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Locally Socle of C(X)

  • June 2014
Authors:
Mehrdad Namdari at Shahid Chamran University of Ahvaz
Mehrdad Namdari
Somayeh Soltanpour at Petroleum University of Technology
Somayeh Soltanpour
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ﻝﺪﻣ ﻪﻠﺠﻣ ﻩﺭﻭﺩ ،ﻲﺿﺎﻳﺭ ﻪﺘﻓﺮﺸﻴﭘ ﻱﺯﺎﺳ4 ﻩﺭﺎﻤﺷ ،2 ،ﺰﻴﻳﺎﭘ ﻭﻥﺎﺘﺴﻣﺯ 1393
ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ
()CX
ﺭﻮﭙﻧﺎﻄﻠﺳ ﻪﻴﻤﺳ
1
ﻭ ﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ
ﻲﺿﺎﻳﺭ ﻩﻭﺮﮔ، ﺯﺍﻮﻫﺍ ﻥﺍﺮﻤﭼ ﺪﻴﻬﺷ ﻩﺎﮕﺸﻧﺍﺩ
ﺖﻓﺎﻳﺭﺩ ﺦﻳﺭﺎﺗ: 12/6/1393 ﺵﺮﻳﺬﭘ ﺦﻳﺭﺎﺗ :6/2/1394
ﻩﺪﻴﻜﭼ :ﻪﻌﻟﺎﻄﻣ ﻭ ﻲﻓﺮﻌﻣ ﻪﺑ ﻪﻟﺎﻘﻣ ﻦﻳﺍ ﺭﺩ ﻱ
()
F
LC X
، ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ
()CX
، ﻲﻣ ﺕﺭﺎﺒﻋ ﻪﻛ ،ﻢﻳﺯﺍﺩﺮﭘ
ﺯﺍ ﺖﺳﺍ
^`
() ():
Ff
LC X f C X S X
ﻥﺁ ﺭﺩ ﻪﻛ
f
S
ﻪﻋﻮﻤﺠﻣ ﻉﺎﻤﺘﺟﺍ ﺎﺑ ﺮﺑﺍﺮﺑ ﺯﺎﺑ ﻱﺎﻫ
UX
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ
\()UZf f
. ﻢﻳﺮﻴﮔ
()
F
CX
ﻞﻛﺎﺳ ﺶﻳﺎﻤﻧ
()CX
،ﺖﺳﺍﻲﻣ ﻥﺎﺸﻧ ﻪﻛ ﻢﻴﻫﺩ
()
F
LC X
ﻚﻳ
z
ﻝﺁﺪﻳﺍ
()CX
ﻞﻣﺎﺷ
()
F
CX
ﺍﺖﺳ. ﻪﻄﺑﺍﺭ ﺭﺩ ﻱﻭﺎﺴﺗ ﻱﺭﺍﺮﻗﺮﺑ ﻂﻳﺍﺮﺷ ﻱ
() () ()
FF
CX LCX CX
ﻲﻣ ﻲﺳﺭﺮﺑ ﺍﺭ ﻲﻣ ﻥﺎﺸﻧ ﻊﻗﺍﻭ ﺭﺩ ﻭ ﻢﻴﻨﻛ ﻪﻛ ﻢﻴﻫﺩ
X
ًﺎﺒﻳﺮﻘﺗ ﻱﺎﻀﻓ ﻚﻳ
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ ﺖﺳﺍ ﻪﺘﺴﺴﮔ
() ()
F
CX LC X
.ﻲﻣ ﻪﺟﻮﺗ ﻩﺎﮔﺮﻫ ﻪﻛ ﻢﻴﻨﻛ
X
ﻲﻫﺎﻨﺘﻣﺎﻧ ﻱﺎﻀﻓ ﻚﻳ
،ﺪﺷﺎﺑ
()CX
ﺮﺑ ﺰﮔﺮﻫ
()
F
CX
ﺖﺴﻴﻧ ﻖﺒﻄﻨﻣ .ﻲﻣ ﺖﺑﺎﺛ ﻦﻴﻨﭽﻤﻫ ﻪﻛ ﻢﻴﻨﻛ
()IX f
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
() ()
FF
CX LCX
. ﻩﺎﮔﺮﻫ ﻩﻭﻼﻋ ﻪﺑ
()IX f
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
()
F
LC X
ﭻﻴﻫ ﺭﺩ ﻪﻘﻠﺣﺮﻳﺯ ﺯﺍ ﻚﻳ ﻱﺎﻫ
()CX
ﻲﻤﻧ ﻲﺳﺎﺳﺍ ﻥﺁ ﻞﻣﺎﺷ ﺪﺷﺎﺑ .ﻲﻣ ﻪﻛ ﻲﻟﺎﺣ ﺭﺩ ﻢﻴﻨﻴﺑ
()
F
LC X
ﻝﺍﺪﻳﺍ ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ ﻲﺳﺎﺳﺍ ﻱﺎﻫ
ﺖﺳﺍ .ﺍﺭ ﻲﻄﻳﺍﺮﺷﻲﻣ ﻥﺎﻴﺑ ﻪﻛ ﻢﻴﻨﻛ
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﭻﻴﻫ ﺭﺩ ﻱ
()CX
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻪﻛ ﻱﺎﻫ
()CX
ﻲﻤﻧ ﻝﻭﺍ ﻝﺍﺪﻳﺍ ﻚﻳ ﺖﺳﺍ ﺪﺷﺎﺑ . ﻥﺩﻮﺑ ﻝﻭﺍ ﻦﻴﻨﭽﻤﻫ
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﺯﺍ ﻲﺧﺮﺑ ﺭﺩ ﺍﺭ ﻱﺎﻫ
()CX
ﻲﻣ ﺺﺨﺸﻣ ﻢﻴﻨﻛ.
ﻩژﺍﻭ ﻱﺪﻴﻠﻛ ﻱﺎﻫ: ،ﻞﻛﺎﺳ
z
ﻲﺳﺎﺳﺍ ﻝﺁﺪﻳﺍ ،ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ،ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﻱﺎﻀﻓ ،ﻝﺁﺪﻳﺍ.
ﻩﺩﺭ ﻲﺿﺎﻳﺭ ﻱﺪﻨﺑ :30C54، 40C54.
1- ﻪﻣﺪﻘﻣ
ﺪﻴﻨﻛ ﺽﺮﻓ
()CX
ﻪﻘﻠﺣ ﻪﺘﺳﻮﻴﭘ ﻊﺑﺍﻮﺗ ﻱ ﻲﻘﻴﻘﺣ ﻱ- ﻒﻧﻮﺨﻴﺗ ﻱﺎﻀﻓ ﻱﻭﺭ ﺭﺍﺪﻘﻣ
X
ﺪﺷﺎﺑ . ﺭﺩ
ﻪﻌﻟﺎﻄﻣ ﻱ
()CX
ﻢﻴﻫﺎﻔﻣ ﺎﺑ ﺩﺭﻮﺧﺮﺑ ﻥﺎﻜﻣﺍ ،ﻱژﻮﻟﻮﭘﻮﺗ ﻭ ﺮﺒﺟ ﻦﻴﺑ ﻱﺎﻫﺪﻧﻮﻴﭘ ﻱﺭﺍﺮﻗﺮﺑ ﻅﺎﺤﻟ ﻪﺑ
ﻩﺯﺎﺗ ﺩﺭﺍﺩ ﺩﻮﺟﻭ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﺎﻳ ﻱﺮﺒﺟ ﻱ .ﻪﺘﺧﺎﻨﺷ ﻱﺎﻫﺎﻀﻓ ﺯﺍ ﻱﺭﺎﻴﺴﺑ ﻩﺪﺷ ﺖﺴﺨﻧ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﻱ
ﻲﮔﮋﻳﻭ ﻚﻤﻛ ﺎﺑ ﻪﻘﻠﺣ ﻱﺮﺒﺟ ﻱﺎﻫ ﻱ
()CX
ﺎﺑ ﺍﺪﺘﺑﺍ ،ﻱﺮﺒﺟ ﻢﻴﻫﺎﻔﻣ ﺯﺍ ﻱﺭﺎﻴﺴﺑ ﻭ ﺪﻧﺪﻣﺁ ﺖﺳﺩ ﻪﺑ
ﻲﮔﮋﻳﻭ ﻚﻤﻛ ﺯﺍ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﻱﺎﻫ
()CX
ﻪﺘﻓﺮﮔ ﻪﻤﺸﭼﺮﺳ ﻪﻘﻠﺣ ﻪﺑ ﺲﭙﺳ ﻭ ﺪﻧﺍ ﻲﻠﻛ ﻱﺎﻫ ﺰﻴﻧ ﺮﺗ
1- ﻲﻜﻴﻧﻭﺮﺘﻜﻟﺍ ﺱﺭﺩﺁ ﻪﻟﺎﻘﻣ ﻝﻮﺌﺴﻣ ﻩﺪﻨﺴﻳﻮﻧ: s-soltanpour@phdstu.scu.ac.ir
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 88
ﻪﺘﻓﺎﻳ ﺫﻮﻔﻧﺪﻧﺍ .ﻪﻨﻴﻣﺯ ﺭﺩ ﺕﺎﻘﻴﻘﺤﺗ ﻱ
()CX
ﻝﺎﺳ ﺭﺩ ﻥﺍﺮﻳﺍ ﺭﺩ ﻪﺘﺷﺬﮔ ﻱﺎﻫ)ﺯﺍﻮﻫﺍ ( ﻭ ﻩﺩﻮﺑ ﻝﺎﻌﻓ
ﺎﺑ ﻱﺎﻫﺭﺎﻛ ،ﺖﺳﺍ ﻪﺘﻓﺮﮔ ﻡﺎﺠﻧﺍ ﻪﻨﻴﻣﺯ ﻦﻳﺍ ﺭﺩ ﻲﺘﻴﻤﻫﺍﻥﺁ ﺭﺩ ﻪﻛ ﻪﺒﻨﺟ ﻪﺑ ،ﺎﻫ ﻖﻴﻤﻋ ﻱﺮﺒﺟ ﻱﺎﻫ ﻱﺮﺗ
ﺯﺍ
()CX
ﻝﺁﺪﻳﺍ ،ﻞﻛﺎﺳ ﻞﻴﺒﻗ ﺯﺍ ﻲﻤﻴﻫﺎﻔﻣ ﻭ ،ﺖﺳﺍ ﻩﺪﺷ ﻪﺘﺧﺍﺩﺮﭘ ﻭ ﻝﺎﻤﻴﻨﻴﻣ ،ﻲﺳﺎﺳﺍ ﻱﺎﻫ
،ﺖﺧﺍﻮﻨﻜﻳ
- ،ﻲﺘﻳﻮﻴﺘﻛﮋﻧﺍ
ue
-ﻪﻘﻠﺣ ﻭ ،ﻱﺪﻠﮔ ﺪﻌﺑ ،ﺎﻫ
z
-ﻝﺁﺪﻳﺍ ﻪﻘﻠﺣ ﻱﺎﻫ ﻲﺘﻤﺴﻗ ﺝﺭﺎﺧ ﻱﺎﻫ
()CX
ﺕﻻﺎﻘﻣ ،ﺖﺳﺍ ﻩﺪﺷ ﻪﻌﻟﺎﻄﻣ]1 ،2 ،3 ،4[ ﺪﻴﻨﻴﺒﺑ ﺍﺭ . ﺭﺎﺘﺷﻮﻧ ﺭﺩ ﻦﻴﻨﭽﻤﻫ"ﺖﻓﺮﺸﻴﭘ ﻱﺎﻫ
ﻱژﻮﻟﻮﭘﻮﺗ ﺭﺩ ﺮﻴﺧﺍ" ) ﻊﺟﺮﻣ]6[ (ﻦﺴﻜﻳﺮﻨﻫ ﻦﻳﻮﻠﻣ ﻂﺳﻮﺗ
1
ﺖﺳﺍ ﻩﺪﺷ ﻩﺭﺎﺷﺍ ﺐﻠﻄﻣ ﻦﻳﺍ ﻪﺑ. ﺭﺩ]7
ﻭ8[،
- ﺮﺒﺟﺮﻳﺯ
()
c
CX
، ﻪﺘﺳﻮﻴﭘ ﻊﺑﺍﻮﺗ ﺯﺍ ﻞﻜﺸﺘﻣ ﻲﻘﻴﻘﺣ ﻱ- ﻭ ﻲﻳﺎﺳﺎﻨﺷ ﺍﺭﺎﻤﺷ ﺩﺮﺑ ﺎﺑ ﺭﺍﺪﻘﻣ
ﻪﻘﻠﺣﺮﻳﺯ ﺮﮕﻳﺩ ﻑﻼﺧ ﺮﺑ ﺮﺒﺟﺮﻳﺯ ﻦﻳﺍ ﻪﻛ ﺖﺳﺍ ﻦﻳﺍ ﺖﻴﻤﻫﺍ ﺰﺋﺎﺣ ﻪﺘﻜﻧ ،ﺖﺳﺍ ﻩﺪﺷ ﻪﻌﻟﺎﻄﻣ ﻱﺎﻫ
()CX
ﻩﺪﺷ ﻲﺳﺭﺮﺑ ﻥﻮﻨﻛﺎﺗ ﻪﻛ ﺎﺑ ًﺎﻣﻭﺰﻟ ،ﺪﻧﺍ
(Y)C
ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﭻﻴﻫ ﻱﺍﺮﺑ
Y
ﺖﺨﻳﺮﻜﻳ
ﻲﻤﻧ ﺪﺷﺎﺑ .ﺍ ﺭﺩ ﻩﮋﻳﻭ ﻪﺑ ،ﺖﺳﺍ ﻩﺪﺷ ﻪﺘﺧﺍﺩﺮﭘ ﺰﻴﻧ ﺞﻳﺎﺘﻧ ﻦﻳﺍ ﻲﻌﺿﻮﻣ ﻢﻴﻫﺎﻔﻣ ﻪﺑ ًﺍﺮﻴﺧ]9[،
- ﻱﺎﻫﺮﺒﺟﺮﻳﺯ
()
c
LX
،
()
F
LX
ﻭ
()LX
1
ﺯﺍ
()CX
ﺖﺳﺍ ﻩﺪﺷ ﻲﺳﺭﺮﺑ ﻭ ﻲﻓﺮﻌﻣ . ﺭﺩ
ﻪﻌﻟﺎﻄﻣ ﻪﻘﻠﺣ ﻱﺪﻠﮔ ﺪﻌﺑ ﻱ ﺾﻳﻮﻌﺗ ﻱﺎﻫ ﻲﻣ ﺡﺮﻄﻣ ﻞﻛﺎﺳ ﻡﻮﻬﻔﻣ ،ﺮﻳﺬﭘﺎﻧ ﺩﻮﺷ . ﻞﻛﺎﺳ
()CX
ﻪﻛ ،
ﺎﺑ ﺍﺭ ﻥﺁ
()
F
CX
ﻲﻣ ﺶﻳﺎﻤﻧ ﻝﺁﺪﻳﺍ ﺯﺍ ﻲﻤﻴﻘﺘﺴﻣ ﻉﻮﻤﺠﻣ ﺎﺑ ﺮﺑﺍﺮﺑ ،ﻢﻴﻫﺩ ﻝﺎﻤﻴﻨﻴﻣ ﻱﺎﻫ
()CX
ﺖﺳﺍ . ﺭﺩ]5[ﻝﺁﺪﻳﺍ ، ﻝﺎﻤﻴﻨﻴﻣ ﻱﺎﻫ
()CX
ﻩﺪﺷ ﻲﻳﺎﺳﺎﻨﺷ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺐﻴﺗﺮﺗ ﻪﺑ ﻥﺁ ﻞﻛﺎﺳ ﻭ ﺪﻧﺍ
ﺪﻨﺘﺴﻫ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﻲﺗﺍﺩﻮﺟﻮﻣ ﻪﻛ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻥﺎﺸﻧ ﻭ .
^`
^`
^`
(): \ ()
() (): \Z()
x
F
mfCXXZf x
CX f CX X f
f
ﻥﺁ ﺭﺩ ﻪﻛ
^`
( ) :f(x)Zf x X
`
ﻲﻣ ﻥﺁ ﻭ ﺪﺷﺎﺑ ﺮﻔﺻ ﺍﺭ-ﻪﻋﻮﻤﺠﻣ ﻱ
f
ﻲﻣ ﻢﻴﻣﺎﻧ . ﻝﺍﺪﻳﺍ
I
ﺯﺍ
()CX
ﻚﻳ ﺍﺭ
z
- ﻩﺎﮔﺮﻫ ،ﻢﻴﻳﻮﮔ ﻝﺍﺪﻳﺍ
[[]]ZZI I
1
ﻩﺎﮔﺮﻫ ؛ﻲﻨﻌﻳ ،
() ( )Zf Zg
،
f
I
ﻭ
()gCX
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
gI
.ﺍ ﻦﻴﻨﭽﻤﻫ ﻝﺁﺪﻳ
E
ﻪﻘﻠﺣ ﺭﺩ ﺍﺭ ﻱ
R
ﻲﺳﺎﺳﺍ
،ﻢﻴﻳﻮﮔ ﺪﻳﺍ ﺮﻫ ﺮﮔﺍ ﻪﻘﻠﺣ ﺮﻔﺻﺮﻴﻏ ﻝﺁ ﻱ
R
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻭ ﺪﻳﺎﻤﻧ ﻊﻄﻗ ﻲﻬﻳﺪﺑﺎﻧ ﺭﻮﻃ ﻪﺑ ﺍﺭ ﻱ
A
ﺯﺍ
ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ
X
ﭻﻴﻫ ﺍﺭ ﻩﺎﮔﺮﻫ ،ﻢﻴﻳﻮﮔ ﻝﺎﮕﭼ ﺎﺟ
int ( )
XX
cl A
. ﺭﺩ
[]1
ﻝﺩﺎﻌﻣ ﻱﺎﻫ
ﺭﺩ ﻝﺁﺪﻳﺍ ﻚﻳ ﻥﺩﻮﺑ ﻲﺳﺎﺳﺍ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ
()CX
ﻩﺪﺷ ﺺﺨﺸﻣ ﻩﮋﻳﻭ ﻪﺑ ،ﺪﻧﺍ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻥﺎﺸﻧ
ﻝﺁﺪﻳﺍ ﻪﻛ
E
ﺭﺩ
()CX
ﺮﮔﺍ ﺎﻬﻨﺗﻭ ﺮﮔﺍ ﺖﺳﺍ ﻲﺳﺎﺳﺍ
[E] ( )
fE
ZZf
[E] ( )
ZZf
[E] ( )()
ﭻﻴﻫ ﺪﺷﺎﺑ ﻝﺎﮕﭼ ﺎﺟ .
ﻲﻣ ﻦﻴﻨﭽﻤﻫ ﻪﻛ ﻢﻴﻨﻴﺑ
()
[()] ()
FfC X
F
ZC X Zf
[()] ()
F
f
[()]
()
()]
()
F
ﺩﺮﻔﻨﻣﺎﻧ ﻁﺎﻘﻧ ﺯﺍ ﻞﻜﺸﺘﻣ
X
ﺖﺑﺎﺛ ﻭ ﺖﺳﺍ
ﻪﻛ ﺖﺳﺍ ﻩﺪﺷ
()
F
CX
ﻪﻋﻮﻤﺠﻣ ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ ﺖﺳﺍ ﻲﺳﺎﺳﺍ ﺩﺮﻔﻨﻣ ﻁﺎﻘﻧ ﻱ
X
ﺭﺩ
X
ﻝﺎﮕﭼ
،ﺪﺷﺎﺑ]1 ﻭ4[ ﺪﻴﻨﻴﺒﺑ ﺍﺭ . ﺶﻘﻧ
()
F
CX
ﻱﺎﻀﻓ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﺹﺍﻮﺧ ﻥﺎﻴﻣ ﺪﻧﻮﻴﭘ ﺭﺩ
X
ﺹﺍﻮﺧ ﻭ
ﻪﻘﻠﺣ ﻱﺮﺒﺟ ﻱ
C( )X
ﺕﻻﺎﻘﻣ ﺭﺩ]3 ،4 ،5 ،8 ،9 ،10[ ، ﻦﻳﺍ ﺭﺩ ﻪﻛ ﺩﺮﻛ ﺩﺎﺠﻳﺍ ﺍﺭ ﻩﺰﻴﮕﻧﺍ ﻦﻳﺍ
1- Melvin Henriksen.
89 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
ﻪﻘﻠﺣ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻪﻟﺎﻘﻣ ﻱ
()CX
ﺎﺑ ﺍﺭ ﻥﺁ ﻪﻛ
()
F
LC X
ﻲﻣ ﺶﻳﺎﻤﻧ ﻪﻌﻟﺎﻄﻣ ﻭ ﻲﻓﺮﻌﻣ ،ﻢﻴﻫﺩ
ﻢﻴﻨﻛ .ﻲﻣ ﻥﺎﺸﻧ ﻡﻭﺩ ﺶﺨﺑ ﺭﺩ ﻪﻛ ﻢﻴﻫﺩ
()
F
LC X
ﻚﻳ
z
ﻝﺁﺪﻳﺍ
()CX
ﻲﻣ ﻪﻛ ﺪﺷﺎﺑ
()
F
CX
ﺖﺳﺍ ﻞﻣﺎﺷ ﺍﺭ .ﻲﻣ ﻥﺎﺸﻧ ﺮﮔﺍ ﻪﻛ ﻢﻴﻫﺩ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫ ﻩﺎﮔ
() () ()
FF
CX LCX
)
.
ﻪﻄﺑﺍﺭ ﺭﺩ ﻱﻭﺎﺴﺗ ﻲﺳﺭﺮﺑ ﻪﺑ ،ﻡﻮﺳ ﺶﺨﺑ ﺭﺩ ﻱ
() () ()
FF
CX LCX CX
ﻲﻣ ﻢﻳﺯﺍﺩﺮﭘ . ﻪﺟﻮﺗ
ﻲﻣ ﺚﺤﺒﻣ ﺭﺩ ﻚﻴﺳﻼﻛ ﻞﻛﺎﺳ ﻪﻛ ﻢﻴﻨﻛ
()CX
ﻪﻄﻘﻧ ﻦﻳﺍ ﻱﺍﺭﺍﺩ ﻪﻛ ﻲﺗﺭﻮﺻ ﺭﺩ ﻪﻛ ﺖﺳﺍ ﻒﻌﺿ
X
ﻩﺭﺍﻮﻤﻫ ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣﺎﻧ ﻱﺎﻀﻓ ﻚﻳ
() C()
F
CX X
C( )
C(
(
ﻲﻣ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﺩﺭﻮﻣ ﺭﺩ ﺎﻣﺍ ، ﻥﺍﻮﺗ
ﺪﺷﺎﺑ ﺭﺍﺮﻗﺮﺑ ﻱﻭﺎﺴﺗ ﻪﻛ ﺩﻮﻤﻧ ﺺﺨﺸﻣ ﺍﺭ ﻲﻳﺎﻫﺎﻀﻓ . ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ
X
ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﺍﺭ
ﻲﻣ ﻪﻋﻮﻤﺠﻣ ﻩﺎﮔﺮﻫ ،ﻢﻴﻣﺎﻧ ﺩﺮﻔﻨﻣ ﻁﺎﻘﻧ ﻱ
X
ﻝﺎﮕﭼ ﻥﺁ ﺭﺩ ، ،ﻲﻨﻌﻳ ؛ﺪﺷﺎﺑ
()IX X
.ﻗﺍﻭ ﺭﺩ ﻊ
ﻲﻣ ﺖﺑﺎﺛ ﻪﻛ ﻢﻴﻨﻛ
X
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ ﺖﺳﺍ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﻱﺎﻀﻓ ﻚﻳ
() ()
F
LC X C X
.
ﻲﻣ ﻥﺎﺸﻧ ﻦﻴﻨﭽﻤﻫ ﻪﻛ ﻢﻴﻫﺩ
()IX f
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
() ()
FF
CX LCX
. ﺭﺩ]3 ﻭ4[
ﻪﻛ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻥﺎﺸﻧ
()
F
CX
ﻚﻳ ﺰﮔﺮﻫﻝﺁﺪﻳﺍ ﻝﻭﺍ
C( )X
ﺖﺴﻴﻧ . ﺎﻣ ،ﻡﺭﺎﻬﭼ ﺶﺨﺑ ﺭﺩ
ﻥﺩﻮﺑ ﻝﻭﺍ ﻪﻛ ﻢﻳﺭﺍﺩ ﻲﻌﺳ
()
F
LC X
ﺭﺩ
()CX
ﻢﻴﻳﺎﻤﻧ ﺺﺨﺸﻣ ﺍﺭǤ ﺍﺭ ﻲﻄﻳﺍﺮﺷ ﺍﺪﺘﺑﺍ ﻥﺎﻴﺑ
ﻲﻣ ﻪﻛ ﻢﻴﻨﻛ
LC ( )
FX
ﻪﻘﻠﺣﺮﻳﺯ ﺭﺩ ﻱ
R
ﺯﺍ
()CX
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻪﻛ ﻱﺎﻫ
()CX
ﻚﻳ ،ﺪﺷﺎﺑ
ﻝﺁﺪﻳﺍ ﺖﺴﻴﻧ ﻝﻭﺍ .ﻲﻣ ﺖﺑﺎﺛ ﻦﻴﻨﭽﻤﻫ ﻪﻛ ﻢﻴﻨﻛ ﺩﺍﺪﻌﺗ ﻩﺎﮔﺮﻫﻪﻔﻟﺆﻣ ﺎﻫ ﻱﺪﻨﺒﻤﻫ ﻱ
X
ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣ
ﻥﺁ ﺯﺍ ﺎﺗ ﻭﺩ ﻞﻗﺍﺪﺣ ﻭ ﺎﻫ ﻥﺁ ،ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣﺎﻧ ﻩﺎﮔ
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﺭﺩ ﻱ
R
ﺯﺍ
()CX
ﻪﻛ
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻱﺎﻫ
()CX
ﻲﻣ ﺖﺴﻴﻧ ﻝﻭﺍ ،ﺪﺷﺎﺑ . ﻥﺩﻮﺑ ﻝﻭﺍ ﻂﻳﺍﺮﺷ
()
F
LC X
ﺯﺍ ﻲﺧﺮﺑ ﺭﺩ ﺍﺭ
ﻪﻘﻠﺣﺮﻳﺯ ﻱﺎﻫ
()CX
ﻲﻣ ﺺﺨﺸﻣ ﺰﻴﻧ ﻢﻴﻨﻛ .ﻲﻣ ﻪﻣﺍﺩﺍ ﺭﺩ ﻪﻛ ﻢﻴﻨﻴﺑ
()
F
LC X
ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ
ﻝﺁﺪﻳﺍ ﻲﻣ ﺖﺑﺎﺛ ﻭ ﺖﺳﺍ ﻲﺳﺎﺳﺍ ﻱﺎﻫ ﻩﺎﮔﺮﻫ ﻪﻛ ﻢﻴﻨﻛ
()IX f
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
()
F
LC X
ﺭﺩ
ﭻﻴﻫ ﻪﻘﻠﺣﺮﻳﺯ ﺯﺍ ﻚﻳ ﻱﺎﻫ
()CX
ﻪﻛ
()
F
LC X
ﻲﻣ ﺮﺑ ﺭﺩ ﺍﺭ ﻲﻤﻧ ﻲﺳﺎﺳﺍ ،ﺩﺮﻴﮔﺪﺷﺎﺑ .
ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻪﻟﺎﻘﻣ ﻦﻳﺍ ﺭﺩ
X
ﻲﻣ ﺮﻈﻧ ﺭﺩ ﻑﺭﻭﺪﺳﺎﻫ ﻭ ﻢﻈﻨﻣ ًﻼﻣﺎﻛ ،ﻲﻫﺎﻨﺘﻣﺎﻧ ﺍﺭ ﺮﮕﻣ ،ﻢﻳﺮﻴﮔ
ﻦﻳﺍ ﺩﻮﺷ ﺮﻛﺫ ﻥﺁ ﻑﻼﺧ ﻪﻛ .ﻪﻋﻮﻤﺠﻣ ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻡﺎﻤﺗ ﻱ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﺯﺎﺑ ﻱﺎﻫ
X
ﺎﺑ ﺍﺭ
O( )X
ﻲﻣ ﺶﻳﺎﻤﻧ ،ﻢﻴﻫﺩﻪﻋﻮﻤﺠﻣ ﺖﻟﻮﻬﺳ ﻱﺍﺮﺑ ﻦﻴﻨﭽﻤﻫ ﻲﻣ ﺯﺎﺒﺘﺴﺑ ﺍﺭ ﺯﺎﺑ ﻭ ﻪﺘﺴﺑ ﻱﺎﻫ ﻢﻴﻣﺎﻧ .
ﻪﻨﻴﻣﺯ ﺭﺩ ﺮﺘﺸﻴﺑ ﺕﺎﻋﻼﻃﺍ ﻱﺍﺮﺑ ﻱ ﻪﺑ ﻪﺘﺳﻮﻴﭘ ﻊﺑﺍﻮﺗ ﻪﻘﻠﺣ ﻭ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻫﺎﻀﻓ]11 ﻭ12[ ﻪﻌﺟﺍﺮﻣ ،
ﺩﻮﺷ.
2 - ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ
ﻲﻣ ﻥﺎﺸﻧ ﻭ ﻪﺘﺧﺍﺩﺮﭘ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻲﻓﺮﻌﻣ ﻪﺑ ﺍﺪﺘﺑﺍ ﺶﺨﺑ ﻦﻳﺍ ﺭﺩ ﻚﻳ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻪﻛ ﻢﻴﻫﺩ
z
ﻝﺁﺪﻳﺍ ﻞﻣﺎﺷ
()
F
CX
ﺖﺳﺍ .
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 90
ﻒﻳﺮﻌﺗ1: ﻢﻴﻨﻛ ﺽﺮﻓ
C( )
f
X
ﻭ
f
S
ﻪﻋﻮﻤﺠﻣ ﻉﺎﻤﺘﺟﺍ ﺎﺑ ﺮﺑﺍﺮﺑ ﺯﺎﺑ ﻱﺎﻫ
UX
ﻱﺭﻮﻃ ﻪﺑ
ﻪﻛ
\()UZf f
ﺪﺷﺎﺑ . ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ
C( )X
ﺎﺑ ﺍﺭ
()
F
LC X
ﻲﻣ ﺶﻳﺎﻤﻧ ﻥﺁ ﻭ ﻢﻴﻫﺩ ﺍﺭ
ﻪﻋﻮﻤﺠﻣ ﻪﻤﻫ ﻱ ﻊﺑﺍﻮﺗ ﻱ
fC(X)
ﻪﻛ
f
S
ﺭﺩ
X
ﻲﻣ ﻒﻳﺮﻌﺗ ،ﺪﺷﺎﺑ ﻝﺎﮕﭼ ﻢﻴﻨﻛ .،ﻲﻨﻌﻳ
()
\()
fUX
UZf
SU
2 f
U
^`
() ():
Ff
LC X f C X S X
ﻢﻟ 1 :
f
SX
ﻪﻋﻮﻤﺠﻣ ﺮﻫ ﻱﺍﺮﺑ ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ ﺯﺎﺑ ﻱ
GX
ﻪﻋﻮﻤﺠﻣ ، ﺯﺎﺑ ﻱ
UX
ﻱﺭﻮﻃ ﻪﺑ ﺪﺷﺎﺑ ﺩﻮﺟﻮﻣ ﻪﻛ
\()UZf f
ﻭ
UGz
ﺮﻫ ﻱﺍﺮﺑ ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
ﻪﻋﻮﻤﺠﻣ ﺯﺎﺑ ﻱ
GX
،
ﻪﻋﻮﻤﺠﻣ ﺯﺎﺑ ﻱ
UX
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺪﺷﺎﺑ ﺩﻮﺟﻮﻣ
\()UZf f
ﻭ
UG
.
ﻩﺎﮔﺮﻫ
U
ﻪﻋﻮﻤﺠﻣ ﻚﻳ ﻲﻬﺗﺎﻧ ﺯﺎﺑ ﻱ ﻑﺭﻭﺪﺳﺎﻫ ﻱﺎﻀﻓ ﺭﺩ ﻲﻫﺎﻨﺘﻣ
X
ﻭ
xU
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
x
ﺖﺳﺍ ﺩﺮﻔﻨﻣ .ﺍﺮﻳﺯ
^` ^`

\\xUUx
ﻭ ﻪﻛ ﺖﺳﺍ ﺢﺿﺍﻭ
()UX
U
UX
U
()
(
U
()
(
)
(
X
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
()IX X
.
ﻩﺭﺍﺰﮔ 1 : ﺮﻫ ﻱﺍﺮﺑ
()
f
CX
،
()
\()
fUX
UZf
SU
2 d
U
1
.
ﺕﺎﺒﺛﺍ: ﺖﺳﺍ ﻲﻬﻳﺪﺑ
() ()
\() 1 \()
f
VOX U X
VZf UZf
VUS
2
df
f
VUS
f
U
U
.ﻢﻴﻨﻛ ﺽﺮﻓ
^`
\ ( ) , , ...,
n
UZf xx x
12
.ﻲﻣ ﻒﻳﺮﻌﺗ ﻢﻴﻨﻛ
^`
\ , ,..., , , ...,
ii n
i
VUxx x x x

12 1 1
ﻪﻛ ﺖﺳﺍ ﺭﺎﻜﺷﺁ
i
V
ﺭﺩ
X
ﻭ ﺖﺳﺍ ﺯﺎﺑ
^`
\ Z(f)
ii
Vx
.ﻲﻣ ﺭﺍﺮﻗ ﻥﻮﻨﻛﺍ ﻢﻴﻫﺩ
1
n
i
i
UV
ﻭ
ﺖﺳﺍ ﻡﺎﻤﺗ ﺕﺎﺒﺛﺍ .ז
ﻢﻟ 2 :ﻩﺎﮔﺮﻫ
,()
f
gCX
ﻥﺁ ،ﺪﻨﺷﺎﺑ ﻩﺭﺍﺰﮔ ﻩﺎﮔ ﺪﻨﺘﺴﻫ ﺭﺍﺮﻗﺮﺑ ﺮﻳﺯ ﻱﺎﻫ.
ﺍﻒﻟ (
fg f g
SSS

.
ﺏ (
fg f g
SSS
.
پ (
f
f
SS
.
91 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
ﺕ ( ﺮﮔﺍ
,()
F
f
gLCX
ﻥﺁ ،ﺪﻨﺷﺎﺑ ﻩﺎﮔ
fg
SSX
.
ﺕﺎﺒﺛﺍ: ﻪﻛ ﺖﺳﺍ ﺢﺿﺍﻭ
,() ,()
|( )\ ( )|
\() |( ) \ ( )|
\()
W() W()
W\ ( ) W\ ( )
W\ ( )
() ()
= W W

f
f f
f

f  f
f

() ()
()
(
WW
W
fg
UV OX UV OX
UVZf
UZ f UVZg
VZg
fg
OX OX
Zf Zf g
Zg
S S UV UV
S
،ﻢﻳﺭﺍﺩ ﻩﺭﺍﻮﻤﻫ ﻪﻛ ﻦﻳﺍ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ
\( ) ( \()) ( \())UZf g UZf UZg 
ﻭ
\()( \()) ( \())UZfg UZf UZg
ﺖﻤﺴﻗ ﺕﺎﺒﺛﺍ ﻱﺎﻫ)ﻒﻟﺍ ( ﻭ)ﺏ (ﺖﺳﺍ ﺢﺿﺍﻭ . ﺖﻤﺴﻗ ﺕﺎﺒﺛﺍ)پ ( ﺯﺍ
\() \()UZf UZf
ﻪﺑ
ﻲﻣ ﺖﺳﺩ ﺪﻳﺁ . ﺖﻤﺴﻗ ﺕﺎﺒﺛﺍ ﻱﺍﺮﺑ)ﺕ (ﻲﻣ ﻱﺭﻭﺁﺩﺎﻳ ﻩﺎﮔﺮﻫ ﻪﻛ ﻢﻴﻨﻛ
YX
ﻭ
GX
ﺯﺎﺑ
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
GY G
. ﻥﻮﭼ
f
S
ﺭﺩ
X
ﻭ ﺖﺳﺍ ﻝﺎﮕﭼ ﻭ ﺯﺎﺑ
fgfg
SSSSX
.ז
ﺖﻤﺴﻗ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ)پ (ﻪﻌﻟﺎﻄﻣ ﺭﺩ ﻞﺒﻗ ﻢﻟ ﺭﺩ ﻱ
()
F
LC X
ﻲﻣ ﺩﺮﻛ ﺭﺎﻛ ﻲﻔﻨﻣﺎﻧ ﻊﺑﺍﻮﺗ ﺎﺑ ﻥﺍﻮﺗ.
ﻒﻳﺮﻌﺗ 2 : ﻪﻛ ﻢﻴﻨﻛ ﺽﺮﻓ
X
ﻭ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻚﻳ
F
f
C
ﻪﻋﻮﻤﺠﻣ ﻉﺎﻤﺘﺟﺍ ﺯﺎﺑ ﻱﺎﻫ
UX
ﺪﺷﺎﺑ ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ
()fU f
. ﻲﻫﺎﻨﺘﻣ ﻲﻌﺿﻮﻣ ﺭﻮﻃ ﻪﺑ ﺮﺒﺟﺮﻳﺯ
()CX
ﺎﺑ ﺍﺭ
()
F
LX
ﺶﻳﺎﻤﻧ
ﻲﻣ ﻒﻳﺮﻌﺗ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﻭ ﻩﺩﺍﺩ ،ﻢﻴﻨﻛ
^`
() ():
F
Ff
LX f CXC X
ﻩﺭﺍﺰﮔ 2 :
() ()
FF
LC X L X
.
ﺕﺎﺒﺛﺍ: ﻢﻴﻨﻛ ﺽﺮﻓ
()
F
f
LC X
ﻭ
GX
ﻪﻋﻮﻤﺠﻣ ﻚﻳ ﺲﭘ ،ﺪﺷﺎﺑ ﻲﻬﺗﺎﻧ ﻩﺍﻮﺨﻟﺩ ﺯﺎﺑ ﻱ
ﻪﻋﻮﻤﺠﻣ ﺯﺎﺑ ﻱ
UX
ﻪﻛ ﺖﺳﺍ ﺩﻮﺟﻮﻣ
\()UZff
ﻭ
UGz
.ﻦﻳﺍ ﺯﺍ ﺎﻣﺍ ﻪﻛ
\()UZff
ﻲﻣ ﻪﺠﻴﺘﻧ ﻪﻛ ﺩﻮﺷ
()fU f
،ﻲﻨﻌﻳ ؛
()
F
f
LX
.ז
ﻩﺭﺍﺰﮔ 3 :
()
F
LC X
ﻝﺁﺪﻳﺍ ﻚﻳ
()CX
ﺖﺳﺍ .
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 92
ﺕﺎﺒﺛﺍ: ﻢﻴﻨﻛ ﺽﺮﻓ
,()
F
fg LC X
ﻲﻣ ﻥﺎﺸﻧ ، ﻪﻛ ﻢﻴﻫﺩ
() F
fgLCX
. ﻞﺒﻗ ﻢﻟ ﻖﺒﻃ
fg f g
SSS

ﺲﭘ ،
fg f g f g
SSSSSX

ﻦﻳﺍﺮﺑﺎﻨﺑ ،
f()
F
gLCX
.
ﻢﻴﻨﻛ ﺽﺮﻓ ﻥﻮﻨﻛﺍ
()
F
fLCX
ﻭ
()gCX
ﻲﻣ ﻥﺎﺸﻧ ، ﻪﻛ ﻢﻴﻫﺩ
()
F
f
gLCX
. ﻖﺒﻃ
ﻞﺒﻗ ﻢﻟ
fg f g
SSS
ﺲﭘ ،
fg f g f g
SSSSSX
. ﻦﻳﺍﺮﺑﺎﻨﺑ
()
F
LC X
ﻚﻳ
ﺖﺳﺍ ﻝﺁﺪﻳﺍ .ז
ﻩﺭﺍﺰﮔ 4 :
()
F
LC X
ﻚﻳ
z
ﺖﺳﺍ ﻝﺁﺪﻳﺍ .
ﺕﺎﺒﺛﺍ :ﻢﻴﻨﻛ ﺽﺮﻓ
()
F
fLCX
ﻭ
() (g)Zf Z
ﻲﻣ ﻥﺎﺸﻧ ، ﻪﻛ ﻢﻴﻫﺩ
()
F
gLCX
. ﻱﺍﺮﺑ
ﻪﻋﻮﻤﺠﻣ ﺮﻫ ﺯﺎﺑ ﻱ
f
US
ﻢﻳﺭﺍﺩ ،
\() \()UZg UZf
ﺲﭘ ،
fg
SS
. ﻦﻳﺍﺮﺑﺎﻨﺑ
fg
XS S
،ﻲﻨﻌﻳ ؛
()
F
gLCX
.ז
ﻥﺁ ﺯﺍ ﺮﻫ ﻪﻛ ﺎﺟ
z
ﻝﺁﺪﻳﺍ ﺲﭘ ،ﺖﺳﺍ ﺏﺪﺤﻣ ًﺎﻘﻠﻄﻣ
()
F
LC X
ﻚﻳﻝﺁﺪﻳﺍ ﺏﺪﺤﻣ ًﺎﻘﻠﻄﻣ
ﻲﻣ ﺪﺷﺎﺑ.
ﻩﺭﺍﺰﮔ 5:
ﻩﺎﮔﺮﻫ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫ ﻱﺎﻀﻓ ﻚﻳ ﻩﺎﮔ
() () ()
FF
CX LCX
)
.
ﺕﺎﺒﺛﺍ : ﻢﻴﻨﻛ ﺽﺮﻓ
()
F
fLCX
ﻭ
()UOX
ﻪﻛ ﺪﺷﺎﺑ ﻲﻤﺴﻗ ﻪﺑ
\()UZff
.
ﻦﻳﺍﺮﺑﺎﻨﺑ
\()UZf
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻚﻳ ﺯﺎﺒﺘﺴﺑ ﻱ
X
ﻲﻣ ﺪﺷﺎﺑ .ﻥﺁ ﺯﺍ ﻪﻛ ﺎﺟ
X
ﺲﭘ ،ﺖﺳﺍ ﺪﻨﺒﻤﻫ
\()UZf
ﺎﻳ
\()UZf X
. ﺮﮔﺍ
\()UZf
ﻥﺁ ، ﻩﺎﮔ
()UZf
، ﺲﭘ
()
f
U
. ﺮﮔﺍ
\()UZf X
ﺲﭘ ،
X
ﺩﺭﺍﺩ ﺾﻗﺎﻨﺗ ﺽﺮﻓ ﺎﺑ ﻪﻛ ﺖﺳﺍ ﻲﻫﺎﻨﺘﻣ . ﻦﻳﺍﺮﺑﺎﻨﺑ
()
f
f
S
ﻥﻮﭼ ﻭ
f
SX
ﺲﭘ ،
f
.ז
3-ﻪﻄﺑﺍﺭ ﺭﺩ ﻱﻭﺎﺴﺗ ﻲﺳﺭﺮﺑ ﻱ
() () ()
FF
CX LCX CX
ﻲﻣ ﻥﺎﻴﺑ ﺍﺭ ﻲﻄﻳﺍﺮﺷ ﺶﺨﺑ ﻦﻳﺍ ﺭﺩ ﻦﻴﻨﭽﻤﻫ ﻭ ﻚﻴﺳﻼﻛ ﻞﻛﺎﺳ ﺎﺑ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻥﺁ ﺭﺩ ﻪﻛ ﻢﻴﻨﻛ
ﻪﻘﻠﺣ ﺎﺑ ﻱ
()CX
ﺖﺳﺍ ﻖﺒﻄﻨﻣ.
ﻩﺭﺍﺰﮔ6:
() ()
FF
CX LCX
.
ﺕﺎﺒﺛﺍ: ﻢﻳﺮﻴﮔ
()
F
f
CX
ﺲﭘ ،
\()XZff
. ﻦﻳﺍﺮﺑﺎﻨﺑ
f
SX
،ﻪﺠﻴﺘﻧ ﺭﺩ
()
F
f
LC X
.ז
93 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
ﻪﻄﺑﺍﺭ ﺭﺩ ﻱﻭﺎﺴﺗ ﻱ
() ()
FF
CX LCX
ﺖﺴﻴﻧ ﺭﺍﺮﻗﺮﺑ ًﺎﻣﻭﺰﻟ . ﻩﺎﮔﺮﻫ ﻝﺎﺜﻣ ﻱﺍﺮﺑ
X
ﻱﺎﻀﻓ ﻚﻳ
ﻥﺁ ،ﺪﺷﺎﺑ ﺍﺭﺎﻤﺷﺎﻧ ﻪﺘﺴﺴﮔ ﻩﺎﮔ
() () C()
FF
CX LCX X
()
F
()
(
. ﺮﮔﺍ ﻲﻓﺮﻃ ﺯﺍ
X
،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫ
() ( ) C( )
F
LC X X
)
(
F
(
(
C(
)
C(
(
.
ﻪﻴﻀﻗ ﺭﺩ ﻲﻣ ﺖﺳﺩ ﻪﺑ ﺍﺭ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻭ ﻚﻴﺳﻼﻛ ﻞﻛﺎﺳ ﻱﻭﺎﺴﺗ ﻝﺩﺎﻌﻣ ﻂﻳﺍﺮﺷ ﺪﻌﺑ ﻱ ﻪﻛ ﻢﻳﺭﻭﺁ
ﻧ ﺩﺍﺪﻌﺗ ﻥﺩﻮﺑ ﻲﻫﺎﻨﺘﻣ ﻝﺩﺎﻌﻣ ﺖﻘﻴﻘﺣ ﺭﺩﺖﺳﺍ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻚﻳ ﺩﺮﻔﻨﻣ ﻁﺎﻘ.
ﻪﻴﻀﻗ 1 :
()IX f
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
() ()
FF
CX LCX
.
ﺕﺎﺒﺛﺍ : ﻢﻴﻨﻛ ﺽﺮﻓ
() ()
FF
CX LCX
.ﺮﮔﺍ
()IX f
ﺲﭘ ،ﺪﺷﺎﺒﻧ
X
ﻚﻳ ﻱﺍﺭﺍﺩ
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻞﺜﻣ ﻲﻫﺎﻨﺘﻣﺎﻧ ﻱﺍﺭﺎﻤﺷ ﻱ
^`
, ,..., ,...
n
Axx x
12
ﺖﺳﺍ . ﻊﺑﺎﺗ
f
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺍﺭ
ﺮﻫ ﻱﺍﺮﺑ
n
xA
،
1
()
n
f
xn
ﺕﺭﻮﺻ ﻦﻳﺍ ﺮﻴﻏ ﺭﺩ ﻭ
()
f
x
ﻲﻣ ﻒﻳﺮﻌﺗ ،ﺪﺷﺎﺑ ﻢﻴﻨﻛ . ﻦﻳﺍ ﺭﺩ
ﻩﺎﮔﺮﻫ ﺕﺭﻮﺻ
H!
،ﺪﺷﺎﺑ
k
ﺮﻫ ﻱﺍﺮﺑ ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺖﺳﺍ ﺩﻮﺟﻮﻣ
nkt
ﻢﻳﺭﺍﺩ
1
nH
.
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻱﺍﺮﺑ ﻥﻮﻨﻛﺍ ﺯﺎﺒﺘﺴﺑ ﻱ
^`
\ , , ...,
k
GX xx x
12
ﺮﻫ ﻱﺍﺮﺑ ﻭ
xG
،ﻢﻳﺭﺍﺩ
()fx H
ﺲﭘ ،
()
f
CX
ﻭ
X\ Z(f)
A
؛ﻲﻨﻌﻳ ،ﺖﺳﺍ ﻲﻫﺎﻨﺘﻣﺎﻧ
()
F
f
CX
. ﻥﺎﺸﻧ
ﻲﻣ ﻪﻛ ﻢﻴﻫﺩ
.()
F
fLCX
ﻢﻳﺮﻴﮔ
GX
ﺪﻳﺎﺑ ،ﺪﺷﺎﺑ ﻩﺍﻮﺨﻟﺩ ﻲﻬﺗﺎﻧ ﺯﺎﺑ ﻪﻋﻮﻤﺠﻣ ﻚﻳ
ﺯﺎﺑ ﻪﻋﻮﻤﺠﻣ
UX
ﻪﻛ ﺪﺷﺎﺑ ﺩﻮﺟﻮﻣ
()UCozff
ﻭ
UG
. ﻭﺩ ﺭﻮﻈﻨﻣ ﻦﻳﺍ ﻱﺍﺮﺑ
ﻲﻣ ﺮﻈﻧ ﺭﺩ ﺖﻟﺎﺣ ﺮﮔﺍ ،ﻢﻳﺮﻴﮔ
()xG IX z
ﻥﺁ ، ﻢﻴﻫﺩ ﺭﺍﺮﻗ ﺖﺳﺍ ﻲﻓﺎﻛ ﻩﺎﮔ
^`
Ux
.
ﺮﮔﺍ
()GIX
ﻥﺁ ، ﻩﺎﮔ
\( ) \AGXIX X
ﺩﻮﺑ ﺪﻫﺍﻮﺧ . ﻦﻳﺍﺮﺑﺎﻨﺑ
() ( \ )GCozf G A XA A
ﻢﻴﻫﺩ ﺭﺍﺮﻗ ﺖﺳﺍ ﻲﻓﺎﻛ ،
UG
. ﺲﭘ
()\ ()
F
F
f
LC X C X
ﺖﺳﺍ ﺾﻗﺎﻨﺗ ﻪﻛ . ﻢﻳﺮﻴﮔ ،ﺲﻜﻋﺮﺑ
()IX f
.ﻲﻣ ﻥﺎﺸﻧ ﻢﻴﻫﺍﻮﺧ
ﻪﻛ ﻢﻴﻫﺩ
() ()
FF
CX LCX
. ﻢﻳﺮﻴﮔ
()
F
f
LC X
ﺲﭘ ،
f
SX
ﻩﺭﺍﺰﮔ ﺮﺑﺎﻨﺑ ﻲﻓﺮﻃ ﺯﺍ ،1
،ﻢﻳﺭﺍﺩ
()
\()1
f
UOX
UZf
SU
d
U
.ﻦﻳﺍﺮﺑﺎﻨﺑ
\ () \ () \ () ( )\ () ( \ ()) ( ) ( )
ff
X Zf S Zf S Zf U Zf U Zf IX IX
ff
)\ () ( \ ())
)\ () ( \ ()
)\ () ( \ ()))\ ( ) ( \ (
(\())
ﻞﻛﺎﺳ ﻲﻜﻳژﻮﻟﻮﭘﻮﺗ ﻒﻳﺮﻌﺗ ﻖﺒﻃ ﺲﭘ
()CX
ﺭﺩ
[]3
ﻲﻣ ﻪﺠﻴﺘﻧ ، ﻢﻳﺮﻴﮔ
()
F
f
CX
.ז
ﻥﺁ ﺯﺍ ﻪﻄﻘﻧ ﺪﻗﺎﻓ ﺪﻨﺒﻤﻫ ﻱﺎﻀﻓ ﻪﻛ ﺎﺟ ﻲﻣ ﻢﻫ ﻞﺒﻗ ﻩﺭﺍﺰﮔ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ ،ﺖﺳﺍ ﺩﺮﻔﻨﻣ ﻱ ﻪﺠﻴﺘﻧ ﻥﺍﻮﺗ
ﻪﻛ ﺖﻓﺮﮔ ﻩﺎﮔﺮﻫ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫ ﻱﺎﻀﻓ ﻚﻳ ﻩﺎﮔ
() () ()
FF
CX LCX
)
.
ﻩﺭﺍﺰﮔ7 : ﻩﺎﮔﺮﻫ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﻪﺘﺴﺴﮔ ﻱﺎﻀﻓ ﻚﻳ ﻩﺎﮔ
() ()
F
LC X C X
.
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 94
ﺕﺎﺒﺛﺍ: ﺮﮔﺍ
()fCX
ﻥﺁ ، ﻩﺎﮔ
^`
fxX
SxX
^
`
xX
^
`
^
`
^
`
xX
x
^
`
xX
. ﺲﭘ
()
F
fLCX
.ז
ﻲﻣ ﻱﺭﻭﺁﺩﺎﻳ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻪﻛ ﻢﻴﻨﻛ
X
ﻲﻣ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﺍﺭ ﻪﻋﻮﻤﺠﻣ ﻩﺎﮔﺮﻫ ،ﻢﻴﻣﺎﻧ ﻁﺎﻘﻧ ﻱ
ﺩﺮﻔﻨﻣ
X
،
()IX
،ﻲﻨﻌﻳ ؛ﺪﺷﺎﺑ ﻝﺎﮕﭼ ﻥﺁ ﺭﺩ
()IX X
.ﻪﻴﻀﻗ ﺭﺩ ﻲﻣ ﻥﺎﺸﻧ ﺪﻌﺑ ﻱ ﻪﻛ ﻢﻴﻫﺩ
ﻦﻳﺍ ﺎﺑ ﺖﺳﺍ ﻝﺩﺎﻌﻣ ﻲﮔﮋﻳﻭ ﻦﻳﺍ ﻪﻛ
()CX
ﻭ
()
F
LC X
ﺪﻨﺷﺎﺑ ﻖﺒﻄﻨﻣ ﻢﻫﺮﺑ .
ﻪﻴﻀﻗ2 :
X
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ ﺖﺳﺍ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﻱﺎﻀﻓ
() ()
F
LC X C X
.
ﺕﺎﺒﺛﺍ : ﻢﻴﻨﻛ ﺽﺮﻓ
X
ﻭ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﻱﺎﻀﻓ ﻚﻳ
()IX
ﻪﻋﻮﻤﺠﻣ ﺩﺮﻔﻨﻣ ﻁﺎﻘﻧ ﻱ
X
ﻭ
()fCX
ﺪﺷﺎﺑ . ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ
^`
()
()
fxIx
SxIX
^
`
)
I
(
x
^
`
^
`
^
`
)
xI
(
x
^
`
()
xI
(
(
ﺲﭘ ،
f
SX
،ﻲﻨﻌﻳ ؛
()
F
fLCX
. ،ﺲﻜﻋﺮﺑ ﻢﻴﻨﻛ ﺽﺮﻓ
rz
z
r
ﻦﻳﺍﺮﺑﺎﻨﺑ ،
()
F
rLCXz
rL
C
z
r
ﻭ
()Zr
،
ﺲﭘ
r
SX
. ﻢﻳﺮﻴﮔ
G
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻚﻳ ﺯﺎﺑ ﻱ
X
ﻥﺎﺸﻧ ﺪﻳﺎﺑ ،ﺪﺷﺎﺑ ﻪﻛ ﻢﻴﻫﺩ
()GIXz
. ﻥﻮﭼ
r
SX
ﻭ
()Zr
،
ﺮﻳﺯ ﺯﺎﺑ ﻪﻋﻮﻤﺠﻣ
r
U
ﺭﺩ
X
ﻪﺑ ﺖﺳﺍ ﺩﻮﺟﻮﻣ
ﻱﺭﻮﻃ ﻪﻛ
r
UGz
ﻭ
\()
r
UZrf
. ﺎﻣﺍ
()Zr
ﻭ
r
U
ﻪﻋﻮﻤﺠﻣ ﺮﻳﺯ ﻚﻳ ﺯﺎﺑ ﻱ
ﺯﺍ ﻲﻫﺎﻨﺘﻣ
X
ﻲﻣ ﺪﺷﺎﺑ . ﻦﻳﺍﺮﺑﺎﻨﺑ
()
r
UIX
،
ﺲﭘ
()
r
UGIXGz  
.ﻦﻳﺍ ﺯﺍ ﻭﺭ
()IX X
،ﻲﻨﻌﻳ ؛
X
ﺖﺳﺍ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ. ז
ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ
X
ﻲﻣ ﻩﺪﻨﻛﺍﺮﭘ ﺍﺭ ﻱﺎﻀﻓﺮﻳﺯ ﺮﻫ ﻩﺎﮔﺮﻫ ،ﻢﻴﻣﺎﻧ
X
ﻪﻄﻘﻧ ﻱﺍﺭﺍﺩ ﺪﺷﺎﺑ ﺩﺮﻔﻨﻣ ﻱ .
ﻲﻣ ﻥﺎﺸﻧ ﺮﻃﺎﺧ ﻪﻛ ﻢﻳﺯﺎﺳﺖﺳﺍ ﻪﺘﺴﺴﮔ ًﺎﺒﻳﺮﻘﺗ ﻩﺪﻨﻛﺍﺮﭘ ﻱﺎﻀﻓ ﺮﻫ . ﻪﻛ ﻢﻴﻨﻛ ﺽﺮﻓ ﺍﺮﻳﺯ
()IX
ﻁﺎﻘﻧﺩﺮﻔﻨﻣ
X
ﻭ
x
U
ﻲﮕﻳﺎﺴﻤﻫ ﻚﻳ
xX
ﺪﺷﺎﺑ . ﻥﻮﭼ
X
،ﺖﺳﺍ ﻩﺪﻨﻛﺍﺮﭘ
x
U
ﻱﺍﺭﺍﺩ
ﻪﻄﻘﻧ ﻚﻳ ﺩﺮﻔﻨﻣ ﻱ
x
ﺖﺳﺍ. ﻲﻓﺮﻃ ﺯﺍ
x
U
ﺲﭘ ،ﺖﺳﺍ ﺯﺎﺑ
x
ﻪﻄﻘﻧ ﺩﺮﻔﻨﻣ ﻱ
X
ﻲﻣ ﺰﻴﻧ ﺪﺷﺎﺑ .
ﻦﻳﺍﺮﺑﺎﻨﺑ
()
x
IxU Xz
x
x
U
x
.ﺲﭘ
()IX X
.
ﻪﺠﻴﺘﻧ1 : ﻩﺎﮔﺮﻫ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺪﻨﻛﺍﺮﭘ ﻱﺎﻀﻓ ﻚﻳ ﻩﺎﮔ
() ()
F
LC X C X
.
ﻪﺠﻴﺘﻧ ﺲﻜﻋ ﺖﺴﻴﻧ ﺭﺍﺮﻗﺮﺑ ﻻﺎﺑ ﻱ .ﻪﻳﺎﭘ ﻢﻴﻨﻛ ﺽﺮﻓ ﻝﺎﺜﻣ ﻱﺍﺮﺑ ﻱژﻮﻟﻮﭘﻮﺗ ﻚﻳ ﻱﺍﺮﺑ ﺍﺭ ﻲﻌﺿﻮﻣ ﻱﺎﻫ
ﻱﻭﺭ
ﻚﺗ ﺕﺭﻮﺻ ﻪﺑ ﺎﻳﻮﮔ ﻁﺎﻘﻧ ﺭﺩ ﻪﻄﻘﻧ ﺎﺴﻤﻫ ﺕﺭﻮﺻ ﻪﺑ ﻢﺻﺍ ﻁﺎﻘﻧ ﺭﺩ ﻭ ﻱﺍﻲﮕﻳ ﻲﻟﻮﻤﻌﻣ ﻱﺎﻫ
ﻢﻳﺮﻴﮕﺑ ﺮﻈﻧ ﺭﺩ . ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ﺎﻣﺍ ،ﺖﺴﻴﻧ ﻩﺪﻨﻛﺍﺮﭘ ﻱژﻮﻟﻮﭘﻮﺗ ﻦﻳﺍ ﺎﺑ
() ()
F
LC X C X
.
4- ﻥﺩﻮﺑ ﻝﻭﺍ ﻲﺳﺭﺮﺑ
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﺯﺍ ﻲﺧﺮﺑ ﺭﺩ ﻱﺎﻫ
()CX
ﻪﻴﻀﻗ 3 : ﺩﺍﺪﻌﺗ ﻩﺎﮔﺮﻫﻪﻔﻟﺆﻣ ﺎﻫ ﻱﺪﻨﺒﻤﻫ ﻱ
X
ﻥﺁ ﺯﺍ ﺎﺗ ﻭﺩ ﻞﻗﺍﺪﺣ ﻭ ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣ ﻲﻫﺎﻨﺘﻣﺎﻧ ﺎﻫ
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﭻﻴﻫ ﺭﺩ ﻱ
A
ﺯﺍ
()CX
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻪﻛ ﻱﺎﻫ
()CX
ﻲﻣ ﺖﺴﻴﻧ ﻝﻭﺍ ،ﺪﺷﺎﺑ.
95 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
ﺕﺎﺒﺛﺍ : ﺩﺍﺪﻌﺗ ﻩﺎﮔﺮﻫﻪﻔﻟﺆﻣ ﺎﻫ ﻱﺪﻨﺒﻤﻫ ﻱ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣ ﺖﺳﺍ ﺯﺎﺒﺘﺴﺑ ﻪﻔﻟﺆﻣ ﺮﻫ ﻩﺎﮔ . ﺽﺮﻓ
ﻢﻴﻨﻛ
{ , ,..., }
n
CXX X 12
ﻪﻋﻮﻤﺠﻣ ﻪﻔﻟﺆﻣ ﻱ ﻱﺪﻨﺒﻤﻫ ﻱﺎﻫ
X
ﻭ ﺪﺷﺎﺑ
X1
ﻭ
X
2
ﺪﻨﺷﺎﺑ ﻲﻫﺎﻨﺘﻣﺎﻧ .ﻲﻣ ﻒﻳﺮﻌﺗ ﻢﻴﻨﻛ:
,\
() ,
xXX
f
xxX
°
®
°
¯
1
1
1
,
x
ﻭ
,
g( ) ,\
xX
xxXX
°
®
°
¯
1
1
1
,
ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ
,()
f
gCX
ﻭ
()
F
f
gLCX
F
LC X
(
(
F
ﺎﻣﺍ ،
,()
F
f
gLCX
. ﺍﺮﻳﺯ
XX
2
ﻭ ﺖﺳﺍ ﺯﺎﺑ
||X!f
2
ﻭ
f
XS
2
. ﺮﮔﺍ ﻪﻛ ﺪﻴﻨﻛ ﻪﺟﻮﺗ
UX
2
ﻭ ﺪﺷﺎﺑ ﺯﺎﺑ
()UCozff
ﻱﺪﻨﺒﻤﻫ ﺎﺑ ﺾﻗﺎﻨﺗ
X
2
ﻲﻣ ﺪﺷﺎﺑ . ﺐﻴﺗﺮﺗ ﻦﻴﻤﻫ ﻪﺑ
g
XS
1
.ז
ﻪﻴﻀﻗ 4 : ﻩﺎﮔﺮﻫ
()IX f
ﻭ
\( )XIX
ﻥﺁ ،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫﺎﻧ ﻩﺎﮔ
()
F
LC X
ﺭﺩ ﻪﻘﻠﺣﺮﻳﺯ ﻱ
R
ﺯﺍ
()CX
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻪﻛ ﻱﺎﻫ
()CX
ﻲﻣ ﺖﺴﻴﻧ ﻝﻭﺍ ،ﺪﺷﺎﺑ .
ﺕﺎﺒﺛﺍ : ﻢﻴﻨﻛ ﺽﺮﻓ
A
ﻭ
B
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻭﺩ ﺯﺎﺒﺘﺴﺑ ﻱ
\( )XIX
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺪﻨﺷﺎﺑ
\( ) A BXIX
.ﻲﻣ ﻒﻳﺮﻌﺗ ﻢﻴﻨﻛǣ
f( ( ))AIX1
،
()
f
B
ﻭ
g(B ( ))IX1
،
g(A)
ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ
,()
f
gRCX
ﻭ
()
F
f
gLCX
F
LC X
(
(
F
ﺎﻣﺍ ،
,()()
FF
f
gCX LCX
.
ﺲﭘ
() ()
FF
LC X C X
ﺭﺩ
R
ﺖﺴﻴﻧ ﻝﻭﺍ .ז
ﺮﮔﺍ ﻪﻛ ﺖﺳﺍ ﺭﺎﻜﺷﺁ
Y
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻚﻳ ﻱ
X
ﺮﻫ ﻱﺍﺮﺑ ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺪﺷﺎﺑ
()
f
CX
،
|
Y
f
ﻥﺁ ،ﺪﺷﺎﺑ ﺖﺑﺎﺛ ﻩﺎﮔ
Y
ﻚﺗ ﺪﻳﺎﺑ ﻪﻄﻘﻧ ﺪﺷﺎﺑ ﻱﺍ. ﻩﺎﮔﺮﻫ ﺍﺮﻳﺯ
,yy Y
12
ﻭ
yyz
12
، ﻥﺁ ﻥﻮﭼ ﻩﺎﮔ
X
ﻊﺑﺎﺗ ،ﺖﺳﺍ ﻢﻈﻨﻣ ًﻼﻣﺎﻛ
()
f
CX
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ
() ()
f
yfyz
12
ﺩﺭﺍﺩ ﺩﻮﺟﻭ .ﻦﻳﺍ ﺭﺩ ﺎﺟ
ﻒﻳﺮﻌﺗ ﻚﻳ ﺯﺍ]9[ ﻡﻮﻬﻔﻣ ﻦﻳﺍ ﺎﺑ ﺮﺘﺸﻴﺑ ﻲﻳﺎﻨﺷﺁ ﺖﻬﺟ ﻪﺑ ﺪﻨﭼ ﻲﺗﺎﻜﻧ ﻩﺍﺮﻤﻫ ﻪﺑﻲﻣ ﺭﺩ ﻭ ﻢﻳﺭﻭﺁ
ﻪﻣﺍﺩﺍ ﺩﻮﻤﻧ ﻢﻴﻫﺍﻮﺧ ﻩﺩﺎﻔﺘﺳﺍ ﻥﺁ ﺯﺍ ﺚﺣﺎﺒﻣ ﻱ.
ﻒﻳﺮﻌﺗ 3 : ﻩﺎﮔﺮﻫ
Y
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻚﻳ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻱ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﻪﻋﻮﻤﺠﻣ ﻩﺎﮔ ﻪﻤﻫ ﻱ ﻱ
ﻊﺑﺍﻮﺗ
()
f
CX
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ
|
Y
f
ﺮﺒﺟﺮﻳﺯ ﻚﻳ ،ﺪﺷﺎﺑ ﺖﺑﺎﺛ
()CX
ﺎﺑ ﻪﻛ ﺖﺳﺍ
()CY
1
ﻲﻣ ﻩﺩﺍﺩ ﺶﻳﺎﻤﻧ ﺩﻮﺷ . ﻲﻌﻴﺒﻃ ﺭﻮﻃ ﻪﺑ
Y
ﻪﻘﻠﺣﺮﻳﺯ ﻪﺑ ﺖﺒﺴﻧ ﺍﺭ ﻱ
A
ﺯﺍ
()CX
،ﻢﻴﻳﻮﮔ ﺖﺑﺎﺛ
ﻩﺎﮔﺮﻫ
()ACY
1
.
ﻩﺭﺍﺰﮔ 8 : ﻩﺎﮔﺮﻫ
X
ﻭ ﻱژﻮﻟﻮﭘﻮﺗ ﻱﺎﻀﻓ ﻚﻳ
Y
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻚﻳ ﺪﻨﺒﻤﻫ ﻱ
X
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
() ()
c
CX CY
1
. ﻩﺎﮔﺮﻫ ،ﻩﮋﻳﻭ ﻪﺑ
\XY
ﻥﺁ ،ﺪﺷﺎﺑ ﺍﺭﺎﻤﺷ ﻩﺎﮔ
()ACY1
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
(X)
c
AC
.
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 96
ﺕﺎﺒﺛﺍ: ﻊﺑﺎﺗ ﺮﻫ ﺍﺭﺎﻜﺷﺁ
()
c
f
CX
ﻪﻋﻮﻤﺠﻣﺮﻳﺯ ﻱﻭﺭ ﺪﻨﺒﻤﻫ ﻱ
YX
ﺲﭘ ،ﺖﺳﺍ ﺖﺑﺎﺛ
() ()
c
CX CY
1
. ﻩﺎﮔﺮﻫ
\XY
ﻭ ﺍﺭﺎﻤﺷ
()
f
ACY
1
ﻥﺁ ،ﺪﺷﺎﺑ ﻩﺎﮔ
()fX d
.ז
ﺩ ﺭﻦﻳﺍ ،ﺎﺟ ﻪﺠﻴﺘﻧ ﻥﺩﻮﺑ ﻢﻈﻨﻣ ًﻼﻣﺎﻛ ﺱﺎﺳﺍ ﺮﺑ ﻪﻛ ﻱﺍ
X
ﻲﻣ ﺖﺑﺎﺛ ﻲﻧﺎﺳﺁ ﻪﺑ ،ﺩﻮﺷﻲﻣ ﻥﺎﻴﺑ ﻢﻴﻨﻛ.
ﻪﺠﻴﺘﻧ 2 : ﻱﺎﻀﻓﺮﻳﺯ ﻚﻳ ﻱﺍﺮﺑ
Y
ﺯﺍ
X
ﻢﻳﺭﺍﺩ ،
() C()CY X
()
C
())
()
C
()
1
. ﻩﻭﻼﻋ ﻪﺑ
() CY
()
C
()
)
1
ﺮﮔﺍ ﺎﻬﻨﺗ ﻭ ﺮﮔﺍ
Y
ﺭﺩ
X
ﺪﺷﺎﺑ ﻝﺎﮕﭼ.
ﺕﺎﺒﺛﺍ: ﻪﺟﻮﺗ ﺮﺧﺁ ﺖﻤﺴﻗ ﺕﺎﺒﺛﺍ ﻱﺍﺮﺑﻢﻳﺭﺍﺩ ﺮﮔﺍ ﻪﻛ
\xXY
ﻥﺁ ، ﻩﺎﮔ
()
f
CX
ﻮﺟﻭﺩ
ﻪﻛ ﺩﺭﺍﺩ
()
f
x
ﻭ
()
f
Y 1
.ﻲﻨﻌﻳ،
()CY
()
C
()
)
1
.ﻲﻣ ﻪﺠﻴﺘﻧ ﻦﻳﺍ ﻪﻛ ﺪﻫﺩ
YX
.
ﻩﺎﮔﺮﻫ ،ﺲﻜﻋﺮﺑ
YX
ﻭ
()
f
CY
1
ﻥﺁ ، ﻩﺎﮔ
()
f
Yc
ﻥﺁ ﺭﺩ ﻪﻛ
c
. ﻪﺠﻴﺘﻧﺭﺩ
f
c
ﺖﺳﺍ ﻡﺎﻤﺗ ﺕﺎﺒﺛﺍ ﻭ. ז
ﻗ ﻪﻴﻀ4 : ﻩﺎﮔﺮﻫ
|( )|IX f
ﻭ
()RCX
ﻭ
(\())RCXIX
1
ﻥﺁ ، ﻩﺎﮔ
()
F
LC X
ﺭﺩ
R
ﺖﺳﺍ ﻝﻭﺍ .
ﺕﺎﺒﺛﺍ: ﻢﻴﻨﻛ ﺽﺮﻓ
()
F
f
gLCX
F
LC X
((
F
ﻭ
,
f
gR
ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ،
(\())
f
XIX
ﺎﻳ
g( \ ( ))XIX
ﺍﺮﻳﺯ ،
(\())RCXIX
1
. ﺲﭘ
f
SX
ﺎﻳ
g
SX
. ﻦﻳﺍﺮﺑﺎﻨﺑ
()
F
f
LC X
ﺎﻳ
()
F
gLCX
،ﻲﻨﻌﻳ ؛
() ()
FF
LC X C X
ﺭﺩ
R
ﺖﺳﺍ ﻝﻭﺍ .ז
ﺭﺩ
[]7
ﻪﻘﻠﺣﺮﻳﺯ ، ﻱﺎﻫ
^`
() ():()
c
CX f CX fX d
`
ﻭ
^`
() ():()
F
CX f CX fX f
ﻩﺪﺷ ﻪﻌﻟﺎﻄﻣ ﻭ ﻲﻓﺮﻌﻣ ﺪﻧﺍ .
ﻪﺠﻴﺘﻧ3 : ﻩﺎﮔﺮﻫ
()IX f
ﻭ
\( )XIX
ﻥﺁ ،ﺪﺷﺎﺑ ﺪﻨﺒﻤﻫ ﻩﺎﮔ
()
F
LC X
ﺭﺩ
()
c
CX
ﻭ
()
F
CX
ﻲﻣ ﻝﻭﺍ ﺪﺷﺎﺑ.
ﺕﺎﺒﺛﺍ : ﺍﺮﻳﺯ
(), () ( \())
F
c
CX C X CX IX1
.ז
ﻲﻣ ﻥﺎﺸﻧ ﺪﻌﺑ ﻝﺎﺜﻣ ﻪﺠﻴﺘﻧ ﻪﻛ ﺪﻫﺩ ﻱﺍﺮﺑ ﻞﺒﻗ ﻱ
()CX
ﺖﺴﻴﻧ ﺖﺳﺭﺩ . ﻩﺎﮔﺮﻫ ﻊﻗﺍﻭ ﺭﺩ
()IX f
ﺭﺩ ﺰﮔﺮﻫ ﻭ ﺖﺳﺍ ﻖﺒﻄﻨﻣ ﻚﻴﺳﻼﻛ ﻞﻛﺎﺳ ﺮﺑ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ،ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣ
()CX
ﺩﻮﺑ ﺪﻫﺍﻮﺨﻧ ﻝﻭﺍ .ﻪﻴﻀﻗ ﻖﺒﻃ ﻱ3 ،ﺪﻌﺑ ﻝﺎﺜﻣ ﺭﺩ ،
()
F
LC X
ﻪﻘﻠﺣﺮﻳﺯ ﺮﮕﻳﺩ ﺭﺩ ﻱﺎﻫ
()CX
ﻪﻛ
ﻥﺍﻮﺗﺩﻮﺧ ﻞﻣﺎﺷ ﻱﺎﻫ
()CX
ﻲﻤﻧ ﻝﻭﺍ ﺰﻴﻧ ،ﺪﻨﺘﺴﻫ ﺪﺷﺎﺑ.
ﻝﺎﺜﻣ1: ﻢﻴﻨﻛ ﺽﺮﻓ
>@>@
^`
,,,,,Xn  
@
^
`
,,
,
@
^
n
@
^
,,
,
@
^
@
@
32 1 12
ﻱﺎﻀﻓﺮﻳﺯ ﻚﻳ ﻱژﻮﻟﻮﭘﻮﺗ ﺎﺑ
ﻭ ﻲﻟﻮﻤﻌﻣ
97 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
>@
>@
,,
() ,,
x
f
xx
 
°
® 
°
¯
,
x
132
32
ﻭ
>@
>@
,,
g( ) ,,
x
xx

°
®
°
¯
@
>
@
,,
>
x

>
>
11
1
ﺪﺷﺎﺑ . ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ
()
F
f
gLCX
F
LC X
((
F
ﺎﻣﺍ ،
,()
F
f
gLCX
.ﺲﭘ
()
F
LC X
ﻚﻳ
z
ﻪﻘﻠﺣﺮﻳﺯ ﺭﺩ ﻝﻭﺍ ﻝﺁﺪﻳﺍ ﻱﺎﻫ
()CX
ﻥﺍﻮﺗﺩﻮﺧ ﻪﻛ ﻱﺎﻫ
()CX
ﻲﻣ ﻞﻣﺎﺷ ﺍﺭ ﺖﺴﻴﻧ ،ﺪﻨﺷﺎﺑ.ﻲﻣ ﻢﻴﻧﺍﺩ
ﻥﺁ ،ﺪﺷﺎﺑ ﻝﻭﺪﻣ ﻞﻛﺎﺳ ﻞﻣﺎﺷ ﻲﻟﻭﺪﻣﺮﻳﺯ ﻩﺎﮔﺮﻫ ﻲﻠﻛ ﺭﻮﻃ ﻪﺑ ﻪﻛ ﻝﻭﺪﻣﺮﻳﺯ ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ ﻩﺎﮔ ﻱﺎﻫ
ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ ،ﻞﻛﺎﺳ ﻞﻣﺎﺷ ﻝﺍﺪﻳﺍ ﺮﻫ ،ﻩﮋﻳﻭ ﻪﺑ ؛ﺖﺳﺍ ﻲﺳﺎﺳﺍﻝﺁﺪﻳﺍ ،ﺖﺳﺍ ﻲﺳﺎﺳﺍ ﻱﺎﻫ]13[ ﺍﺭ
ﺪﻴﻨﻴﺒﺑ .ﻦﻳﺍ ﻪﺑ ﻪﺟﻮﺗ ﺎﺑ ﺎﻣﺍ ﻦﻳﺍ ﺭﺩ ﻭ ﺖﺴﻴﻧ ﻩﺩﺎﺳ ﻥﺁ ﻱﺮﺒﺟ ﺕﺎﺒﺛﺍ ﻪﻛ ﻱﻮﻗ ﺭﺍﺰﺑﺍ ﺎﺟ ﺍﺭ ﻊﺑﺍﻮﺗ ﺮﺗ ،ﻢﻳﺭﺍﺩ
ﻪﻨﻴﻣﺯ ﺭﺩ ﻲﺠﻳﺎﺘﻧ ﺯﺍ ﻩﺩﺎﻔﺘﺳﺍ ﺎﺑ ﻪﻣﺍﺩﺍ ﺭﺩ ﻱ
()CX
ﻩﺩﺎﺳ ﺕﺎﺒﺛﺍ ﻪﺑ ﺕﺭﺩﺎﺒﻣ ﻲﻣ ﻥﺁ ﻱﺍﺮﺑ ﻱﺮﺗ ﻢﻴﻨﻛ.
ﻩﺭﺍﺰﮔ9:
()
F
LC X
ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍﻝﺁﺪﻳﺍ ﺖﺳﺍ ﻲﺳﺎﺳﺍ ﻱﺎﻫ.
ﺕﺎﺒﺛﺍ : ﻥﻮﭼ
()
F
LC X
ﻚﻳ
z
-ﻝﺁﺪﻳﺍ ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ ﺕﺭﻮﺻ ﻪﺑ ﺲﭘ ،ﺖﺳﺍﻝﺁﺪﻳﺍ ،ﺖﺳﺍ ﻝﻭﺍ ﻱﺎﻫ
ﻪﻴﻀﻗ ﺮﺑﺎﻨﺑ ﺭﺩ ﻱﺍ]3[ ﻞﻣﺎﺷ ﻝﻭﺍ ﻝﺍﺪﻳﺍ ﺮﻫ ،
()
F
CX
ﺭﺩ
()CX
ﺖﺳﺍ ﻲﺳﺎﺳﺍ . ﻦﻳﺍﺮﺑﺎﻨﺑ
()
F
LC X
ﺯﺍ ﻲﻛﺍﺮﺘﺷﺍ ﺕﺭﻮﺻ ﻪﺑﻝﺁﺪﻳﺍ ﺖﺳﺍ ﻲﺳﺎﺳﺍ ﻱﺎﻫ .ז
ﺎﻀﻓ ﺩﺮﻔﻨﻣ ﻁﺎﻘﻧ ﺩﺍﺪﻌﺗ ﻲﺘﻗﻭ ﺍﺭ ﻚﻴﺳﻼﻛ ﻞﻛﺎﺳ ﻦﻴﻨﭽﻤﻫ ﻭ ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ ﻥﺩﻮﺑ ﻲﺳﺎﺳﺍ ﻪﻣﺍﺩﺍ ﺭﺩ
ﻲﻣ ﻲﺳﺭﺮﺑ ،ﺪﺷﺎﺑ ﻲﻫﺎﻨﺘﻣ ﻢﻴﻨﻛ .
ﻩﺭﺍﺰﮔ 10:
ﻩﺎﮔﺮﻫ
()IX f
ﻥﺁ ، ﻩﺎﮔ
()
F
LC X
ﭻﻴﻫ ﺭﺩ ﻪﻘﻠﺣﺮﻳﺯ ﺯﺍ ﻚﻳ ﻱﺎﻫ
()CX
ﻥﺁ ﻞﻣﺎﺷ
ﻲﻤﻧ ﻲﺳﺎﺳﺍ ﺪﺷﺎﺑ.
ﺕﺎﺒﺛﺍ :ﻢﻳﺮﻴﮔ
^`
( ) , ,...,
n
IX x x x
12
ﺲﭘ ،
() ()
n
Fi
i
CX eCX
¦
1
.ﺍﺬﻟ
() () ()
FF
LC X C X eC X
ﻥﺁ ﺭﺩ ﻪﻛ
()eCX
ﻪﻛ ﺖﺳﺍ ﻥﺍﻮﺗﺩﻮﺧ ﻚﻳ
() ( )Coz e I X
. ﺲﭘ
() () (1 )C()CX eCX e X
. ﻢﻳﺮﻴﮔ
()
F
CX R
ﻚﻳ
ﻪﻘﻠﺣﺮﻳﺯ ﻱ
()CX
ﺕﺭﻮﺻ ﻦﻳﺍ ﺭﺩ ،ﺪﺷﺎﺑ
(()(1)()) ReCX eCX R
. ﻥﻮﻨﻛﺍ
(1 ) ( ) REeCX
ﻚﻳﻝﺁﺪﻳﺍ ﺭﺩ
R
ﻪﻛ ﻱﺭﻮﻃ ﻪﺑ ﺖﺳﺍ
() I
F
CX
.،ﻲﻨﻌﻳ
() ()
FF
LC X C X
ﺭﺩ
R
ﺖﺴﻴﻧ ﻲﺳﺎﺳﺍ .ז
ﻲﻧﺍﺩﺭﺪﻗ ﻭ ﺮﻜﺸﺗ
ﻲﻠﻋﺪﻴﻣﺍ ﺮﺘﻛﺩ ﻱﺎﻗﺁ ﺏﺎﻨﺟ ﻲﻳﺎﻤﻨﻫﺍﺭ ﺖﺤﺗ ﻝﻭﺍ ﻩﺪﻨﺴﻳﻮﻧ ﻱﺮﺘﻛﺩ ﻪﻣﺎﻧ ﻥﺎﻳﺎﭘ ﺯﺍ ﻪﺘﻓﺮﮔﺮﺑ ﻪﻟﺎﻘﻣ ﻦﻳﺍ
ﻡﺮﻛ ﻲﻨﻬﺷﻲﻣ ﻩﺩﺍﺯﺪﺷﺎﺑ . ﺯﺍ ﻥﺎﮔﺪﻨﺴﻳﻮﻧ ﻥﺎﺸﻳﺍﻩﺪﻧﺯﺭﺍ ﻱﺎﻫﺩﺎﻬﻨﺸﻴﭘ ﻭ ﻡﻮﻬﻔﻣ ﻦﻳﺍ ﻲﻓﺮﻌﻣ ﺮﻃﺎﺧ ﻪﺑ
ﺍﺭ ﺮﻜﺸﺗ ﻝﺎﻤﻛ ﻥﺎﺸﻳﺍﺪﻧﺭﺍﺩ.
ﻥﺎﻄﻠﺳ ﻪﻴﻤﺳﻱﺭﺍﺪﻣﺎﻧ ﺩﺍﺩﺮﻬﻣ ﻭ ﺭﻮﭘ 98
ﻊﺟﺍﺮﻣ
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Henriksen, M. (2002).Topology related to rings of real-valued
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Recent Progress In General Topology II, eds M. Husek and J. Van Mill,
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99 ﻲﻌﺿﻮﻣ ﻞﻛﺎﺳ C(X)
Locally Socle of
C(X)
Somayeh Soltanpour, Mehrdad Namdari
Department of Mathematics, Shahid Chamran University, Ahvaz, Iran
Abstract
Let
()
F
CX
be the socle of
()CX
and
^`
() ():
Ff
LC X f C X S X
, where
f
S
is the union of all open subsets
U
in
X
such that
|\()|UZff
,
()
F
LC X
is called the locally socle of
()CX
and it is a z-ideal of
()CX
containing
()
F
CX
. We characterize spaces
X
for which the equality in the
relation
() () ()
FF
CX LCX CX
is hold. In fact, we show that
X
is an
almost discrete space if and only if
() ()
F
LC X C X
. We note that if
X
is an
infinite space, then
() ()
F
CX CX
()
(
(
. We also observe that
()IX f
if and
only if
() ()
FF
CX LCX
. Moreover, it is shown that if
()IX f
, then
()
F
LC X
is never essential in any subring of
()CX
, while
()
F
LC X
is an
intersection of essential ideals of
()CX
We determine the conditions such
that
()
F
LC X
is not prime in any subring of
()CX
which contains the
idempotent of
X
. We investigate the primness of
()
F
LC X
in some
subrings of
()CX
.
Keywords: Socle, z-ideal, Almost discrete space, Locally socle, Essential
ideal.
Mathematics Subject Classification (2010): 54C30, 54C40.
ResearchGate has not been able to resolve any citations for this publication.
On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)
Article
Full-text available
  • Oct 2015
p> Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|<\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered. We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective. We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$ are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$. </p
On C(X) modulo its socle
Article
Full-text available
  • Jan 2003
Let CF(X) denote the socle of C(X). It is shown that X is a P-space if and only if C(X) is a א0-selnnjective ring or equivalently, if and only if C(X)/CF(X) is א0-selfmjective. We also prove that X is an extremally disconnected P-space with only a finite number of isolated points if and only if C(X)/CF(X) is selfinjective. Consequently, if X is a P-space, then X is either an extremally disconnected space with at most a countable number of isolated points or both C(X) and C(X)/CF(X) have uncountable Goldie-dimensions. Prime ideals of C(X)/CF(X) are also studied.
Contracting the Socle in Rings of Continuous Functions
Article
Full-text available
Not every ring homomorphism contracts the socle of its codomain to an ideal contained in the socle of its domain. Rarer still does a homomorphism contract the socle to the socle. We find conditions on a frame homomorphism that ensure that the induced ring homomorphism contracts the socle of the codomain to an ideal contained in the socle of the domain. The surjective among these frame homomorphisms induce ring homomorphisms that contract the socle to the socle. These homomorphisms characterize P -frames as those L for which every frame homomorphism with L as domain is of this kind.
On the Intrinsic Topology and Some Related Ideals of C(X)
Article
Full-text available
  • Jan 1985
The above topology is defined and studied on C(X), the ring of real-valued continuous functions on a completely regular Hausdorff space X. The minimal ideals and the socle of C(X) are characterized via their corresponding z-filters. We observe that these ideals are z-ideals and X is discrete if and only if the socle of C(X) is a free ideal. It is also shown that for a class of topological spaces, containing all P-spaces, the family Ck(X) of functions with compact support is identical with the intersection of the free maximal ideals of C(X).
C(X) vs. C(X) modulo its socle
Article
Full-text available
Abstract. Let CF (X) be the socle of C(X). It is shown that each prime ideal in C(X)/CF (X) is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. intZ(h) = ∅). It is proved that dim(C(X)/CF (X)) ≥ dim C(X), where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential ideal E in C(X), we observe that E/CF (X) is essential in C(X)/CF (X) if and only if the set of isolated points of X is finite. Finally, we characterize topological spaces X for which the Jacobson radical of C(X)/CF (X) is zero, and as a consequence, we observe that the cardinality of a discrete space X is nonmeasurable if and only if υX, the realcompactification of X, is first countable
On the Functionally Countable Subalgebra of C(X)
Article
Full-text available
Let Cc(X) = {f ∈ C(X):f(X)is countable}. Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring Cc(X) is either Cc(X) or a semiprime (resp.prime) ideal in Cc(X). For an ideal I in Cc(X), it is observed that I and √I have the same largest zc-ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that Cc(X) ≅ Cc(Y). Consequently, if X has only countable number of components, then Cc(X) ≅ C(Y) for some zero-dimensional space Y. Spaces X for which Cc(X) is regular (called CP-spaces) are characterized both algebraically and topologically and it is shown that P-spaces and CP-spaces coincide when X is zero-dimensional. In contrast to C*(X), we observe that Cc(X) enjoys the algebraic properties of regularity, No-selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that Cc(X) is not isomorphic to any C(Y) is given.
Intersection of essential ideals in C(X)
Article
  • Jan 1997
The innite intersection of essential ideals in any ring may not be an essential ideal, this intersection may even be zero. By the topological characterization of the socle by Karamzadeh and Rostami (Proc. Amer. Math. Soc. 93 (1985), 179{184), and the topological characterization of essential ideals in Proposition 2.1, it is easy to see that every intersection of essential ideals of C(X) is an essential ideal if and only if the set of isolated points of X is dense in X. Motivated by this result in C(X), we study the essentiallity of the intersection of essential ideals for topological spaces which may have no isolated points. In particular, some important ideals CK(X )a nd C 1 ( X), which are the intersection of essential ideals, are studied further and their essentiallity is characterized. Finally a question raised by Karamzadeh and Rostami, namely when the socle of C(X) and the ideal of CK(X) coincide, is answered.
Topology related to rings of real-valued continuous functions. Where it has been and where it might be going, Recent Progress In General Topology II
  • Jan 2002
  • 553-556
  • M Henriksen
Henriksen, M. (2002).Topology related to rings of real-valued continuous functions. Where it has been and where it might be going, Recent Progress In General Topology II, eds M. Husek and J. Van Mill, (Elsevier Science), pp: 553-556.