studied based on the results obtained in
[2]. Particularly, the presence of Lyapunov function that satisfies sufficient
conditions of asymptotic dissipativity of the initial system. In this article,
© A. V. Kinash, U. T. Khimka, 2016
Серія: Фізико-математичні науки. Випуск 13
79
such Lyapunov function is constructed, that allows us to state the
dissipativity of the considered Lorentz model.
Statement of the problem. Considering the generalized Lorenz system,
which describes economical model of the city system. Economic activity of
the city is assumed to be very small compared with the outside world [3].
1 2 3
1 2 3 4
1 2
= ( ),
= ( ) ,
= ,
dX
a a Y a X
dt
dY
b b X b Y b XZ
dt
dZ
c XY c Z
dt
(1)
where X — volume of products manufactured by the city system; Y —
the number of indigenous people; Z — land rent.
Constants are defined as follows 1a — rate of formation speed;
2a — parameter, which represents the magnitude of demand for the
products of the city, normalized per person; 3a — level of supply of
products within the city; 2b — the demand for labor on the part of firms
for the production of a unit of output; 3b — defined as a part of residents
who choose work in the city; 1c та 2c — determine the level of
production and population of the city positive impact, as well as the
negative impact of rent growth on its current value, respectively.
System (1) can not fully describe the system of the city, because it
does not include the impact of the world on the economic situation in the
city. Thus, it is worthwhile to consider perturbed system. In particular, we
introduce a diffusional perturbed term ( , , ) ( )X Y Z dw t in the third
equation, which describes the influence of the world on the rate of land
rent change in the city[4, p. 101].
1 2 3
1 2 3 4
1 2
= ( ),
= ( ) ,
= ( , , ) ( ),
dX
a a Y a X
dt
dY
b b X b Y b XZ
dt
dZ
c XY c Z X Y Z dw t
dt
(2)
Let us also consider the impact of external factors on the coefficient
2 ,c which defines the change of the city land rent. This influence is given
as ergodic Markov process ( )x t in phase space of states { 0.2,0.2} with
stationary distribution {0.5,0.5}, namely
Математичне та комп’ютерне моделювання
80
0.2, = 0.5,
=
0.2, = 0.5,
p
x
p
due to random rent changes.
3
1 2 3
1 2 4
1 2 2
= ( ),
= ( ) ,
= ( ( ) ) ( , , ) ( ),
dX
a a Y a X
dt
dY
b b X b Y b XZ
dt
dZ
c XY c x t c Z X Y Z dw t
dt
(3)
According to the theorem on asymptotic dissipativity [2] in order that
the process was asymptotically dissipative, dissipative should be its
limited process. Specifically, conditions must be satisfied
1( ) ( ) ( ),C u V u AV u (4)
2sup || ( ) ||< ,
du R
u A
(5)
where 1 2> 0, > 0A A і 3( , , ) ( )dV X Y Z C R Lyapunov function
= ( ).
du
C u
dt
Example. Let us consider specific model of the city
= 6 ,
= 3 3 7 ,
= 6 4(1 ( )) ( ).
dX
Y X
dt
dY
X Y XZ
dt
dZ
XY x t Z dw t
dt
The average system has the form
= 6 ,
= 3 3 7 ,
= 6 4 ( ).
dX
Y X
dt
dY
X Y XZ
dt
dZ
XY Z dw t
dt
(6)
In this case ( , , )C X Y Z has the presentation
6
( , , ) = 3 3 7 .
6 4
T
Y X
C X Y Z X Y XZ
XY Z
Серія: Фізико-математичні науки. Випуск 13
81
We verify the conditions of the theorem for the system (6).
Considering Lyapunov function in the form
2 2 2( , , ) .V X Y Z X Y Z (7)
Hence
( , , ) = (2 , 2 , 2 ) .TV X Y Z X Y Z
Particularly for condition (4), have
6 2
( , , ) ( , , ) = 3 3 7 2 =
6 4 2
T
Y X X
C X Y Z V X Y Z X Y XZ Y
XY Z Z
2
= ( 6 ,3 3 7 ,6 4 ) 2 =
2
X
Y X X Y XZ XY Z Y
Z
2 2 2= 2 12 6 6 14 12 8 =XY X XY Y XYZ XYZ Z
2 2 2= 12 8 6 8 2 .X XY Y Z XYZ
Under 1 = 1A condition (4) takes the form
2 2 2 2 2 212 8 6 8 2 < ,X XY Y Z XYZ X Y Z
2 2 2 2 2 212 8 6 8 2 < 0,X XY Y Z XYZ X Y Z
2 2 211 8 5 7 2 < 0,X XY Y Z XYZ
2 2 2 2 216 16
11 8 5 7 2 < 0,
11 11
X XY Y Y Y Z XYZ
2
2 2 24 16
11 5 7 2 < 0,
1111
X Y Y Y Z XYZ
2
2 24 39
11 7 2 < 0.
1111
X Y Y Z XYZ
And, as, ( , , ) = 1,X Y Z that condition (5) is satisfied.
Thus, system (6) is dissipative. Therefore, initial system is
asymptotically dissipative.
The asymptotic dissipativity of initial system. Set the conditions of
asymptotic dissipativity for the problem (3). Namely, for (4) have
1 2 3 1 2 1 3
1 2 3 4 1 2 1 3 4
1 2 1 2
( )
( , , ) = ( ) .
T T
a a Y a X a a Y a a X
C X Y Z b b X b Y b XZ b b X b b Y b XZ
c XY c Z c XY c Z
Математичне та комп’ютерне моделювання
82
So, for the Lyapunov function (7) get
1 2 1 3
1 2 1 3 4
1 2
2
( , , ) ( , , ) = 2 =
2
T
a a Y a a X X
C X Y Z V X Y Z b b X b b Y b XZ Y
c XY c Z Z
1 2 1 3 1 2 1 3 4 1 2
2
= ( , , ) 2 =
2
X
a a Y a a X b b X b b Y b XZ c XY c Z Y
Z
2 2 2
1 3 1 2 1 2 1 3 4 1 2= 2 2( ) 2 2( ) 2 .a a X a a b b XY b b Y b c XYZ c Z
Substituting the obtained result in (4)
2 2 2
1 3 1 2 1 2 1 3 4 1 2
2 2 2
1 1 1
2 2( ) 2 2( ) 2 <
,
a a X a a b b XY b b Y b c XYZ c Z
A X A Y A Z
2 2 2
1 3 1 2 1 2 1 3 4 1 2
2 2 2
1 1 1
2 2( ) 2 2( ) 2
< 0,
a a X a a b b XY b b Y b c XYZ c Z
A X A Y A Z
2
1 2 1 2
1 3 1 4 1
1 3 1
2
2 21 2 1 2
1 3 1 2 1
1 3 1
2 2( )
2
( )
2 (2 ) < 0.
2
a a b b
a a A X Y b c XYZ
a a A
a a b b
b b A Y c A Z
a a A
Therefore, the following conditions must be satisfied
1 3 1
2
1 2 1 2
1 3 1
1 3 1
4 1
2 1
2 > 0,
( )
2 0,
2
0,
2 0.
a a A
a a b b
b b A
a a A
b c
c A
1 1 3
2
1 2 1 2
1 3 1
1 3 1
4 1
1 2
< 2 ,
( )
2 0,
2
,
2 .
A a a
a a b b
b b A
a a A
b c
A c
(8)
Consider separately the second inequality
2
1 2 1 2
1 3 1
1 3 1
( )
2 0,
2
a a b b
b b A
a a A
Серія: Фізико-математичні науки. Випуск 13
83
2
1 3 1 3 1 1 2 1 2 1 1 3 1
1 3 1
2 (2 ) ( ) (2 )
0.
2
b b a a A a a b b A a a A
a a A
Since, 1 1 3< 2 ,A a a then the numerator of left side of inequality
should be nonnegative
2
1 3 1 3 1 1 2 1 2 1 1 3 12 (2 ) ( ) (2 ) 0.b b a a A a a b b A a a A
Solve the inequality relative to 1A
2 2 2 2 2
1 1 3 1 3 1 1 3 1 3 1 2 1 2 1 2 1 2(2 2 ) (4 2 ) 0,A a a b b A b b a a a a a a b b b b
2 2
1 3 1 3 1 2 1 2(2 2 ) (2 2 ) ,D a a b b a a b b
1,2 2 2
1 3 1 3 1 3 1 3 1 2 1 21
1
= 2 2 (2 2 ) (2 2 ) .
2
A a a b b a a b b a a b b
Therefore, system (8) has repredentation
1 1 3
2 2
1 1 3 1 3 1 3 1 3 1 2 1 2
4 1
1 2
1 1 3
2 2
1 1 3 1 3 1 3 1 3 1 2 1 2
4 1
1 2
< 2 ,
( ) ( ) ,
,
2 .
< 2 ,
( ) ( ) ,
,
2 .
A a a
A a a b b a a b b a a b b
b c
A c
A a a
A a a b b a a b b a a b b
b c
A c
(9)
Consider separately second system in the (9). It is obvious that in
order to have a solution of the system the inequality should be fulfilled
2 2
1 3 1 3 1 3 1 3 1 2 1 2 1 3( ) ( ) 2 .a a b b a a b b a a b b a a
Herefrom
2 2
1 3 1 3 1 2 1 2 1 3 1 3( ) ( ) .a a b b a a b b a a b b
Since all coefficients are positive, then bring to square both sides of
inequality, we get
2
1 2 1 2( ) 0.a a b b
Then, system (9) have a form
1 1 3
2 2
1 1 3 1 3 1 3 1 3 1 2 1 2
4 1
1 2
2 ,
( ) ( ) ,
,
2 .
A a a
A a a b b a a b b a a b b
b c
A c
(10)
Математичне та комп’ютерне моделювання
84
According to fulfillment of these conditions and the condition (5)
system (2) is dissipative, and therefore (3) asymptotically dissipative.
Hence, for the system (6) we obtain the following interval 1 (0;4].A
Stationary points. Let us find the stationary points of the system (1).
From the equating left sides of equations to zero, we get
1 2 3
1 2 3 4
1 2
( ) = 0,
( ) = 0,
= 0,
a a Y a X
b b X b Y b XZ
c XY c Z
Solve obtained system
3
2
3
1 2 3 4
2
23
1 2
2
= ,
( ) = 0,
= 0,
a
Y X
a
a
b b X b X b XZ
a
a
c X c Z
a
3
2
3 1 3
1 2 4
2
23
1 2
2
= ,
( ) = 0,
= 0.
a
Y X
a
a b b
X b b b Z
a
a
c X c Z
a
From second equation = 0X or 3 1 3
1 2 4
2
= 0.
a b b
b b b Z
a
Thus point 1(0,0,0)K — stationary point of the system (1).
Under 3 1 3
1 2 4
2
= 0,
a b b
b b b Z
a
have
3
2
2 1 2 3 1 3
4 2
23 2 1 2 3 1 3
1 2
2 4 2
= ,
= ,
= ,
a
Y X
a
a b b a b b
Z
b a
a a b b a b b
c X c
a b a
Серія: Фізико-математичні науки. Випуск 13
85
3
2
2 1 2 3 1 3
4 2
2 2 1 2 3 1 3
2
4 1 3
= ,
= ,
= .
a
Y X
a
a b b a b b
Z
b a
a b b a b b
X c
b c a
Taking into accunt, that all coefficients of the system are positive, the
following condition should be held
2 1 2 3 1 3 > 0a b b a b b (11)
(under 2 1 2 3 1 3=a b b a b b get stationary point 1K ).
Under this condition, for the considered system there exist two more
stationary points
3 2 1 2 3 1 3
2
2 4 1 3
2 1 2 3 1 3
4 2
2 1 2 3 1 3
2
4 1 3
= ,
= ,
= .
a a b b a b b
Y c
a b c a
a b b a b b
Z
b a
a b b a b b
X c
b c a
Therefore, stationary points of the system are 1(0,0,0)K and, under
condition (11),
2 1 2 3 1 3 3 2 1 2 3 1 3 2 1 2 3 1 3
2,3 2 2
4 1 3 2 4 1 3 4 2
= , , .
a b b a b b a a b b a b b a b b a b b
K c c
b c a a b c a b a
Now, from second inequality of asymptotic dissipativity conditions
(10), the following condition is obviously holds
2 2
1 3 1 3 1 3 1 3 1 2 1 2( ) ( ) > 0.a a b b a a b b a a b b
Hence,
2 2
1 3 1 3 1 2 1 2 1 3 1 3( ) ( ) < ,a a b b a a b b a a b b
2 2 2
1 3 1 3 1 2 1 2 1 3 1 3( ) ( ) < ( ) ,a a b b a a b b a a b b
2 2 2 2 2 2 2 2 2
1 3 1 3 1 3 1 3 1 2 1 2 1 3 1 3 1 3 1 32 ( ) < ,a a a a b b b b a a b b a a a a b b b b
2
1 2 1 2 1 3 1 3( ) < 4 .a a b b a a b b
According to (11)
2
1 2 1 2 1 3 1 3 1 2 1 2( ) < 4 < 4 ,a a b b a a b b a a b b
Математичне та комп’ютерне моделювання
86
2 2 2 2
1 2 1 2 1 2 1 2 1 2 1 22 < 4 ,a a a a b b b b a a b b
2 2 2 2
1 2 1 2 1 2 1 22 < 0.a a a a b b b b
Therefore
2
1 2 1 2( ) < 0.a a b b
Thus, under conditions of asymptotic stability of (3), the only
stationary point is 1(0,0,0).K
Conclusions. In the article the dissipativity conditions were obtain
for the generalized Lorenz model, which describes the city economics. The
city system is considered provided the existence of the external perturba-
tions on the rent value. Also, the stationary points were studied for the
considered system under condition of the asymptotic diddipativity.
References:
1. Olems’koi О. І. Synergetic model of the economic structure of society /
О. І. Olems’koi, О. V. Yushchenko, S. V. Kokhan // Journal of Physical Re-
search. — 2004. — Vol. 8, № 3. — P. 268–278. (in Ukrainian).
2. Kinash A. V. Asymptotic dissipativity of the diffusion process. процесу /
A. V. Kinash, Ya. M. Chabanyuk, U. T. Khimka // Mathematical and computer
modeling. A series of physical and mathematical sciences. Collected Works.
V.M. Glushkov Institute of Cybernetics, Kamyanets-Podilsky national univer-
sity of I. Ogiienko. — 2014. — Issue. 11. — P. 77–87. (in Ukrainian).
3. Beliavskii S. S. The generalized model of urban system (generalized Lorenz
system) / S. S. Beliavskii, T. A. Orlianin // Third International Scientific Con-
ference «Mathematical Modeling and Differential Equations»: a collection of
articles of the Third International Scientific Conference. Brest, September 17–
22, 2012. — Minsk : BGU, 2012. — P. 70-78. (in Russian).
4. Bondarev B. V. Stochastic calculus in problems of financial and actuarial
mathematics. Risk assessment for insurance / B. V. Bondarev, О. Е. Sosnitz-
kii. — 2013. — 228 p. (in Russian).
У роботі розглянуто діяльність системи міста, що описується уза-
гальненою моделлю Лоренца. Припускається наявність зовнішнього
впливу на зміну величини земельної ренти в місті, який описується
дифузійним доданком та ергодичним марковським процесом. За зроб-
лених припущень, встановлено умови асимптотичної дисипативності
вихідної системи. Досліджено стаціонарні точки моделі економічної
діяльності міста на основі результатів, отриманих для асимптотичної
дисипативності узагальненої моделі Лоренца.
Ключові слова: модель Лоренца, система міста, асимптотична
дисипативність, стаціонарна точка.
Отримано: 17.03.2016
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/HRV (Za stvaranje Adobe PDF dokumenata pogodnih za pouzdani prikaz i ispis poslovnih dokumenata koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 5.0 e versioni successive.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|