UoAMathCorpus is a collection of representative mathematical expressions in MathML developed by the Speech and Accessibility Lab., National and Kapodistrian University of Athens, Greece. It includes a section with a mix of math and text, both for the English and Greek languages. UoAMathCorpus has been designed as a research tool for eAccessiblity.
You can view UoAMathCorpus using one of the following browsers: a) Mozilla Firefox, b) Internet Explorer 11 with the MathPlayer plugin, c) Google Chrome with the extension MathJax for Chrome
1−
1
2
+
1
3
−
1
4
+
1
5
−⋯=ln2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk
HiTmaalaaabaGaaGymaaqaaiaaikdaaaGaey4kaSYaaSaaaeaacaaI
XaaabaGaaG4maaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaai
abgUcaRmaalaaabaGaaGymaaqaaiaaiwdaaaGaeyOeI0IaeS47IWKa
eyypa0JaciiBaiaac6gacaaIYaaaaa@46F3@
|
1−
1
3
+
1
5
−
1
7
+
1
9
−⋯=
π
4
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk
HiTmaalaaabaGaaGymaaqaaiaaiodaaaGaey4kaSYaaSaaaeaacaaI
XaaabaGaaGynaaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI3aaaai
abgUcaRmaalaaabaGaaGymaaqaaiaaiMdaaaGaeyOeI0IaeS47IWKa
eyypa0ZaaSaaaeaacqaHapaCaeaacaaI0aaaaaaa@46E8@
|
1
2
−
1
5
+
1
8
−
1
11
+
1
14
−⋯=
π
3
9
+
1
3
ln2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGOmaaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI1aaa
aiabgUcaRmaalaaabaGaaGymaaqaaiaaiIdaaaGaeyOeI0YaaSaaae
aacaaIXaaabaGaaGymaiaaigdaaaGaey4kaSYaaSaaaeaacaaIXaaa
baGaaGymaiaaisdaaaGaeyOeI0IaeS47IWKaeyypa0ZaaSaaaeaacq
aHapaCdaGcaaqaaiaaiodaaSqabaaakeaacaaI5aaaaiabgUcaRmaa
laaabaGaaGymaaqaaiaaiodaaaGaciiBaiaac6gacaaIYaaaaa@4F15@
|
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+⋯=
π
2
6
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWc
aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaIYaaaaaaakiabgU
caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaikdaaaaa
aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG
OmaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa
CaaaleqabaGaaGOmaaaaaOqaaiaaiAdaaaaaaa@49DD@
|
1
1
4
+
1
2
4
+
1
3
4
+
1
4
4
+⋯=
π
4
90
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaaGccqGHRaWkdaWc
aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaaaakiabgU
caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaisdaaaaa
aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG
inaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa
CaaaleqabaGaaGinaaaaaOqaaiaaiMdacaaIWaaaaaaa@4AA4@
|
1
1
6
+
1
2
6
+
1
3
6
+
1
4
6
+⋯=
π
6
945
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaGOnaaaaaaGccqGHRaWkdaWc
aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI2aaaaaaakiabgU
caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaiAdaaaaa
aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG
OnaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa
CaaaleqabaGaaGOnaaaaaOqaaiaaiMdacaaI0aGaaGynaaaaaaa@4B71@
|
1
1
6
+
1
2
6
+
1
3
6
+
1
4
6
+⋯=
31
π
6
30,240
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaGOnaaaaaaGccqGHRaWkdaWc
aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI2aaaaaaakiabgU
caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaiAdaaaaa
aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG
OnaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiaaiodacaaI
XaGaeqiWda3aaWbaaSqabeaacaaI2aaaaaGcbaGaaG4maiaaicdaca
GGSaGaaGOmaiaaisdacaaIWaaaaaaa@4F04@
|
1
1
3
+
1
3
3
−
1
5
3
−
1
7
3
+⋯=
3
π
3
2
128
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaG4maaaaaaGccqGHRaWkdaWc
aaqaaiaaigdaaeaacaaIZaWaaWbaaSqabeaacaaIZaaaaaaakiabgk
HiTmaalaaabaGaaGymaaqaaiaaiwdadaahaaWcbeqaaiaaiodaaaaa
aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4namaaCaaaleqabaGaaG
4maaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiaaiodacqaH
apaCdaahaaWcbeqaaiaaiodaaaGcdaGcaaqaaiaaikdaaSqabaaake
aacaaIXaGaaGOmaiaaiIdaaaaaaa@4D15@
|
1
1×3
+
1
3×5
+
1
5×7
+
1
7×9
+⋯=
1
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymaiabgEna0kaaiodaaaGaey4kaSYaaSaaaeaacaaI
XaaabaGaaG4maiabgEna0kaaiwdaaaGaey4kaSYaaSaaaeaacaaIXa
aabaGaaGynaiabgEna0kaaiEdaaaGaey4kaSYaaSaaaeaacaaIXaaa
baGaaG4naiabgEna0kaaiMdaaaGaey4kaSIaeS47IWKaeyypa0ZaaS
aaaeaacaaIXaaabaGaaGOmaaaaaaa@4F7A@
|
1
1
2
×
3
2
+
1
3
2
×
5
2
+
1
5
2
×
7
2
+
1
7
2
×
9
2
+⋯=
π
2
−8
16
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaaaakiabgEna0kaaioda
daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaaba
GaaG4mamaaCaaaleqabaGaaGOmaaaakiabgEna0kaaiwdadaahaaWc
beqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGynam
aaCaaaleqabaGaaGOmaaaakiabgEna0kaaiEdadaahaaWcbeqaaiaa
ikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4namaaCaaale
qabaGaaGOmaaaakiabgEna0kaaiMdadaahaaWcbeqaaiaaikdaaaaa
aOGaey4kaSIaeS47IWKaeyypa0ZaaSaaaeaacqaHapaCdaahaaWcbe
qaaiaaikdaaaGccqGHsislcaaI4aaabaGaaGymaiaaiAdaaaaaaa@5B75@
|
β+
γ
δ
−ε
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey
4kaSYaaSaaaeaacqaHZoWzaeaacqaH0oazaaGaeyOeI0IaeqyTduga
aa@3E6A@
|
β+γ
δ+ε
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHYoGycqGHRaWkcqaHZoWzaeaacqaH0oazcqGHRaWkcqaH1oqzaaaa
aa@3E5F@
|
β−γ
δ
+ε
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHYoGycqGHsislcqaHZoWzaeaacqaH0oazaaGaey4kaSIaeqyTduga
aa@3E6A@
|
α+β+γ+δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey
4kaSIaeqOSdiMaey4kaSIaeq4SdCMaey4kaSIaeqiTdqgaaa@3F29@
|
α+
β
γ
+δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey
4kaSYaaSaaaeaacqaHYoGyaeaacqaHZoWzaaGaey4kaSIaeqiTdqga
aa@3E57@
|
α
β+γ+δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHXoqyaeaacqaHYoGycqGHRaWkcqaHZoWzcqGHRaWkcqaH0oazaaaa
aa@3E57@
|
α+β
γ+δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHXoqycqGHRaWkcqaHYoGyaeaacqaHZoWzcqGHRaWkcqaH0oazaaaa
aa@3E57@
|
(
α+β
)⋅(
γ+δ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaacqGHflY1daqadaqa
aiabeo7aNjabgUcaRiabes7aKbGaayjkaiaawMcaaaaa@43A3@
|
χ+
ψ
2
κ+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHhpWycqGHRaWkcqaHipqEdaahaaWcbeqaaiaaikdaaaaakeaacqaH
6oWAcqGHRaWkcaaIXaaaaaaa@3EB0@
|
χ+
ψ
2
κ
+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
4kaSYaaSaaaeaacqaHipqEdaahaaWcbeqaaiaaikdaaaaakeaacqaH
6oWAaaGaey4kaSIaaGymaaaa@3EB0@
|
β×(
γ+δ
)
ε
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
aHYoGycqGHxdaTdaqadaqaaiabeo7aNjabgUcaRiabes7aKbGaayjk
aiaawMcaaaqaaiabew7aLnaaCaaaleqabaGaaGOmaaaaaaaaaa@4206@
|
∂Ω=
Ω
¯
Ω
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaeu
yQdCLaeyypa0ZaaSaaaeaacuqHPoWvgaqeaaqaaiabfM6axbaaaaa@3D35@
|
υ
τ
(
χ,τ
)=∇(
Λ(
χ
)∇υ(
χ,τ
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS
baaSqaaiabes8a0bqabaGcdaqadaqaaiabeE8aJjaacYcacqaHepaD
aiaawIcacaGLPaaacqGH9aqpcqGHhis0daqadaqaaiabfU5amnaabm
aabaGaeq4XdmgacaGLOaGaayzkaaGaey4bIeTaeqyXdu3aaeWaaeaa
cqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaaacaGLOaGaayzkaa
aaaa@513A@
|
∇
Δ
υ(
χ
)=
∇
Κ,σ
υ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaS
baaSqaaiabfs5aebqabaGccqaHfpqDdaqadaqaaiabeE8aJbGaayjk
aiaawMcaaiabg2da9iabgEGirpaaBaaaleaacqqHAoWscaGGSaGaeq
4Wdmhabeaakiabew8a1baa@468F@
|
∂
0
υ
ν
=
υ
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW
baaSqabeaacaaIWaaaaOGaeqyXdu3aaSbaaSqaaiabe27aUbqabaGc
cqGH9aqpcqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaaa@40B4@
|
∂
2
υ
ν
=
1
κ
(
∂
1
υ
ν
−
∂
1
υ
ν−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW
baaSqabeaacaaIYaaaaOGaeqyXdu3aaSbaaSqaaiabe27aUbqabaGc
cqGH9aqpdaWcaaqaaiaaigdaaeaacqaH6oWAaaWaaeWaaeaacqGHci
ITdaahaaWcbeqaaiaaigdaaaGccqaHfpqDdaWgaaWcbaGaeqyVd4ga
beaakiabgkHiTiabgkGi2oaaCaaaleqabaGaaGymaaaakiabew8a1n
aaBaaaleaacqaH9oGBcqGHsislcaaIXaaabeaaaOGaayjkaiaawMca
aaaa@4FC0@
|
α‖
∇
Δ
η
Δ
ν
‖
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aau
WaaeaacqGHhis0daWgaaWcbaGaeuiLdqeabeaakiabeE7aOnaaDaaa
leaacqqHuoaraeaacqaH9oGBaaaakiaawMa7caGLkWoaaaa@42E0@
|
d
2
υ
d
λ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqyXduhabaGaamizaiabeU7a
SnaaCaaaleqabaGaaGOmaaaaaaaaaa@3D30@
|
∇(
Φ(
τ,υ(
τ
)
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aae
WaaeaacqqHMoGrdaqadaqaaiabes8a0jaacYcacqaHfpqDdaqadaqa
aiabes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawM
caaaaa@4393@
|
υ
¨
(
τ
)=∇Φ(
τ,υ(
τ
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyXduNbam
aadaqadaqaaiabes8a0bGaayjkaiaawMcaaiabg2da9iabgEGirlab
fA6agnaabmaabaGaeqiXdqNaaiilaiabew8a1naabmaabaGaeqiXdq
hacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@482F@
|
|
∇Η(
τ,χ
) |≤φ(
τ
)
| χ |
α
+γ(
τ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq
GHhis0cqqHxoasdaqadaqaaiabes8a0jaacYcacqaHhpWyaiaawIca
caGLPaaaaiaawEa7caGLiWoacqGHKjYOcqaHgpGAdaqadaqaaiabes
8a0bGaayjkaiaawMcaamaaemaabaGaeq4XdmgacaGLhWUaayjcSdWa
aWbaaSqabeaacqaHXoqyaaGccqGHRaWkcqaHZoWzdaqadaqaaiabes
8a0bGaayjkaiaawMcaaaaa@5507@
|
φ
′
(
υ
ν
)→0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOXdOMbau
aadaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaaGccaGLOaGa
ayzkaaGaeyOKH4QaaGimaaaa@3FA5@
|
‖
υ
˜
ν
‖≥|
〈
φ
′
(
υ
ν
),
υ
˜
ν
〉 |
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacu
aHfpqDgaacamaaBaaaleaacqaH9oGBaeqaaaGccaGLjWUaayPcSdGa
eyyzIm7aaqWaaeaadaaadaqaaiqbeA8aQzaafaWaaeWaaeaacqaHfp
qDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMcaaiaacYcacuaH
fpqDgaacamaaBaaaleaacqaH9oGBaeqaaaGccaGLPmIaayPkJaaaca
GLhWUaayjcSdaaaa@4F15@
|
∂
ξ
υ
1
∂
τ
ξ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
GHciITdaahaaWcbeqaaiabe67a4baakiabew8a1naaBaaaleaacaaI
XaaabeaaaOqaaiabgkGi2kabes8a0naaCaaaleqabaGaeqOVdGhaaa
aaaaa@413A@
|
ιΔυ−
υ
τ
φ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyUdKMaey
iLdqKaeqyXduNaeyOeI0IaeqyXdu3aaSbaaSqaaiabes8a0bqabaGc
cqaHgpGAaaa@4140@
|
Δ
ω
υ+κ×υ=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdq0aaS
baaSqaaiabeM8a3bqabaGccqaHfpqDcqGHRaWkcqaH6oWAcqGHxdaT
cqaHfpqDcqGH9aqpcaaIWaaaaa@435A@
|
χ=ρ×
χ
′
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
ypa0JaeqyWdiNaey41aqRafq4XdmMbauaaaaa@3E4E@
|
∂υ
∂ω
|
ω=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqGaaeaada
WcaaqaaiabgkGi2kabew8a1bqaaiabgkGi2kabeM8a3baaaiaawIa7
amaaBaaaleaacqaHjpWDcqGH9aqpcaaIWaaabeaaaaa@41B6@
|
∂
2
υ
∂
v
2
−
λ
2
υ=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
GHciITdaahaaWcbeqaaiaaikdaaaGccqaHfpqDaeaacqGHciITcaWG
2bWaaWbaaSqabeaacaaIYaaaaaaakiabgkHiTiabeU7aSnaaCaaale
qabaGaaGOmaaaakiabew8a1jabg2da9iaaicdaaaa@4496@
|
∂
∂
χ
ξ
γ
2
(
τ
)lnρ=
γ
2
(
τ
)
χ
ξ
ρ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
GHciITaeaacqGHciITcqaHhpWydaWgaaWcbaGaeqOVdGhabeaaaaGc
cqaHZoWzdaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiabes8a0bGaay
jkaiaawMcaaiGacYgacaGGUbGaeqyWdiNaeyypa0Jaeq4SdC2aaSba
aSqaaiaaikdaaeqaaOWaaeWaaeaacqaHepaDaiaawIcacaGLPaaada
WcaaqaaiabeE8aJnaaBaaaleaacqaH+oaEaeqaaaGcbaGaeqyWdi3a
aWbaaSqabeaacaaIYaaaaaaaaaa@5364@
|
1
ρ
sin
ωπ
ω
0
∂
∂ω
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaeqyWdihaaiGacohacaGGPbGaaiOBamaalaaabaGaeqyY
dCNaeqiWdahabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaakmaala
aabaGaeyOaIylabaGaeyOaIyRaeqyYdChaaaaa@465A@
|
υ(
0
)=
υ
′
(
1
)=
υ
″
(
0
)=
υ
‴
(
1
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aae
WaaeaacaaIWaaacaGLOaGaayzkaaGaeyypa0JafqyXduNbauaadaqa
daqaaiaaigdaaiaawIcacaGLPaaacqGH9aqpcuaHfpqDgaGbamaabm
aabaGaaGimaaGaayjkaiaawMcaaiabg2da9iqbew8a1zaasaWaaeWa
aeaacaaIXaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4B25@
|
υ
(
4
)
(
τ
)=φ(
τ,υ(
τ
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW
baaSqabeaadaqadaqaaiaaisdaaiaawIcacaGLPaaaaaGcdaqadaqa
aiabes8a0bGaayjkaiaawMcaaiabg2da9iabeA8aQnaabmaabaGaeq
iXdqNaaiilaiabew8a1naabmaabaGaeqiXdqhacaGLOaGaayzkaaaa
caGLOaGaayzkaaaaaa@4960@
|
∂
2
∂
u
2
Ξ(
τ,σ
)=−Γ(
τ,σ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
GHciITdaahaaWcbeqaaiaaikdaaaaakeaacqGHciITcaWG1bWaaWba
aSqabeaacaaIYaaaaaaakiabf65aynaabmaabaGaeqiXdqNaaiilai
abeo8aZbGaayjkaiaawMcaaiabg2da9iabgkHiTiabfo5ahnaabmaa
baGaeqiXdqNaaiilaiabeo8aZbGaayjkaiaawMcaaaaa@4C14@
|
|
‖ υ ‖ |=max{
‖ υ ‖,‖
υ
′
‖,‖
υ
″
‖ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada
qbdaqaaiabew8a1bGaayzcSlaawQa7aaGaay5bSlaawIa7aiabg2da
9iGac2gacaGGHbGaaiiEamaacmaabaWaauWaaeaacqaHfpqDaiaawM
a7caGLkWoacaGGSaWaauWaaeaacuaHfpqDgaqbaaGaayzcSlaawQa7
aiaacYcadaqbdaqaaiqbew8a1zaagaaacaGLjWUaayPcSdaacaGL7b
GaayzFaaaaaa@5455@
|
(
υ
*
)
′′
(
τ
)=
(
Τ
υ
*
)
′′
(
τ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
aHfpqDdaahaaWcbeqaaiaacQcaaaaakiaawIcacaGLPaaadaahaaWc
beqaaOGamai4gkdiIUGaaGzaVRGamai4gkdiIcaadaqadaqaaiabes
8a0bGaayjkaiaawMcaaiabg2da9maabmaabaGaeuiPdqLaeqyXdu3a
aWbaaSqabeaacaGGQaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaaki
adacUHYaIOliaaygW7kiadacUHYaIOaaWaaeWaaeaacqaHepaDaiaa
wIcacaGLPaaaaaa@56D5@
|
δ
Κ,σ
>0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS
baaSqaaiabfQ5aljaacYcacqaHdpWCaeqaaOGaeyOpa4JaaGimaaaa
@3D7A@
|
β
σ
Κ
≠0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aa0
baaSqaaiabeo8aZbqaaiabfQ5albaakiabgcMi5kaaicdaaaa@3D86@
|
χ
σ
=
∑
Κ∈Μ
β
σ
Κ
χ
Κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS
baaSqaaiabeo8aZbqabaGccqGH9aqpdaaeqbqaaiabek7aInaaDaaa
leaacqaHdpWCaeaacqqHAoWsaaGccqaHhpWydaWgaaWcbaGaeuOMdS
eabeaaaeaacqqHAoWscqGHiiIZcqqHCoqtaeqaniabggHiLdaaaa@4997@
|
{
χ
Κ
}
Κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq
aHhpWydaWgaaWcbaGaeuOMdSeabeaaaOGaay5Eaiaaw2haamaaBaaa
leaacqqHAoWsaeqaaaaa@3D27@
|
υ
Κ
=φ(
χ
Κ
),∀Κ∈Μ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS
baaSqaaiabfQ5albqabaGccqGH9aqpcqaHgpGAdaqadaqaaiabeE8a
JnaaBaaaleaacqqHAoWsaeqaaaGccaGLOaGaayzkaaGaaiilaiabgc
GiIiabfQ5aljabgIGiolabfY5anbaa@4701@
|
∂
1
υ
ν
=
υ
ν
−
υ
ν−1
κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW
baaSqabeaacaaIXaaaaOGaeqyXdu3aaWbaaSqabeaacqaH9oGBaaGc
cqGH9aqpdaWcaaqaaiabew8a1naaCaaaleqabaGaeqyVd4gaaOGaey
OeI0IaeqyXdu3aaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaGc
baGaeqOUdSgaaaaa@48CE@
|
Φ
Κ,σ
(
υ
)=
∑
σ
′
∈
Ε
κ
Α
σ
σ
′
(
υ
κ
−
υ
σ
′
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS
baaSqaaiabfQ5aljaacYcacqaHdpWCaeqaaOWaaeWaaeaacqaHfpqD
aiaawIcacaGLPaaacqGH9aqpdaaeqbqaaiabfg5abnaaCaaaleqaba
Gaeq4WdmNafq4WdmNbauaaaaGcdaqadaqaaiabew8a1naaBaaaleaa
cqaH6oWAaeqaaOGaeyOeI0IaeqyXdu3aaSbaaSqaaiqbeo8aZzaafa
aabeaaaOGaayjkaiaawMcaaaWcbaGafq4WdmNbauaacqGHiiIZcqqH
voqrdaWgaaadbaGaeqOUdSgabeaaaSqab0GaeyyeIuoaaaa@57C5@
|
α
| υ |
χ
2
≤
〈
υ,υ 〉
Φ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaq
WaaeaacqaHfpqDaiaawEa7caGLiWoadaqhaaWcbaGaeq4XdmgabaGa
aGOmaaaakiabgsMiJoaaamaabaGaeqyXduNaaiilaiabew8a1bGaay
zkJiaawQYiamaaBaaaleaacqqHMoGraeqaaaaa@4892@
|
Γ
6
=
δ
3
θ+
δ
7
2
θ
2
+
δ
5
2
θ+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS
baaSqaaiaaiAdaaeqaaOGaeyypa0JaeqiTdq2aaWbaaSqabeaacaaI
ZaaaaOGaeqiUdeNaey4kaSIaeqiTdq2aaWbaaSqabeaadaWcaaqaai
aaiEdaaeaacaaIYaaaaaaakiabeI7aXnaaCaaaleqabaGaaGOmaaaa
kiabgUcaRiabes7aKnaaCaaaleqabaWaaSaaaeaacaaI1aaabaGaaG
OmaaaaaaGccqaH4oqCcqGHRaWkcaaIXaaaaa@4C3A@
|
υ
¯
Δ
0
=
υ
Δ
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyXduNbae
badaqhaaWcbaGaeuiLdqeabaGaaGimaaaakiabg2da9iabew8a1naa
DaaaleaacqqHuoaraeaacaaIWaaaaaaa@3F47@
|
θ
2
υ
ν
=
κ
−1
(
∂
1
υ
ν
−
∂
1
υ
ν−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaW
baaSqabeaacaaIYaaaaOGaeqyXdu3aaWbaaSqabeaacqaH9oGBaaGc
cqGH9aqpcqaH6oWAdaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqada
qaaiabgkGi2oaaCaaaleqabaGaaGymaaaakiabew8a1naaCaaaleqa
baGaeqyVd4gaaOGaeyOeI0IaeyOaIy7aaWbaaSqabeaacaaIXaaaaO
GaeqyXdu3aaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaGccaGL
OaGaayzkaaaaaa@5127@
|
‖ χ ‖
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq
aHhpWyaiaawMa7caGLkWoaaaa@3AD5@
|
α‖
η
Δ
ν+1
‖χ≤
Γ
π
Μ
Δ
ν+1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aau
WaaeaacqaH3oaAdaqhaaWcbaGaeuiLdqeabaGaeqyVd4Maey4kaSIa
aGymaaaaaOGaayzcSlaawQa7aiabeE8aJjabgsMiJkabfo5ahnaaBa
aaleaacqaHapaCaeqaaOGaeuiNd00aa0baaSqaaiabfs5aebqaaiab
e27aUjabgUcaRiaaigdaaaaaaa@4E81@
|
Φ
2
(
τ,χ
)=|
sinωτ |
| χ |
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS
baaSqaaiaaikdaaeqaaOWaaeWaaeaacqaHepaDcaGGSaGaeq4Xdmga
caGLOaGaayzkaaGaeyypa0ZaaqWaaeaaciGGZbGaaiyAaiaac6gacq
aHjpWDcqaHepaDaiaawEa7caGLiWoadaabdaqaaiabeE8aJbGaay5b
SlaawIa7amaaCaaaleqabaGaaG4maaaaaaa@4E6D@
|
Η(
τ,χ
)=−
| χ |
1+α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4LdG0aae
WaaeaacqaHepaDcaGGSaGaeq4XdmgacaGLOaGaayzkaaGaeyypa0Ja
eyOeI0YaaqWaaeaacqaHhpWyaiaawEa7caGLiWoadaahaaWcbeqaai
aaigdacqGHRaWkcqaHXoqyaaaaaa@474E@
|
γ=
log
2λ
(
2μ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey
ypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaaikdacqaH7oaBaeqa
aOWaaeWaaeaacaaIYaGaeqiVd0gacaGLOaGaayzkaaaaaa@4215@
|
1
2
≤μ<
2
π
−
−1
λ
π
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGOmaaaacqGHKjYOcqaH8oqBcqGH8aapcaaIYaWaaWba
aSqabeaacqaHapaCdaahaaadbeqaaiabgkHiTaaaliabgkHiTiaaig
daaaGccqaH7oaBdaahaaWcbeqaaiabec8aWnaaCaaameqabaGaeyOe
I0caaaaaaaa@4624@
|
| υ |
π(
τ
)
>1⇒
| υ |
π(
τ
)
π
−
≤ρ(
υ
)≤
| υ |
π(
τ
)
π
+
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq
aHfpqDaiaawEa7caGLiWoadaWgaaWcbaGaeqiWda3aaeWaaeaacqaH
epaDaiaawIcacaGLPaaaaeqaaOGaeyOpa4JaaGymaiabgkDiEpaaem
aabaGaeqyXduhacaGLhWUaayjcSdWaa0baaSqaaiabec8aWnaabmaa
baGaeqiXdqhacaGLOaGaayzkaaaabaGaeqiWda3aaWbaaWqabeaacq
GHsislaaaaaOGaeyizImQaeqyWdi3aaeWaaeaacqaHfpqDaiaawIca
caGLPaaacqGHKjYOdaabdaqaaiabew8a1bGaay5bSlaawIa7amaaDa
aaleaacqaHapaCdaqadaqaaiabes8a0bGaayjkaiaawMcaaaqaaiab
ec8aWnaaCaaameqabaGaey4kaScaaaaaaaa@66AC@
|
Β
Τ
1,π(
τ
)
=
Β
˜
Τ
1,π(
τ
)
⊕
ℝ
Ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aa0
baaSqaaiabfs6aubqaaiaaigdacaGGSaGaeqiWda3aaeWaaeaacqaH
epaDaiaawIcacaGLPaaaaaGccqGH9aqpcuqHsoGqgaacamaaDaaale
aacqqHKoavaeaacaaIXaGaaiilaiabec8aWnaabmaabaGaeqiXdqha
caGLOaGaayzkaaaaaOGaeyyLIuSaeSyhHe6aaWbaaSqabeaacqqHDo
Gtaaaaaa@4F54@
|
υ(
χ,τ
)=
γ
1
(
τ
)+
γ
2
(
τ
)lnρ+
υ
0
(
χ,τ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aae
WaaeaacqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaGaeyypa0Ja
eq4SdC2aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacqaHepaDaiaawI
cacaGLPaaacqGHRaWkcqaHZoWzdaWgaaWcbaGaaGOmaaqabaGcdaqa
daqaaiabes8a0bGaayjkaiaawMcaaiGacYgacaGGUbGaeqyWdiNaey
4kaSIaeqyXdu3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacqaHhpWy
caGGSaGaeqiXdqhacaGLOaGaayzkaaaaaa@581A@
|
|
υ
1
|≤Γ
| χ |
Im
λ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq
aHfpqDdaWgaaWcbaGaaGymaaqabaaakiaawEa7caGLiWoacqGHKjYO
cqqHtoWrdaabdaqaaiabeE8aJbGaay5bSlaawIa7amaaCaaaleqaba
Gaciysaiaac2gacqaH7oaBdaWgaaadbaGaaGymaaqabaaaaaaa@4850@
|
Imλ(
τ
)=β+α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2
gacqaH7oaBdaqadaqaaiabes8a0bGaayjkaiaawMcaaiabg2da9iab
ek7aIjabgUcaRiabeg7aHbaa@41E1@
|
Im
λ
Ν
0
(
τ
0
)=−
β
1
+2μ+
λ
1
−
ν
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2
gacqaH7oaBdaWgaaWcbaGaeuyNd40aaSbaaWqaaiaaicdaaeqaaaWc
beaakmaabmaabaGaeqiXdq3aaSbaaSqaaiaaicdaaeqaaaGccaGLOa
GaayzkaaGaeyypa0JaeyOeI0IaeqOSdi2aaSbaaSqaaiaaigdaaeqa
aOGaey4kaSIaaGOmaiabeY7aTjabgUcaRiabeU7aSnaaBaaaleaaca
aIXaaabeaakiabgkHiTmaalaaabaGaeqyVd4gabaGaaGOmaaaaaaa@4F1B@
|
φ
^
=
κ
0
φ+
Λ
1
υ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOXdOMbaK
aacqGH9aqpcqaH6oWAdaWgaaWcbaGaaGimaaqabaGccqaHgpGAcqGH
RaWkcqqHBoatdaWgaaWcbaGaaGymaaqabaGccqaHfpqDaaa@4238@
|
Λ
Σ
δ
=
Α
δ
+
Ρ
δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdWKaeu
4Odm1aaSbaaSqaaiabes7aKbqabaGccqGH9aqpcqqHroqqdaWgaaWc
baGaeqiTdqgabeaakiabgUcaRiabfg6asnaaBaaaleaacqaH0oazae
qaaaaa@4340@
|
ι
(
−1
)
μ−1
Λυ−
υ
τ
=φ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyUdK2aae
WaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaH
8oqBcqGHsislcaaIXaaaaOGaeu4MdWKaeqyXduNaeyOeI0IaeqyXdu
3aaSbaaSqaaiabes8a0bqabaGccqGH9aqpcqaHgpGAaaa@491A@
|
χ
0
+χ+
χ
2
+
χ
3
+
χ
4
+
χ
5
+
χ
6
+…+
χ
ν−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW
baaSqabeaacaaIWaaaaOGaey4kaSIaeq4XdmMaey4kaSIaeq4Xdm2a
aWbaaSqabeaacaaIYaaaaOGaey4kaSIaeq4Xdm2aaWbaaSqabeaaca
aIZaaaaOGaey4kaSIaeq4Xdm2aaWbaaSqabeaacaaI0aaaaOGaey4k
aSIaeq4Xdm2aaWbaaSqabeaacaaI1aaaaOGaey4kaSIaeq4Xdm2aaW
baaSqabeaacaaI2aaaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2a
aWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaaa@5528@
|
(
χ+ψ+ζ
)
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
aHhpWycqGHRaWkcqaHipqEcqGHRaWkcqaH2oGEaiaawIcacaGLPaaa
daahaaWcbeqaaiaaikdaaaaaaa@3F6F@
|
(
χ+ψ
)
2
+ζ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
aHhpWycqGHRaWkcqaHipqEaiaawIcacaGLPaaadaahaaWcbeqaaiaa
ikdaaaGccqGHRaWkcqaH2oGEaaa@3F79@
|
χ+
(
ψ+ζ
)
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
4kaSYaaeWaaeaacqaHipqEcqGHRaWkcqaH2oGEaiaawIcacaGLPaaa
daahaaWcbeqaaiaaikdaaaaaaa@3F6F@
|
χ+ψ+
ζ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
4kaSIaeqiYdKNaey4kaSIaeqOTdO3aaWbaaSqabeaacaaIYaaaaaaa
@3DE6@
|
χ
0
+
χ
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS
baaSqaaiaaicdaaeqaaOGaey4kaSIaeq4Xdm2aaSbaaSqaaiaaigda
aeqaaaaa@3C1E@
|
χ
˙
+
χ
¨
+
χ
˜
+
χ
^
+
χ
″
+
χ
‴
+
χ
′
+
χ
*
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4XdmMbai
aacqGHRaWkcuaHhpWygaWaaiabgUcaRiqbeE8aJzaaiaGaey4kaSIa
fq4XdmMbaKaacqGHRaWkcuaHhpWygaGbaiabgUcaRiqbeE8aJzaasa
Gaey4kaSIafq4XdmMbauaacqGHRaWkcqaHhpWydaahaaWcbeqaaiaa
cQcaaaaaaa@4B1B@
|
α
1,1
+
α
2,2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaW
baaSqabeaacaaIXaGaaiilaiaaigdaaaGccqGHRaWkcqaHXoqydaah
aaWcbeqaaiaaikdacaGGSaGaaGOmaaaaaaa@3EC9@
|
〈
υ,υ 〉
φ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq
aHfpqDcaGGSaGaeqyXduhacaGLPmIaayPkJaWaaSbaaSqaaiabeA8a
Qbqabaaaaa@3DEE@
|
∑
κ∈ℝ
α
κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaacq
aHXoqydaWgaaWcbaGaeqOUdSgabeaaaeaacqaH6oWAcqGHiiIZcqWI
DesOaeqaniabggHiLdaaaa@4032@
|
∑
ι=1
Ν
α
2
+β+
γ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq
aHXoqydaahaaWcbeqaaiaaikdaaaaabaGaeqyUdKMaeyypa0JaaGym
aaqaaiabf25aobqdcqGHris5aOGaey4kaSIaeqOSdiMaey4kaSIaeq
4SdC2aaWbaaSqabeaacaaIYaaaaaaa@459E@
|
ε
ε
ε
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW
baaSqabeaacqaH1oqzdaahaaadbeqaaiabew7aLnaaCaaabeqaaiab
eE8aJbaaaaaaaaaa@3D20@
|
ε
ε
ε
χ
⋅
ε
ε
χ
⋅
ε
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW
baaSqabeaacqaH1oqzdaahaaadbeqaaiabew7aLnaaCaaabeqaaiab
eE8aJbaaaaaaaOGaeyyXICTaeqyTdu2aaWbaaSqabeaacqaH1oqzda
ahaaadbeqaaiabeE8aJbaaaaGccqGHflY1cqaH1oqzdaahaaWcbeqa
aiabeE8aJbaaaaa@4AB3@
|
ε
(
ε
ε
χ
+
ε
χ
+χ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW
baaSqabeaadaqadaqaaiabew7aLnaaCaaameqabaGaeqyTdu2aaWba
aeqabaGaeq4XdmgaaaaaliabgUcaRiabew7aLnaaCaaameqabaGaeq
4XdmgaaSGaey4kaSIaeq4XdmgacaGLOaGaayzkaaaaaaaa@45C6@
|
∑
ι=1
ν
α
ι
=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq
aHXoqydaWgaaWcbaGaeqyUdKgabeaaaeaacqaH5oqAcqGH9aqpcaaI
XaaabaGaeqyVd4ganiabggHiLdGccqGH9aqpcaaIXaaaaa@429B@
|
∑
ι=1
ν
α
ι
+
β
ι
=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq
aHXoqydaWgaaWcbaGaeqyUdKgabeaakiabgUcaRiabek7aInaaBaaa
leaacqaH5oqAaeqaaaqaaiabeM7aPjabg2da9iaaigdaaeaacqaH9o
GBa0GaeyyeIuoakiabg2da9iaaigdaaaa@4703@
|
1+χ+
χ
2
+
χ
3
+
χ
4
+…+
χ
ν−1
+…=
1
1+χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU
caRiabeE8aJjabgUcaRiabeE8aJnaaCaaaleqabaGaaGOmaaaakiab
gUcaRiabeE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiabeE8aJn
aaCaaaleqabaGaaGinaaaakiabgUcaRiablAciljabgUcaRiabeE8a
JnaaCaaaleqabaGaeqyVd4MaeyOeI0IaaGymaaaakiabgUcaRiablA
ciljabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHRaWkcqaHhpWy
aaaaaa@534F@
|
χ−
χ
2
2
+
χ
3
3
−
χ
4
4
+
χ
5
5
±…=log(
1+χ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
OeI0YaaSaaaeaacqaHhpWydaahaaWcbeqaaiaaikdaaaaakeaacaaI
YaaaaiabgUcaRmaalaaabaGaeq4Xdm2aaWbaaSqabeaacaaIZaaaaa
GcbaGaaG4maaaacqGHsisldaWcaaqaaiabeE8aJnaaCaaaleqabaGa
aGinaaaaaOqaaiaaisdaaaGaey4kaSYaaSaaaeaacqaHhpWydaahaa
WcbeqaaiaaiwdaaaaakeaacaaI1aaaaiabgglaXkablAciljabg2da
9iGacYgacaGGVbGaai4zamaabmaabaGaaGymaiabgUcaRiabeE8aJb
GaayjkaiaawMcaaaaa@54F3@
|
(
α+β
)
3
=
α
3
+3⋅
α
2
⋅β+3⋅α⋅
β
2
+
β
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaadaahaaWcbeqaaiaa
iodaaaGccqGH9aqpcqaHXoqydaahaaWcbeqaaiaaiodaaaGccqGHRa
WkcaaIZaGaeyyXICTaeqySde2aaWbaaSqabeaacaaIYaaaaOGaeyyX
ICTaeqOSdiMaey4kaSIaaG4maiabgwSixlabeg7aHjabgwSixlabek
7aInaaCaaaleqabaGaaGOmaaaakiabgUcaRiabek7aInaaCaaaleqa
baGaaG4maaaaaaa@5868@
|
χ
1
κ
+
χ
2
κ
+
χ
3
κ
+…+
χ
ν
κ
=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aa0
baaSqaaiaaigdaaeaacqaH6oWAaaGccqGHRaWkcqaHhpWydaqhaaWc
baGaaGOmaaqaaiabeQ7aRbaakiabgUcaRiabeE8aJnaaDaaaleaaca
aIZaaabaGaeqOUdSgaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2a
a0baaSqaaiabe27aUbqaaiabeQ7aRbaakiabg2da9iaaicdaaaa@4ECD@
|
χ
κ
1
+
χ
κ
2
+
χ
κ
3
+…
χ
κ
ν
=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS
baaSqaaiabeQ7aRnaaCaaameqabaGaaGymaaaaaSqabaGccqGHRaWk
cqaHhpWydaWgaaWcbaGaeqOUdS2aaWbaaWqabeaacaaIYaaaaaWcbe
aakiabgUcaRiabeE8aJnaaBaaaleaacqaH6oWAdaahaaadbeqaaiaa
iodaaaaaleqaaOGaey4kaSIaeSOjGSKaeq4Xdm2aaSbaaSqaaiabeQ
7aRnaaCaaameqabaGaeqyVd4gaaaWcbeaakiabg2da9iaaicdaaaa@4ECB@
|
χ
κ
1
+
χ
κ
2
+
χ
κ
3
+…+
χ
κ
ν
=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS
baaSqaaiabeQ7aRnaaBaaameaacaaIXaaabeaaaSqabaGccqGHRaWk
cqaHhpWydaWgaaWcbaGaeqOUdS2aaSbaaWqaaiaaikdaaeqaaaWcbe
aakiabgUcaRiabeE8aJnaaBaaaleaacqaH6oWAdaWgaaadbaGaaG4m
aaqabaaaleqaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2aaSbaaS
qaaiabeQ7aRnaaBaaameaacqaH9oGBaeqaaaWcbeaakiabg2da9iaa
icdaaaa@4FA9@
|
χ
2
ψ
≠
χ
2
ψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW
baaSqabeaacaaIYaWaaWbaaWqabeaacqaHipqEaaaaaOGaeyiyIKRa
eq4Xdm2aaWbaaSqabeaacaaIYaWaaWbaaWqabeaacqaHipqEaaaaaa
aa@4100@
|
α⋅
χ
2
+β⋅χ+γ=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey
yXICTaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqOSdiMa
eyyXICTaeq4XdmMaey4kaSIaeq4SdCMaeyypa0JaaGimaaaa@4757@
|
log3
(χ+ψ)
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+
gacaGGNbGaaG4maiaacIcacqaHhpWycqGHRaWkcqaHipqEcaGGPaWa
aWbaaSqabeaacaaIZaaaaaaa@402E@
|
log
2
χ≠2⋅logχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+
gacaGGNbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaeyiyIKRaaGOm
aiabgwSixlGacYgacaGGVbGaai4zaiabeE8aJbaa@44C5@
|
logχ
logα
=
log
α
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaci
GGSbGaai4BaiaacEgacqaHhpWyaeaaciGGSbGaai4BaiaacEgacqaH
XoqyaaGaeyypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiabeg7aHb
qabaGccqaHhpWyaaa@465F@
|
χ
¯
+χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4XdmMbae
bacqGHRaWkcqaHhpWyaaa@3A5F@
|
arg(
ζ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaack
hacaGGNbWaaeWaaeaacqaH2oGEaiaawIcacaGLPaaaaaa@3C05@
|
υ
(
0
)
=
∑
ζ=1
ν
ξ
ζ
(
1
)
ρ
ζ
(
υ
(
ι
)
)
=
∑
ζ=1
ν
ξ
ζ
(
2
)
ρ
ζ
(
υ
(
2
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW
baaSqabeaadaqadaqaaiaaicdaaiaawIcacaGLPaaaaaGccqGH9aqp
daaeWbqaaiabe67a4naaBaaaleaacqaH2oGEaeqaaOWaaWbaaSqabe
aadaqadaqaaiaaigdaaiaawIcacaGLPaaaaaGccqaHbpGCdaWgaaWc
baGaeqOTdOhabeaakmaabmaabaGaeqyXdu3aaWbaaSqabeaadaqada
qaaiabeM7aPbGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaaWcbaGa
eqOTdONaeyypa0JaaGymaaqaaiabe27aUbqdcqGHris5aOGaeyypa0
ZaaabCaeaacqaH+oaEdaWgaaWcbaGaeqOTdOhabeaakmaaCaaaleqa
baWaaeWaaeaacaaIYaaacaGLOaGaayzkaaaaaOGaeqyWdi3aaSbaaS
qaaiabeA7a6bqabaGcdaqadaqaaiabew8a1naaCaaaleqabaWaaeWa
aeaacaaIYaaacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaaaleaacq
aH2oGEcqGH9aqpcaaIXaaabaGaeqyVd4ganiabggHiLdaaaa@6B92@
|
| υ |
=
ορισμος
(
∑
κ=1
ν
υ
κ
2
)
1
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq
aHfpqDaiaawEa7caGLiWoadaWfGaqaaiabg2da9aWcbeqaaiabe+7a
Vjabeg8aYjabeM7aPjabeo8aZjabeY7aTjabe+7aVjabek8awbaakm
aabmaabaWaaabCaeaacqaHfpqDdaqhaaWcbaGaeqOUdSgabaGaaGOm
aaaaaeaacqaH6oWAcqGH9aqpcaaIXaaabaGaeqyVd4ganiabggHiLd
aakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaa
ikdaaaaaaaaa@5747@
|
∫
0
∞
|
ξ
ι
(
1
)
(
τ,χ
) |dχ
≤
Γ(
‖
φ
(
0
)
‖
Λ
1
(
ℝ
+
)
+
‖
α
(
0
)
‖
Λ
1
(
ℝ
+
)
+
‖
η
(
0
)
‖
Λ
1
(
ℝ
+
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXb
qaamaaemaabaGaeqOVdG3aaSbaaSqaaiabeM7aPbqabaGcdaahaaWc
beqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaaakmaabmaabaGaeq
iXdqNaaiilaiabeE8aJbGaayjkaiaawMcaaaGaay5bSlaawIa7aiaa
dsgacqaHhpWyaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakiabgs
MiJcqaaiabfo5ahnaabmaabaWaauWaaeaacqaHgpGAdaahaaWcbeqa
amaabmaabaGaaGimaaGaayjkaiaawMcaaaaaaOGaayzcSlaawQa7am
aaBaaaleaacqqHBoatdaahaaadbeqaaiaaigdaaaWcdaqadaqaaiab
l2riHoaaCaaameqabaGaey4kaScaaaWccaGLOaGaayzkaaaabeaaki
abgUcaRmaafmaabaGaeqySde2aaWbaaSqabeaadaqadaqaaiaaicda
aiaawIcacaGLPaaaaaaakiaawMa7caGLkWoadaWgaaWcbaGaeu4MdW
0aaWbaaWqabeaacaaIXaaaaSWaaeWaaeaacqWIDesOdaahaaadbeqa
aiabgUcaRaaaaSGaayjkaiaawMcaaaqabaGccqGHRaWkdaqbdaqaai
abeE7aOnaaCaaaleqabaWaaeWaaeaacaaIWaaacaGLOaGaayzkaaaa
aaGccaGLjWUaayPcSdWaaSbaaSqaaiabfU5amnaaCaaameqabaGaaG
ymaaaalmaabmaabaGaeSyhHe6aaWbaaWqabeaacqGHRaWkaaaaliaa
wIcacaGLPaaaaeqaaaGccaGLOaGaayzkaaaaaaa@7CA4@
|
Β
˜
1
(
Τ
)=
max
ι=1,…ν
max
ξ≠ι
sup
Γ
ξ
∫
Γ
ξ
|
υ
ι
|dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOKdiKbaG
aadaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabfs6aubGaayjkaiaa
wMcaaiabg2da9maaxababaGaciyBaiaacggacaGG4baaleaacqaH5o
qAcqGH9aqpcaaIXaGaaiilaiablAciljabe27aUbqabaGcdaWfqaqa
aiGac2gacaGGHbGaaiiEaaWcbaGaeqOVdGNaeyiyIKRaeqyUdKgabe
aakmaaxababaGaci4CaiaacwhacaGGWbaaleaacqqHtoWrdaWgaaad
baGaeqOVdGhabeaaaSqabaGcdaWdrbqaamaaemaabaGaeqyXdu3aaS
baaSqaaiabeM7aPbqabaaakiaawEa7caGLiWoacaWGKbGaeqiXdqha
leaacqqHtoWrdaWgaaadbaGaeqOVdGhabeaaaSqab0Gaey4kIipaaa
a@6484@
|
{
ε
λ
′′
}
λ=1
ν+μ−2κ
={
{
ε
ι
}
ι∉Α
,
{
ε
ξ
′
}
ξ∉
Α
′
}
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq
aH1oqzdaWgaaWcbaGaeq4UdWgabeaakmaaCaaaleqabaGccWaGGBOm
Gi6ccaaMb8UccWaGGBOmGikaaaGaay5Eaiaaw2haamaaDaaaleaacq
aH7oaBcqGH9aqpcaaIXaaabaGaeqyVd4Maey4kaSIaeqiVd0MaeyOe
I0IaaGOmaiabeQ7aRbaakiabg2da9maacmaabaWaaiWaaeaacqaH1o
qzdaWgaaWcbaGaeqyUdKgabeaaaOGaay5Eaiaaw2haamaaBaaaleaa
cqaH5oqAcqGHjiYZcqqHroqqaeqaaOGaaiilamaacmaabaGaeqyTdu
2aaSbaaSqaaiabe67a4bqabaGcdaahaaWcbeqaaOGamai4gkdiIcaa
aiaawUhacaGL9baadaWgaaWcbaGaeqOVdGNaeyycI8SafuyKdeKbau
aaaeqaaaGccaGL7bGaayzFaaaaaa@6B00@
|
∇
υ
ε
→
(ν+1,μ+1)
∇υ+
∑
κ=1
ν
∇
ψ
κ
υ
κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIeTaeq
yXdu3aaSbaaSqaaiabew7aLbqabaGcdaGdKaWcbaGaaiikaiabe27a
UjabgUcaRiaaigdacaGGSaGaeqiVd0Maey4kaSIaaGymaiaacMcaae
qakiaawkziaiabgEGirlabew8a1jabgUcaRmaaqahabaGaey4bIe9a
aSbaaSqaaiabeI8a5naaBaaameaacqaH6oWAaeqaaaWcbeaakiabew
8a1naaBaaaleaacqaH6oWAaeqaaaqaaiabeQ7aRjabg2da9iaaigda
aeaacqaH9oGBa0GaeyyeIuoaaaa@5A14@
|
ℑ
wsep,1
μ∼κ
={
(
ε,
{
ε
′
ξ
}
ξ=1
μ
)∈
ℑ
wsep
μ∼κ
:
ε
2
ε
μ
′
→0 }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHe8aa0
baaSqaaiaadEhacaWGZbGaamyzaiaadchacaGGSaGaaGymaaqaaiab
eY7aTjablYJi6iabeQ7aRbaakiabg2da9maacmaabaWaaeWaaeaacq
aH1oqzcaGGSaWaaiWaaeaacuaH1oqzgaqbamaaBaaaleaacqaH+oaE
aeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiabe67a4jabg2da9iaaig
daaeaacqaH8oqBaaaakiaawIcacaGLPaaacqGHiiIZcqGHresWdaqh
aaWcbaGaam4DaiaadohacaWGLbGaamiCaaqaaiabeY7aTjablYJi6i
abeQ7aRbaakiaacQdadaWcaaqaaiabew7aLnaaCaaaleqabaGaaGOm
aaaaaOqaaiabew7aLnaaBaaaleaacqaH8oqBaeqaaOWaaWbaaSqabe
aakiadacUHYaIOaaaaaiabgkziUkaaicdaaiaawUhacaGL9baaaaa@6C40@
|
ε
1
={
ε,{
ε
0,2
,
ε
0,5
,ε,
ε
1,2
,
ε
1,5
|
logε |
} }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS
baaSqaaiaaigdaaeqaaOGaeyypa0ZaaiWaaeaacqaH1oqzcaGGSaWa
aiWaaeaacqaH1oqzdaahaaWcbeqaaiaaicdacaGGSaGaaGOmaaaaki
aacYcacqaH1oqzdaahaaWcbeqaaiaaicdacaGGSaGaaGynaaaakiaa
cYcacqaH1oqzcaGGSaGaeqyTdu2aaWbaaSqabeaacaaIXaGaaiilai
aaikdaaaGccaGGSaWaaSaaaeaacqaH1oqzdaahaaWcbeqaaiaaigda
caGGSaGaaGynaaaaaOqaamaaemaabaGaciiBaiaac+gacaGGNbGaeq
yTdugacaGLhWUaayjcSdaaaaGaay5Eaiaaw2haaaGaay5Eaiaaw2ha
aaaa@5C76@
|
υ
μ
ν
+
ν
μ
ν
+κ
d
υ
μ
ν
dχ
=κ×
φ
μ
ν
+
υ
μ
ν−1
+
ν
μ
ν−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aa0
baaSqaaiabeY7aTbqaaiabe27aUbaakiabgUcaRiabe27aUnaaDaaa
leaacqaH8oqBaeaacqaH9oGBaaGccqGHRaWkcqaH6oWAdaWcaaqaai
aadsgacqaHfpqDdaqhaaWcbaGaeqiVd0gabaGaeqyVd4gaaaGcbaGa
amizaiabeE8aJbaacqGH9aqpcqaH6oWAcqGHxdaTcqaHgpGAdaqhaa
WcbaGaeqiVd0gabaGaeqyVd4gaaOGaey4kaSIaeqyXdu3aa0baaSqa
aiabeY7aTbqaaiabe27aUjabgkHiTiaaigdaaaGccqGHRaWkcqaH9o
GBdaqhaaWcbaGaeqiVd0gabaGaeqyVd4MaeyOeI0IaaGymaaaaaaa@673F@
|
∫
Ω
|
κ
μ
ξ
|
2
dχ+κ
∑
ν=1
ξ
|
κ
μ
ν
|
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada
abdaqaaiabeQ7aRnaaDaaaleaacqaH8oqBaeaacqaH+oaEaaaakiaa
wEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccaWGKbGaeq4XdmMaey
4kaSIaeqOUdS2aaabCaeaadaabdaqaaiabeQ7aRnaaDaaaleaacqaH
8oqBaeaacqaH9oGBaaaakiaawEa7caGLiWoadaahaaWcbeqaaiaaik
daaaaabaGaeqyVd4Maeyypa0JaaGymaaqaaiabe67a4bqdcqGHris5
aaWcbaGaeuyQdCfabeqdcqGHRiI8aaaa@594F@
|
sgn
∑
ξ∈Ι(
δ
1
,
δ
2
)
α
ξ
∫
Γ
ι
1
ε
ξ
dχ
=−sgn
∑
ξ∈Ι(
δ
1
,
δ
2
)
α
ξ
∫
Γ
ι
2
ε
ξ
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacE
gacaGGUbWaaabuaeaacqaHXoqydaWgaaWcbaGaeqOVdGhabeaakmaa
pefabaGaeqyTdu2aaSbaaSqaaiabe67a4bqabaGccaWGKbGaeq4Xdm
galeaacqqHtoWrdaWgaaadbaGaeqyUdK2aaSbaaeaacaaIXaaabeaa
aeqaaaWcbeqdcqGHRiI8aaWcbaGaeqOVdGNaeyicI4SaeuyMdK0aae
WaaeaacqaH0oazdaWgaaadbaGaaGymaaqabaWccaGGSaGaeqiTdq2a
aSbaaWqaaiaaikdaaeqaaaWccaGLOaGaayzkaaaabeqdcqGHris5aO
Gaeyypa0JaeyOeI0Iaci4CaiaacEgacaGGUbWaaabuaeaacqaHXoqy
daWgaaWcbaGaeqOVdGhabeaakmaapefabaGaeqyTdu2aaSbaaSqaai
abe67a4bqabaGccaWGKbGaeq4XdmgaleaacqqHtoWrdaWgaaadbaGa
eqyUdK2aaSbaaeaacaaIYaaabeaaaeqaaaWcbeqdcqGHRiI8aaWcba
GaeqOVdGNaeyicI4SaeuyMdK0aaeWaaeaacqaH0oazdaWgaaadbaGa
aGymaaqabaWccaGGSaGaeqiTdq2aaSbaaWqaaiaaikdaaeqaaaWcca
GLOaGaayzkaaaabeqdcqGHris5aaaa@7AC3@
|
φ(
Φ
σ
κ
)=
φ(
Φ
σ
κ
)−
φ
κ
(
Φ
σ
κ
)
︸
→
→
0
+
φ
κ
(
Φ
σ
κ
)
︸
χ
κ
→χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aae
WaaeaacqqHMoGrdaqhaaWcbaGaeq4WdmhabaGaeqOUdSgaaaGccaGL
OaGaayzkaaGaeyypa0ZaaGbaaeaacqaHgpGAdaqadaqaaiabfA6agn
aaDaaaleaacqaHdpWCaeaacqaH6oWAaaaakiaawIcacaGLPaaacqGH
sislcqaHgpGAdaWgaaWcbaGaeqOUdSgabeaakmaabmaabaGaeuOPdy
0aa0baaSqaaiabeo8aZbqaaiabeQ7aRbaaaOGaayjkaiaawMcaaaWc
baWaa4akaWqabeaacqGHsgIRaSGaayPKHaGaaGimaaGccaGL44pacq
GHRaWkdaagaaqaaiabeA8aQnaaBaaaleaacqaH6oWAaeqaaOWaaeWa
aeaacqqHMoGrdaqhaaWcbaGaeq4WdmhabaGaeqOUdSgaaaGccaGLOa
GaayzkaaaaleaacqaHhpWydaahaaadbeqaaiabeQ7aRbaaaOGaayjo
+dGaeyOKH4Qaeq4Xdmgaaa@6DCC@
|
φ(χ)=φ(α)+φ'(α)×(χ−α)+…+
φ
ν
(α)
ν
×
(χ−α)
ν
+
φ
(ν+1)
(
γ
)
ν+1
×
(
χ−α
)
(ν+1)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai
ikaiabeE8aJjaacMcacqGH9aqpcqaHgpGAcaGGOaGaeqySdeMaaiyk
aiabgUcaRiabeA8aQjaacEcacaGGOaGaeqySdeMaaiykaiabgEna0k
aacIcacqaHhpWycqGHsislcqaHXoqycaGGPaGaey4kaSIaeSOjGSKa
ey4kaSYaaSaaaeaacqaHgpGAdaahaaWcbeqaaiabe27aUbaakiaacI
cacqaHXoqycaGGPaaabaGaeqyVd4gaaiabgEna0kaacIcacqaHhpWy
cqGHsislcqaHXoqycaGGPaWaaWbaaSqabeaacqaH9oGBaaGccqGHRa
WkdaWcaaqaaiabeA8aQnaaCaaaleqabaGaaiikaiabe27aUjabgUca
RiaaigdacaGGPaaaaOWaaeWaaeaacqaHZoWzaiaawIcacaGLPaaaae
aacqaH9oGBcqGHRaWkcaaIXaaaaiabgEna0oaabmaabaGaeq4XdmMa
eyOeI0IaeqySdegacaGLOaGaayzkaaWaaWbaaSqabeaacaGGOaGaeq
yVd4Maey4kaSIaaGymaiaacMcaaaaaaa@7E3F@
|
∂
2
υ
∂χ∂ψ
=(
χ
2
−
ψ
2
)×{
τ×φ"(τ)+3×φ'(τ) }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq
GHciITdaahaaWcbeqaaiaaikdaaaGccqaHfpqDaeaacqGHciITcqaH
hpWycqGHciITcqaHipqEaaGaeyypa0ZaaeWaaeaacqaHhpWydaahaa
WcbeqaaiaaikdaaaGccqGHsislcqaHipqEdaahaaWcbeqaaiaaikda
aaaakiaawIcacaGLPaaacqGHxdaTdaGadaqaaiabes8a0jabgEna0k
abeA8aQjaackcacaGGOaGaeqiXdqNaaiykaiabgUcaRiaaiodacqGH
xdaTcqaHgpGAcaGGNaGaaiikaiabes8a0jaacMcaaiaawUhacaGL9b
aaaaa@6040@
|
log(
1+χ
)−log(
1−χ
)=log
1+χ
1−χ
=
∑
ι=1
∞
χ
2⋅ι−1
2⋅ι−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+
gacaGGNbWaaeWaaeaacaaIXaGaey4kaSIaeq4XdmgacaGLOaGaayzk
aaGaeyOeI0IaciiBaiaac+gacaGGNbWaaeWaaeaacaaIXaGaeyOeI0
Iaeq4XdmgacaGLOaGaayzkaaGaeyypa0JaciiBaiaac+gacaGGNbWa
aSaaaeaacaaIXaGaey4kaSIaeq4XdmgabaGaaGymaiabgkHiTiabeE
8aJbaacqGH9aqpdaaeWbqaamaalaaabaGaeq4Xdm2aaWbaaSqabeaa
caaIYaGaeyyXICTaeqyUdKMaeyOeI0IaaGymaaaaaOqaaiaaikdacq
GHflY1cqaH5oqAcqGHsislcaaIXaaaaaWcbaGaeqyUdKMaeyypa0Ja
aGymaaqaaiabg6HiLcqdcqGHris5aaaa@67C3@
|
Δ
χ
ν
ω=
∑
0≤λ≤ν
∑
κ
1
+…+
κ
ν
=λ
κ
1
+2
κ
2
+…+ν
κ
ν
=ν
κ
1
,…,
κ
ν
≥0
Δ
υ
λ
ω×
n!
(
Δ
χ
1
υ
)
κ
1
…
(
Δ
χ
ν
υ
)
κ
ν
κ
1
!
(1!)
κ
1
…
κ
ν
!
(
ν!
)
κ
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aa0
baaSqaaiabeE8aJbqaaiabe27aUbaakiabeM8a3jabg2da9maaqaba
baWaaabeaeaacqqHuoardaqhaaWcbaGaeqyXduhabaGaeq4UdWgaaO
GaeqyYdCNaey41aq7aaSaaaeaacaWGUbGaaiyiamaabmaabaGaeuiL
dq0aa0baaSqaaiabeE8aJbqaaiaaigdaaaGccqaHfpqDaiaawIcaca
GLPaaadaahaaWcbeqaaiabeQ7aRnaaCaaameqabaGaaGymaaaaaaGc
cqWIMaYsdaqadaqaaiabfs5aenaaDaaaleaacqaHhpWyaeaacqaH9o
GBaaGccqaHfpqDaiaawIcacaGLPaaadaahaaWcbeqaaiabeQ7aRnaa
BaaameaacqaH9oGBaeqaaaaaaOqaaiabeQ7aRnaaBaaaleaacaaIXa
aabeaakiaacgcacaGGOaGaaGymaiaacgcacaGGPaWaaWbaaSqabeaa
cqaH6oWAdaWgaaadbaGaaGymaaqabaaaaOGaeSOjGSKaeqOUdS2aaS
baaSqaaiabe27aUbqabaGccaGGHaWaaeWaaeaacqaH9oGBcaGGHaaa
caGLOaGaayzkaaWaaWbaaSqabeaacqaH6oWAdaWgaaadbaGaeqyVd4
gabeaaaaaaaaWceaqabeaacqaH6oWAdaWgaaadbaGaaGymaaqabaWc
cqGHRaWkcqWIMaYscqGHRaWkcqaH6oWAdaWgaaadbaGaeqyVd4gabe
aaliabg2da9iabeU7aSbqaaiabeQ7aRnaaBaaameaacaaIXaaabeaa
liabgUcaRiaaikdacqaH6oWAdaWgaaadbaGaaGOmaaqabaWccqGHRa
WkcqWIMaYscqGHRaWkcqaH9oGBcqaH6oWAdaWgaaadbaGaeqyVd4ga
beaaliabg2da9iabe27aUbqaaiabeQ7aRnaaBaaameaacaaIXaaabe
aaliaacYcacqWIMaYscaGGSaGaeqOUdS2aaSbaaWqaaiabe27aUbqa
baWccqGHLjYScaaIWaaaaeqaniabggHiLdaaleaacaaIWaGaeyizIm
Qaeq4UdWMaeyizImQaeqyVd4gabeqdcqGHris5aaaa@AA4F@
|
∑
1≤ι≤ν
α
ι
=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq
aHXoqydaWgaaWcbaGaeqyUdKgabeaaaeaacaaIXaGaeyizImQaeqyU
dKMaeyizImQaeqyVd4gabeqdcqGHris5aOGaeyypa0JaaGymaaaa@44A0@
|
∑
κ=1
∞
(
−1
)
κ+1
(
κ+1
)×ln(κ+1)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada
WcaaqaamaabmaabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa
leqabaGaeqOUdSMaey4kaSIaaGymaaaaaOqaamaabmaabaGaeqOUdS
Maey4kaSIaaGymaaGaayjkaiaawMcaaiabgEna0kGacYgacaGGUbGa
aiikaiabeQ7aRjabgUcaRiaaigdacaGGPaaaaaWcbaGaeqOUdSMaey
ypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaaa@515F@
|
∑
κ=11
28
(κ−10)×sin[
π
κ−10
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca
GGOaGaeqOUdSMaeyOeI0IaaGymaiaaicdacaGGPaGaey41aqRaci4C
aiaacMgacaGGUbWaamWaaeaadaWcaaqaaiabec8aWbqaaiabeQ7aRj
abgkHiTiaaigdacaaIWaaaaaGaay5waiaaw2faaaWcbaGaeqOUdSMa
eyypa0JaaGymaiaaigdaaeaacaaIYaGaaGioaaqdcqGHris5aaaa@5014@
|
∫
0
Τ
Φ
2
(
τ,χ
)dτ
≥Δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq
qHMoGrdaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiabes8a0jaacYca
cqaHhpWyaiaawIcacaGLPaaacaWGKbGaeqiXdqhaleaacaaIWaaaba
GaeuiPdqfaniabgUIiYdGccqGHLjYScqqHuoaraaa@48A5@
|
∫
0
Τ
η(
τ
)dτ
=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq
aH3oaAdaqadaqaaiabes8a0bGaayjkaiaawMcaaiaadsgacqaHepaD
aSqaaiaaicdaaeaacqqHKoava0Gaey4kIipakiabg2da9iaaicdaaa
a@4412@
|
〈
υ
Δ,υ
0
〉
Φ
=−
∑
Κ∈Μ
υ
Κ
∫
Κ
Δ
υ
0
(
χ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq
aHfpqDdaqhaaWcbaGaeuiLdqKaaiilaiabew8a1bqaaiaaicdaaaaa
kiaawMYicaGLQmcadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iabgk
HiTmaaqafabaGaeqyXdu3aaSbaaSqaaiabfQ5albqabaGcdaWdrbqa
aiabgs5aejabew8a1naaCaaaleqabaGaaGimaaaakmaabmaabaGaeq
4XdmgacaGLOaGaayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqd
cqGHRiI8aaWcbaGaeuOMdSKaeyicI4SaeuiNd0eabeqdcqGHris5aa
aa@59BC@
|
〈
υ
τ
,υ 〉+
∫
Ω
∇υ(
χ,τ
)×∇υ(
χ
)dχ
=
∫
Ω
φ(
χ,τ
)υ(
χ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaaada
qaaiabew8a1naaBaaaleaacqaHepaDaeqaaOGaaiilaiabew8a1bGa
ayzkJiaawQYiaiabgUcaRmaapefabaGaey4bIeTaeqyXdu3aaeWaae
aacqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaGaey41aqRaey4b
IeTaeqyXdu3aaeWaaeaacqaHhpWyaiaawIcacaGLPaaacaWGKbGaeq
4XdmgaleaacqGHPoWvaeqaniabgUIiYdaakeaacqGH9aqpdaWdrbqa
aiabeA8aQnaabmaabaGaeq4XdmMaaiilaiabes8a0bGaayjkaiaawM
caaiabew8a1naabmaabaGaeq4XdmgacaGLOaGaayzkaaGaamizaiab
eE8aJbWcbaGaeyyQdCfabeqdcqGHRiI8aaaaaa@6B09@
|
α(
β,υ
)=
∫
Ω
∇β(
χ
)⋅∇υ(
χ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae
WaaeaacqaHYoGycaGGSaGaeqyXduhacaGLOaGaayzkaaGaeyypa0Za
a8quaeaacqGHhis0cqaHYoGydaqadaqaaiabeE8aJbGaayjkaiaawM
caaiabgwSixlabgEGirlabew8a1naabmaabaGaeq4XdmgacaGLOaGa
ayzkaaGaamizaiabeE8aJbWcbaGaeuyQdCfabeqdcqGHRiI8aaaa@53F3@
|
∫
Ω
φ(
χ,τ
)υ(
χ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq
aHgpGAdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaawIcacaGLPaaa
cqaHfpqDdaqadaqaaiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHhp
WyaSqaaiabfM6axbqab0Gaey4kIipaaaa@48E8@
|
∫
Κ
(
υ(
χ,
τ
ν+1
)−υ(
χ,
τ
ν
)
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada
qadaqaaiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBaaa
leaacqaH9oGBcqGHRaWkcaaIXaaabeaaaOGaayjkaiaawMcaaiabgk
HiTiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBaaaleaa
cqaH9oGBaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGaamizai
abeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aaaa@533B@
|
−
∫
Κ
Δ
υ
0
(
χ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Yaa8
quaeaacqGHuoarcqaHfpqDdaahaaWcbeqaaiaaicdaaaGcdaqadaqa
aiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHhpWyaSqaaiabfQ5alb
qab0Gaey4kIipaaaa@44A0@
|
Λ
Κ,
σ
″
=
∫
Κ,
σ
″
Ιdχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS
baaSqaaiabfQ5aljaacYcacuaHdpWCgaGbaaqabaGccqGH9aqpdaWd
rbqaaiabfM5ajjaadsgacqaHhpWyaSqaaiabfQ5aljaacYcacuaHdp
WCgaGbaaqab0Gaey4kIipaaaa@46E8@
|
∫
τ
ν
τ
ν+1
Δυ(
χ,τ
)dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq
GHuoarcqaHfpqDdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaawIca
caGLPaaacaWGKbGaeqiXdqhaleaacqaHepaDdaWgaaadbaGaeqyVd4
gabeaaaSqaaiabes8a0naaBaaameaacqaH9oGBcqGHRaWkcaaIXaaa
beaaa0Gaey4kIipaaaa@4CED@
|
∫
0
Τ
|
υ(
τ
) |
2
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
abdaqaaiabew8a1naabmaabaGaeqiXdqhacaGLOaGaayzkaaaacaGL
hWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaamizaiabes8a0bWcba
GaaGimaaqaaiabfs6aubqdcqGHRiI8aaaa@4678@
|
∫
0
Τ
|
υ
˙
(
τ
) |
2
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
abdaqaaiqbew8a1zaacaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaa
aiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccaWGKbGaeqiXdq
haleaacaaIWaaabaGaeuiPdqfaniabgUIiYdaaaa@4681@
|
∫
0
Τ
υ
′
φdτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacu
aHfpqDgaqbaiabeA8aQjaadsgacqaHepaDaSqaaiaaicdaaeaacqqH
Koava0Gaey4kIipaaaa@40DE@
|
∫
0
Τ
|
υ
˜
(
τ
) |
π
−
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
abdaqaaiqbew8a1zaaiaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaa
aiaawEa7caGLiWoadaahaaWcbeqaaiabec8aWnaaCaaameqabaGaey
OeI0caaaaakiaadsgacqaHepaDaSqaaiaaicdaaeaacqqHKoava0Ga
ey4kIipaaaa@48A3@
|
∫
0
Τ
Φ(
τ,χ
)dτ
→∞
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq
qHMoGrdaqadaqaaiabes8a0jaacYcacqaHhpWyaiaawIcacaGLPaaa
caWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgUIiYdGccq
GHsgIRcqGHEisPaaa@47E5@
|
∫
0
Τ
∫
0
1
γ(
τ
)|
υ(
τ
) |dσ
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
WdXbqaaiabeo7aNnaabmaabaGaeqiXdqhacaGLOaGaayzkaaWaaqWa
aeaacqaHfpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSl
aawIa7aiaadsgacqaHdpWCaSqaaiaaicdaaeaacaaIXaaaniabgUIi
YdGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgUIiYd
aaaa@510E@
|
1
μ
∫
0
Τ
Φ
1
(
τ,λ
υ
¯
)dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaeqiVd0gaamaapehabaGaeuOPdy0aaSbaaSqaaiaaigda
aeqaaOWaaeWaaeaacqaHepaDcaGGSaGaeq4UdWMafqyXduNbaebaai
aawIcacaGLPaaacaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfa
niabgUIiYdaaaa@49CB@
|
∫
Ω
ρ
−2
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq
aHbpGCdaahaaWcbeqaaiabgkHiTiaaikdaaaGccaWGKbGaeq4Xdmga
leaacqqHPoWvaeqaniabgUIiYdaaaa@400F@
|
1
6
∫
σ
1
σ(
3−
σ
2
)η(
σ
)α(
σ
)dσ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGOnaaaadaWdXbqaaiabeo8aZnaabmaabaGaaG4maiab
gkHiTiabeo8aZnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaai
abeE7aOnaabmaabaGaeq4WdmhacaGLOaGaayzkaaGaeqySde2aaeWa
aeaacqaHdpWCaiaawIcacaGLPaaacaWGKbGaeq4WdmhaleaacqaHdp
WCaeaacaaIXaaaniabgUIiYdaaaa@50A4@
|
λ
max
0≤τ≤1
∫
0
1
Γ(
τ,σ
)η(
σ
)ψ(
σ,β
)dσ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaC
beaeaaciGGTbGaaiyyaiaacIhaaSqaaiaaicdacqGHKjYOcqaHepaD
cqGHKjYOcaaIXaaabeaakmaapehabaGaeu4KdC0aaeWaaeaacqaHep
aDcaGGSaGaeq4WdmhacaGLOaGaayzkaaGaeq4TdG2aaeWaaeaacqaH
dpWCaiaawIcacaGLPaaacqaHipqEdaqadaqaaiabeo8aZjaacYcacq
aHYoGyaiaawIcacaGLPaaacaWGKbGaeq4WdmhaleaacaaIWaaabaGa
aGymaaqdcqGHRiI8aaaa@5B7B@
|
1
2
λ
sup
υ∈Υ(
ρ
)
∫
0
1
|
υ(
σ
) |dσ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaaGOmaaaacqaH7oaBdaWfqaqaaiGacohacaGG1bGaaiiC
aaWcbaGaeqyXduNaeyicI4SaeuyPdu1aaeWaaeaacqaHbpGCaiaawI
cacaGLPaaaaeqaaOWaa8qCaeaadaabdaqaaiabew8a1naabmaabaGa
eq4WdmhacaGLOaGaayzkaaaacaGLhWUaayjcSdGaamizaiabeo8aZb
WcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaa@5336@
|
∫
0
1
∂
∂τ
Ξ(
τ,σ
)υ(
σ
)dσ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
WcaaqaaiabgkGi2cqaaiabgkGi2kabes8a0baacqqHEoawdaqadaqa
aiabes8a0jaacYcacqaHdpWCaiaawIcacaGLPaaacqaHfpqDdaqada
qaaiabeo8aZbGaayjkaiaawMcaaiaadsgacqaHdpWCaSqaaiaaicda
aeaacaaIXaaaniabgUIiYdaaaa@4D7A@
|
∫
τ
1
‖
υ
″
‖σdσ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
qbdaqaaiqbew8a1zaagaaacaGLjWUaayPcSdGaeq4WdmNaamizaiab
eo8aZbWcbaGaeqiXdqhabaGaaGymaaqdcqGHRiI8aaaa@444A@
|
∫
Η
η
(
χ+2
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq
aH3oaAaSqaaiabfE5aibqab0Gaey4kIipakmaabmaabaGaeq4XdmMa
ey4kaSIaaGOmaaGaayjkaiaawMcaaiaadsgacqaHhpWyaaa@42E2@
|
∫
1
∞
ε
χ
2
−χ−1
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq
aH1oqzdaahaaWcbeqaaiabeE8aJnaaCaaameqabaGaaGOmaaaaliab
gkHiTiabeE8aJjabgkHiTiaaigdaaaGccaWGKbGaeq4Xdmgaleaaca
aIXaaabaGaeyOhIukaniabgUIiYdaaaa@45C2@
|
∫
dχ
χ
=logχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada
WcaaqaaiaadsgacqaHhpWyaeaacqaHhpWyaaaaleqabeqdcqGHRiI8
aOGaeyypa0JaciiBaiaac+gacaGGNbGaeq4Xdmgaaa@41F0@
|
∫
τ
ν
τ
ν+1
∫
Κ
υ
τ
(
χ,τ
)dχ
dτ
−
∑
σ∈
Ε
Κ
∫
τ
ν
τ
ν+1
∫
σ
∇υ(
χ,τ
)×
ν
Κ,σ
(
χ
)
δ
γ
(
χ
)dτ
=
∫
τ
ν
τ
ν+1
∫
Κ
φ(
χ,τ
)dχ
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXb
qaamaapefabaGaeqyXdu3aaSbaaSqaaiabes8a0bqabaGcdaqadaqa
aiabeE8aJjaacYcacqaHepaDaiaawIcacaGLPaaacaWGKbGaeq4Xdm
galeaacqqHAoWsaeqaniabgUIiYdGccaWGKbGaeqiXdqhaleaacqaH
epaDdaWgaaadbaGaeqyVd4gabeaaaSqaaiabes8a0naaBaaameaacq
aH9oGBcqGHRaWkcaaIXaaabeaaa0Gaey4kIipakiabgkHiTaqaamaa
qafabaWaa8qCaeaadaWdrbqaaiabgEGirlabew8a1naabmaabaGaeq
4XdmMaaiilaiabes8a0bGaayjkaiaawMcaaiabgEna0kabe27aUnaa
BaaaleaacqqHAoWscaGGSaGaeq4WdmhabeaakmaabmaabaGaeq4Xdm
gacaGLOaGaayzkaaGaeqiTdq2aaSbaaSqaaiabeo7aNbqabaGcdaqa
daqaaiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHepaDaSqaaiabeo
8aZbqab0Gaey4kIipaaSqaaiabes8a0naaBaaameaacqaH9oGBaeqa
aaWcbaGaeqiXdq3aaSbaaWqaaiabe27aUjabgUcaRiaaigdaaeqaaa
qdcqGHRiI8aaWcbaGaeq4WdmNaeyicI4SaeuyLdu0aaSbaaWqaaiab
fQ5albqabaaaleqaniabggHiLdGccqGH9aqpaeaadaWdXbqaamaape
fabaGaeqOXdO2aaeWaaeaacqaHhpWycaGGSaGaeqiXdqhacaGLOaGa
ayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aOGaam
izaiabes8a0bWcbaGaeqiXdq3aaSbaaWqaaiabe27aUbqabaaaleaa
cqaHepaDdaWgaaadbaGaeqyVd4Maey4kaSIaaGymaaqabaaaniabgU
IiYdaaaaa@A76F@
|
∂
2
(
Δ
α
υ(
χ,
τ
ν
)
)=
1
κ
2
∫
τ
ν−1
τ
ν
∫
τ−η
τ
(
Δ
α
υ
)
ττ
(
χ,τ
)dσ
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW
baaSqabeaacaaIYaaaaOWaaeWaaeaacqqHuoardaahaaWcbeqaaiab
eg7aHbaakiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBa
aaleaacqaH9oGBaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGa
eyypa0ZaaSaaaeaacaaIXaaabaGaeqOUdS2aaWbaaSqabeaacaaIYa
aaaaaakmaapehabaWaa8qCaeaadaqadaqaaiabfs5aenaaCaaaleqa
baGaeqySdegaaOGaeqyXduhacaGLOaGaayzkaaWaaSbaaSqaaiabes
8a0jabes8a0bqabaGcdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaa
wIcacaGLPaaacaWGKbGaeq4WdmhaleaacqaHepaDcqGHsislcqaH3o
aAaeaacqaHepaDa0Gaey4kIipakiaadsgacqaHepaDaSqaaiabes8a
0naaBaaameaacqaH9oGBcqGHsislcaaIXaaabeaaaSqaaiabes8a0n
aaBaaameaacqaH9oGBaeqaaaqdcqGHRiI8aaaa@7456@
|
1
κ
∫
Κ
∂
1
(
∫
τ
ν
τ
ν+1
Δυ(
χ,τ
)dτ
)dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaeqOUdSgaamaapefabaGaeyOaIy7aaWbaaSqabeaacaaI
XaaaaOWaaeWaaeaadaWdXbqaaiabgs5aejabew8a1naabmaabaGaeq
4XdmMaaiilaiabes8a0bGaayjkaiaawMcaaiaadsgacqaHepaDaSqa
aiabes8a0naaBaaameaacqaH9oGBaeqaaaWcbaGaeqiXdq3aaSbaaW
qaaiabe27aUjabgUcaRiaaigdaaeqaaaqdcqGHRiI8aaGccaGLOaGa
ayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aaaa@59B2@
|
1
κ
2
∫
Κ
∫
τ
ν−1
τ
ν
∫
τ
ν−1
τ
∫
σ
σ+κ
Δ
d
2
υ
d
λ
2
(
χ,λ
)dλ
dσ
dτ
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca
aIXaaabaGaeqOUdS2aaWbaaSqabeaacaaIYaaaaaaakmaapefabaWa
a8qCaeaadaWdXbqaamaapehabaGaeyiLdq0aaSaaaeaacaWGKbWaaW
baaSqabeaacaaIYaaaaOGaeqyXduhabaGaamizaiabeU7aSnaaCaaa
leqabaGaaGOmaaaaaaGcdaqadaqaaiabeE8aJjaacYcacqaH7oaBai
aawIcacaGLPaaacaWGKbGaeq4UdWgaleaacqaHdpWCaeaacqaHdpWC
cqGHRaWkcqaH6oWAa0Gaey4kIipakiaadsgacqaHdpWCaSqaaiabes
8a0naaBaaameaacqaH9oGBcqGHsislcaaIXaaabeaaaSqaaiabes8a
0bqdcqGHRiI8aOGaamizaiabes8a0bWcbaGaeqiXdq3aaSbaaWqaai
abe27aUjabgkHiTiaaigdaaeqaaaWcbaGaeqiXdq3aaSbaaWqaaiab
e27aUbqabaaaniabgUIiYdGccaWGKbGaeq4XdmgaleaacqqHAoWsae
qaniabgUIiYdaaaa@73A5@
|
‖ υ ‖≤
Γ
1
(
(
∫
0
Τ
|
υ
˙
|
π(
τ
)
dτ
)
1
π
−
+1+|
υ
¯
|
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq
aHfpqDaiaawMa7caGLkWoacqGHKjYOcqqHtoWrdaWgaaWcbaGaaGym
aaqabaGcdaqadaqaamaabmaabaWaa8qCaeaadaabdaqaaiqbew8a1z
aacaaacaGLhWUaayjcSdWaaWbaaSqabeaacqaHapaCdaqadaqaaiab
es8a0bGaayjkaiaawMcaaaaakiaadsgacqaHepaDaSqaaiaaicdaae
aacqqHKoava0Gaey4kIipaaOGaayjkaiaawMcaamaaCaaaleqabaWa
aSaaaeaacaaIXaaabaGaeqiWda3aaWbaaWqabeaacqGHsislaaaaaa
aakiabgUcaRiaaigdacqGHRaWkdaabdaqaaiqbew8a1zaaraaacaGL
hWUaayjcSdaacaGLOaGaayzkaaaaaa@5EF4@
|
〈
Ξ
′
(
υ
),ν 〉=
∫
0
Τ
(
|
υ
˙
(
τ
) |
π(
τ
)−2
υ
˙
(
τ
),
υ
˙
(
τ
)
)dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacu
qHEoawgaqbamaabmaabaGaeqyXduhacaGLOaGaayzkaaGaaiilaiab
e27aUbGaayzkJiaawQYiaiabg2da9maapehabaWaaeWaaeaadaabda
qaaiqbew8a1zaacaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaa
wEa7caGLiWoadaahaaWcbeqaaiabec8aWnaabmaabaGaeqiXdqhaca
GLOaGaayzkaaGaeyOeI0IaaGOmaaaakiqbew8a1zaacaWaaeWaaeaa
cqaHepaDaiaawIcacaGLPaaacaGGSaGafqyXduNbaiaadaqadaqaai
abes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaiaadsgacqaHepaD
aSqaaiaaicdaaeaacqqHKoava0Gaey4kIipaaaa@630C@
|
Γ
5
(
∫
0
Τ
|
υ
˙
(
τ
) |
π(
τ
)
dτ
)
α+1
π
−
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS
baaSqaaiaaiwdaaeqaaOWaaeWaaeaadaWdXbqaamaaemaabaGafqyX
duNbaiaadaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSlaawI
a7amaaCaaaleqabaGaeqiWda3aaeWaaeaacqaHepaDaiaawIcacaGL
PaaaaaGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgU
IiYdaakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeqySdeMa
ey4kaSIaaGymaaqaaiabec8aWnaaCaaameqabaGaeyOeI0caaaaaaa
aaaa@5511@
|
〈
φ
′
(
υ
ν
),
υ
ν
−υ 〉=
∫
0
Τ
[
(
|
υ
˙
ν
(
τ
) |
π(
τ
)−2
υ
˙
ν
(
τ
),
υ
˙
ν
(
τ
)−
υ
˙
(
τ
)
)
+(
∇Φ(
τ,
υ
ν
(
τ
)
),
υ
ν
(
τ
)−υ(
τ
)
)
]dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacu
aHgpGAgaqbamaabmaabaGaeqyXdu3aaSbaaSqaaiabe27aUbqabaaa
kiaawIcacaGLPaaacaGGSaGaeqyXdu3aaSbaaSqaaiabe27aUbqaba
GccqGHsislcqaHfpqDaiaawMYicaGLQmcacqGH9aqpdaWdXbqaamaa
dmaaeaqabeaadaqadaqaamaaemaabaGafqyXduNbaiaadaWgaaWcba
GaeqyVd4gabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaacaGL
hWUaayjcSdWaaWbaaSqabeaacqaHapaCdaqadaqaaiabes8a0bGaay
jkaiaawMcaaiabgkHiTiaaikdaaaGccuaHfpqDgaGaamaaBaaaleaa
cqaH9oGBaeqaaOWaaeWaaeaacqaHepaDaiaawIcacaGLPaaacaGGSa
GafqyXduNbaiaadaWgaaWcbaGaeqyVd4gabeaakmaabmaabaGaeqiX
dqhacaGLOaGaayzkaaGaeyOeI0IafqyXduNbaiaadaqadaqaaiabes
8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiabgUcaRmaabmaa
baGaey4bIeTaeuOPdy0aaeWaaeaacqaHepaDcaGGSaGaeqyXdu3aaS
baaSqaaiabe27aUbqabaGcdaqadaqaaiabes8a0bGaayjkaiaawMca
aaGaayjkaiaawMcaaiaacYcacqaHfpqDdaWgaaWcbaGaeqyVd4gabe
aakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaeyOeI0IaeqyXdu3a
aeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaGaay
5waiaaw2faaiaadsgacqaHepaDaSqaaiaaicdaaeaacqqHKoava0Ga
ey4kIipaaaa@95D3@
|
∫
0
Τ
∫
0
1
(
∇Γ(
συ(
τ
)
)−∇Γ(
0
),υ(
τ
)
)dσ
dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada
WdXbqaamaabmaabaGaey4bIeTaeu4KdC0aaeWaaeaacqaHdpWCcqaH
fpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaai
abgkHiTiabgEGirlabfo5ahnaabmaabaGaaGimaaGaayjkaiaawMca
aiaacYcacqaHfpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay
jkaiaawMcaaiaadsgacqaHdpWCaSqaaiaaicdaaeaacaaIXaaaniab
gUIiYdGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgU
IiYdaaaa@5C9D@
|
(
∫
Ω
∞
(
∑
| α |+ξ=1
λ
ρ
2(
β+| α |+ξ−λ
)
|
Δ
α
λ
υ
τ
ξ
|
2
+
| υ |
2
)
ε
−2γτ
dχdτ
)
1
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada
WdrbqaamaabmaabaWaaabCaeaacqaHbpGCdaahaaWcbeqaaiaaikda
daqadaqaaiabek7aIjabgUcaRmaaemaabaGaeqySdegacaGLhWUaay
jcSdGaey4kaSIaeqOVdGNaeyOeI0Iaeq4UdWgacaGLOaGaayzkaaaa
aOWaaqWaaeaacqqHuoardaahaaWcbeqaaiabeg7aHbaakiabeU7aSj
abew8a1naaBaaaleaacqaHepaDdaahaaadbeqaaiabe67a4baaaSqa
baaakiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqGHRaWkda
abdaqaaiabew8a1bGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaa
aeaadaabdaqaaiabeg7aHbGaay5bSlaawIa7aiabgUcaRiabe67a4j
abg2da9iaaigdaaeaacqaH7oaBa0GaeyyeIuoaaOGaayjkaiaawMca
aiabew7aLnaaCaaaleqabaGaeyOeI0IaaGOmaiabeo7aNjabes8a0b
aakiaadsgacqaHhpWycaWGKbGaeqiXdqhaleaacqqHPoWvdaWgaaad
baGaeyOhIukabeaaaSqab0Gaey4kIipaaOGaayjkaiaawMcaamaaCa
aaleqabaWaaSaaaeaacaaIXaaabaGaaGOmaaaaaaaaaa@7FE2@
|
∑
| π |,| ρ |=0
μ
(
−1
)
| π |
∫
Ω
α
πρ
(
·,τ
)
Δ
ρ
υ
Δ
Π
υ
¯
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada
qadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaamaa
emaabaGaeqiWdahacaGLhWUaayjcSdaaaOWaa8quaeaacqaHXoqyda
WgaaWcbaGaeqiWdaNaeqyWdihabeaakmaabmaabaGaeS4JPFMaaiil
aiabes8a0bGaayjkaiaawMcaaiabfs5aenaaCaaaleqabaGaeqyWdi
haaOGaeqyXdu3aa0aaaeaacqqHuoardaahaaWcbeqaaiabfc6aqbaa
kiabew8a1baacaWGKbGaeq4XdmgaleaacqqHPoWvaeqaniabgUIiYd
aaleaadaabdaqaaiabec8aWbGaay5bSlaawIa7aiaacYcadaabdaqa
aiabeg8aYbGaay5bSlaawIa7aiabg2da9iaaicdaaeaacqaH8oqBa0
GaeyyeIuoaaaa@6AB6@
|
∑
ξ=0
μ−1
∫
Γ
Β
ξ
υ
∂
ξ
υ
¯
∂
ν
ξ
dσ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada
Wdrbqaaiabfk5acnaaBaaaleaacqaH+oaEaeqaaOGaeqyXdu3aaSaa
aeaacqGHciITdaahaaWcbeqaaiabe67a4baakiqbew8a1zaaraaaba
GaeyOaIyRaeqyVd42aaWbaaSqabeaacqaH+oaEaaaaaOGaamizaiab
eo8aZbWcbaGaeu4KdCeabeqdcqGHRiI8aaWcbaGaeqOVdGNaeyypa0
JaaGimaaqaaiabeY7aTjabgkHiTiaaigdaa0GaeyyeIuoaaaa@5501@
|
Γ
1
∫
Κ
′
[
υ
2
+
|
gradυ |
2
+
∑
| α |=2
|
Δ
α
υ
1
|
2
]d
χ
′
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS
baaSqaaiaaigdaaeqaaOWaa8quaeaadaWadaqaaiabew8a1naaCaaa
leqabaGaaGOmaaaakiabgUcaRmaaemaabaGaam4zaiaadkhacaWGHb
Gaamizaiabew8a1bGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaa
kiabgUcaRmaaqafabaWaaqWaaeaacqqHuoardaahaaWcbeqaaiabeg
7aHbaakiabew8a1naaBaaaleaacaaIXaaabeaaaOGaay5bSlaawIa7
amaaCaaaleqabaGaaGOmaaaaaeaadaabdaqaaiabeg7aHbGaay5bSl
aawIa7aiabg2da9iaaikdaaeqaniabggHiLdaakiaawUfacaGLDbaa
caWGKbGafq4XdmMbauaaaSqaaiqbfQ5alzaafaaabeqdcqGHRiI8aa
aa@6167@
|
∫
0
1
(
χ
ψ
−
ζ
10
)dζ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada
qadaqaaiabeE8aJnaaCaaaleqabaGaeqiYdKhaaOGaeyOeI0YaaSaa
aeaacqaH2oGEaeaacaaIXaGaaGimaaaaaiaawIcacaGLPaaacaWGKb
GaeqOTdOhaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaaa@45AF@
|
∫
0
1
∫
0
1−
ψ
2
1dχdψ
=
∫
0
π
2
∫
0
1
ρdρdθ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada
WdXaqaaiaaigdacaWGKbGaeq4XdmMaamizaiabeI8a5bWcbaGaaGim
aaqaamaakaaabaGaaGymaiabgkHiTiabeI8a5naaCaaameqabaGaaG
OmaaaaaeqaaaqdcqGHRiI8aaWcbaGaaGimaaqaaiaaigdaa0Gaey4k
Iipakiabg2da9maapedabaWaa8qmaeaacqaHbpGCcaWGKbGaeqyWdi
NaamizaiabeI7aXbWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaSqa
aiaaicdaaeaadaWcaaqaaiabec8aWbqaaiaaikdaaaaaniabgUIiYd
aaaa@581C@
|
∫
[
cos
χ
2×
χ
]
dχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada
WadaqaamaalaaabaGaci4yaiaac+gacaGGZbWaaOaaaeaacqaHhpWy
aSqabaaakeaacaaIYaGaey41aq7aaOaaaeaacqaHhpWyaSqabaaaaa
GccaGLBbGaayzxaaaaleqabeqdcqGHRiI8aOGaamizaiabeE8aJbaa
@45FC@
|
lim
ν→∞
|
υ
ν
−υ |
π(
τ
)
=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc
daabdaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaOGaeyOeI0Iaeq
yXduhacaGLhWUaayjcSdWaaSbaaSqaaiabec8aWnaabmaabaGaeqiX
dqhacaGLOaGaayzkaaaabeaakiabg2da9iaaicdaaaa@4EAC@
|
lim
ν→∞
ρ(
υ
ν
−υ
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc
cqaHbpGCdaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaOGaey
OeI0IaeqyXduhacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4992@
|
lim
ν→∞
ρ(
υ
ν
)=ρ(
υ
ν
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc
cqaHbpGCdaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaaGcca
GLOaGaayzkaaGaeyypa0JaeqyWdi3aaeWaaeaacqaHfpqDdaWgaaWc
baGaeqyVd4gabeaaaOGaayjkaiaawMcaaaaa@4D22@
|
limsup
ν→∞
〈
Ξ
′
(
υ
ν
)−
Ξ
′
(
υ
),
υ
ν
−υ 〉=0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaciGGZbGaaiyDaiaacchaaSqaaiabe27aUjab
gkziUkabg6HiLcqabaGcdaaadaqaaiqbf65ayzaafaWaaeWaaeaacq
aHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMcaaiabgkHi
Tiqbf65ayzaafaWaaeWaaeaacqaHfpqDaiaawIcacaGLPaaacaGGSa
GaeqyXdu3aaSbaaSqaaiabe27aUbqabaGccqGHsislcqaHfpqDaiaa
wMYicaGLQmcacqGH9aqpcaaIWaaaaa@5849@
|
[
Α
lim
χ→+0
φ(
χ
)
χ
]
−1
<λ<
[
Β
lim
χ→∞
φ(
χ
)
χ
]
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq
qHroqqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMaeyOK
H4Qaey4kaSIaaGimaaqabaGcdaWcaaqaaiabeA8aQnaabmaabaGaeq
4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2faamaa
CaaaleqabaGaeyOeI0IaaGymaaaakiabgYda8iabeU7aSjabgYda8m
aadmaabaGaeuOKdi0aaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiab
eE8aJjabgkziUkabg6HiLcqabaGcdaWcaaqaaiabeA8aQnaabmaaba
Gaeq4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2fa
amaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@6210@
|
[
Α
lim
χ→∞
φ(
χ
)
χ
]
−1
<λ<
[
Β
lim
χ→+0
φ(
χ
)
χ
]
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq
qHroqqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMaeyOK
H4QaeyOhIukabeaakmaalaaabaGaeqOXdO2aaeWaaeaacqaHhpWyai
aawIcacaGLPaaaaeaacqaHhpWyaaaacaGLBbGaayzxaaWaaWbaaSqa
beaacqGHsislcaaIXaaaaOGaeyipaWJaeq4UdWMaeyipaWZaamWaae
aacqqHsoGqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMa
eyOKH4Qaey4kaSIaaGimaaqabaGcdaWcaaqaaiabeA8aQnaabmaaba
Gaeq4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2fa
amaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@6210@
|
λ
lim
Δτ→0
1
Δτ
[
(
Τυ
)(
τ+Δτ
)−(
Τυ
)(
τ
) ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaC
beaeaaciGGSbGaaiyAaiaac2gaaSqaaiabgs5aejabes8a0jabgkzi
UkaaicdaaeqaaOWaaSaaaeaacaaIXaaabaGaeyiLdqKaeqiXdqhaam
aadmaabaWaaeWaaeaacqqHKoavcqaHfpqDaiaawIcacaGLPaaadaqa
daqaaiabes8a0jabgUcaRiabgs5aejabes8a0bGaayjkaiaawMcaai
abgkHiTmaabmaabaGaeuiPdqLaeqyXduhacaGLOaGaayzkaaWaaeWa
aeaacqaHepaDaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@5BBD@
|
lim
Δτ→0
(
Τ
2
υ
)(
τ+Δτ
)−(
Τ
2
υ
)(
τ
)
Δτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabfs5aejabes8a0jabgkziUkaaicda
aeqaaOWaaSaaaeaadaqadaqaaiabfs6aunaaBaaaleaacaaIYaaabe
aakiabew8a1bGaayjkaiaawMcaamaabmaabaGaeqiXdqNaey4kaSIa
euiLdqKaeqiXdqhacaGLOaGaayzkaaGaeyOeI0YaaeWaaeaacqqHKo
avdaWgaaWcbaGaaGOmaaqabaGccqaHfpqDaiaawIcacaGLPaaadaqa
daqaaiabes8a0bGaayjkaiaawMcaaaqaaiabfs5aejabes8a0baaaa
a@593D@
|
lim
ν→∞
∫
0
1
|
(
Τ
1
υ
ν
)(
τ
)−(
Τ
1
υ
0
)(
τ
) |dτ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc
daWdXbqaamaaemaabaWaaeWaaeaacqqHKoavdaWgaaWcbaGaaGymaa
qabaGccqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMca
amaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaeyOeI0YaaeWaaeaacq
qHKoavdaWgaaWcbaGaaGymaaqabaGccqaHfpqDdaWgaaWcbaGaaGim
aaqabaaakiaawIcacaGLPaaadaqadaqaaiabes8a0bGaayjkaiaawM
caaaGaay5bSlaawIa7aiaadsgacqaHepaDaSqaaiaaicdaaeaacaaI
XaaaniabgUIiYdaaaa@5DC3@
|
lim
χ→+0
φ(
χ
)
χ
<
(
λΒ
)
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkabgUcaRiaaicda
aeqaaOWaaSaaaeaacqaHgpGAdaqadaqaaiabeE8aJbGaayjkaiaawM
caaaqaaiabeE8aJbaacqGH8aapdaqadaqaaiabeU7aSjabfk5acbGa
ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@4C86@
|
λΑ
lim
χ→∞
φ(
χ
)
χ
>1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeu
yKde0aaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkzi
Ukabg6HiLcqabaGcdaWcaaqaaiabeA8aQnaabmaabaGaeq4Xdmgaca
GLOaGaayzkaaaabaGaeq4Xdmgaaiabg6da+iaaigdaaaa@49BA@
|
lim
χ→∞
φ(
χ
)
χ
≥
(
λ
Α
¯
)
−1
+ε
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkabg6HiLcqabaGc
daWcaaqaaiabeA8aQnaabmaabaGaeq4XdmgacaGLOaGaayzkaaaaba
Gaeq4XdmgaaiabgwMiZoaabmaabaGaeq4UdWMafuyKdeKbaebaaiaa
wIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGHRaWkcq
aH1oqzaaa@4FC6@
|
lim
χ→
0
−
1
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkaaicdadaahaaad
beqaaiabgkHiTaaaaSqabaGcdaWcaaqaaiaaigdaaeaacqaHhpWyaa
aaaa@4110@
|
lim
x→
0
−
(
1
χ
+
1
χ
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqa
beaacqGHsislaaaaleqaaOWaaeWaaeaadaWcaaqaaiaaigdaaeaacq
aHhpWyaaGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Xdm2aaWbaaSqa
beaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4636@
|
lim
x→3
1−
χ−2
χ−3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaaabeaakmaa
laaabaGaaGymaiabgkHiTmaakaaabaGaeq4XdmMaeyOeI0IaaGOmaa
WcbeaaaOqaaiabeE8aJjabgkHiTiaaiodaaaaaaa@454F@
|
lim
δx→0
φ(
χ
0
+δχ)−φ(
χ
0
)
δχ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabes7aKjaadIhacqGHsgIRcaaIWaaa
beaakmaalaaabaGaeqOXdOMaaiikaiabeE8aJnaaBaaaleaacaaIWa
aabeaakiabgUcaRiabes7aKjabeE8aJjaacMcacqGHsislcqaHgpGA
caGGOaGaeq4Xdm2aaSbaaSqaaiaaicdaaeqaaOGaaiykaaqaaiabes
7aKjabeE8aJbaaaaa@5264@
|
lim
β→1
β
(
γ−δ
)
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci
GGSbGaaiyAaiaac2gaaSqaaiabek7aIjabgkziUkaaigdaaeqaaOWa
aSaaaeaacqaHYoGyaeaadaqadaqaaiabeo7aNjabgkHiTiabes7aKb
GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaaaa@45AF@
|
(
β
1
β
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGabaaabaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOS
di2aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@3CB2@
|
(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGacaaabaGaeu4Odm1aaSbaaSqaaiaaigdacaGGSaGaaGymaaqa
baaakeaacqqHJoWudaWgaaWcbaGaaGymaiaacYcacaaIYaaabeaaaO
qaaiabfo6atnaaBaaaleaacaaIYaGaaiilaiaaigdaaeqaaaGcbaGa
eu4Odm1aaSbaaSqaaiaaikdacaGGSaGaaGOmaaqabaaaaaGccaGLOa
Gaayzkaaaaaa@4714@
|
(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,2
)=
(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,2
)
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGacaaabaGaeu4Odm1aaWbaaSqabeaacaaIXaGaaiilaiaaigda
aaaakeaacqqHJoWudaahaaWcbeqaaiaaigdacaGGSaGaaGOmaaaaaO
qaaiabfo6atnaaCaaaleqabaGaaGOmaiaacYcacaaIXaaaaaGcbaGa
eu4Odm1aaWbaaSqabeaacaaIYaGaaiilaiaaikdaaaaaaaGccaGLOa
GaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeu4Odm1aaSba
aSqaaiaaigdacaGGSaGaaGymaaqabaaakeaacqqHJoWudaWgaaWcba
GaaGymaiaacYcacaaIYaaabeaaaOqaaiabfo6atnaaBaaaleaacaaI
YaGaaiilaiaaigdaaeqaaaGcbaGaeu4Odm1aaSbaaSqaaiaaikdaca
GGSaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGH
sislcaaIXaaaaaaa@5B10@
|
(
Α
Β
Β
′
Γ
)
−1
=(
αα
αβ
βα
ββ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGacaaabaGaeuyKdeeabaGaeuOKdieabaGafuOKdiKbauaaaeaa
cqqHtoWraaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXa
aaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeqySdeMaeqySdega
baGaeqySdeMaeqOSdigabaGaeqOSdiMaeqySdegabaGaeqOSdiMaeq
OSdigaaaGaayjkaiaawMcaaaaa@4EA6@
|
(
Υ
1
Υ
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP
du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@3C80@
|
(
σ
1
2
γ
γ
σ
2
2
)
Ι
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGacaaabaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaaIYaaaaaGc
baGaeq4SdCgabaGaeq4SdCgabaGaeq4Wdm3aa0baaSqaaiaaikdaae
aacaaIYaaaaaaaaOGaayjkaiaawMcaaiabfM5ajnaaBaaaleaacqaH
9oGBaeqaaaaa@4515@
|
Β
1
=(
1
0
0
0
)⊗
Ι
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS
baaSqaaiaaigdaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa
aGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaaaaiaawIcacaGLPa
aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@432F@
|
Β
2
=(
0
0
0
1
)⊗
Ι
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS
baaSqaaiaaikdaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa
aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaaaaiaawIcacaGLPa
aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@4330@
|
Β
3
=(
0
1
1
0
)⊗
Ι
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS
baaSqaaiaaiodaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa
aGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawIcacaGLPa
aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@4332@
|
Μ
(
Χ
0
0
Χ
)
=
Ι
2
⊗
Μ
Χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiNd00aaS
baaSqaamaabmaabaqbaeqabiGaaaqaaiabfE6adbqaaiaaicdaaeaa
caaIWaaabaGaeu4PdmeaaaGaayjkaiaawMcaaaqabaGccqGH9aqpcq
qHzoqsdaWgaaWcbaGaaGOmaaqabaGccqGHxkcXcqqHCoqtdaWgaaWc
baGaeu4Pdmeabeaaaaa@462D@
|
var(
vec(
Υ
_
^
1
)
vec(
Υ
_
^
2
)
)=(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,2
)⊗
Ι
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg
gacaGGYbWaaeWaaeaafaqabeGabaaabaGaamODaiaadwgacaWGJbWa
aeWaaeaacuqHLoqvgaqhgaqcamaaBaaaleaacaaIXaaabeaaaOGaay
jkaiaawMcaaaqaaiaadAhacaWGLbGaam4yamaabmaabaGafuyPduLb
a0HbaKaadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaaaca
GLOaGaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeu4Odm1a
aSbaaSqaaiaaigdacaGGSaGaaGymaaqabaaakeaacqqHJoWudaWgaa
WcbaGaaGymaiaacYcacaaIYaaabeaaaOqaaiabfo6atnaaBaaaleaa
caaIYaGaaiilaiaaigdaaeqaaaGcbaGaeu4Odm1aaSbaaSqaaiaaik
dacaGGSaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaey4LIqSaeuyM
dK0aaSbaaSqaaiabe27aUbqabaaaaa@5FE9@
|
Ι⊗ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyMdKKaey
4LIqSaeqyVd4gaaa@3B28@
|
(
Υ
1
Υ
2
)∼[
(
Χ
0
0
Χ
)(
β
1
β
2
),(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,2
) ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP
du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiablYJi6m
aadmaabaWaaeWaaeaafaqabeGacaaabaGaeu4PdmeabaGaaGimaaqa
aiaaicdaaeaacqqHNoWqaaaacaGLOaGaayzkaaWaaeWaaeaafaqabe
GabaaabaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOSdi2a
aSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiaacYcadaqada
qaauaabeqaciaaaeaacqqHJoWudaWgaaWcbaGaaGymaiaacYcacaaI
XaaabeaaaOqaaiabfo6atnaaBaaaleaacaaIXaGaaiilaiaaikdaae
qaaaGcbaGaeu4Odm1aaSbaaSqaaiaaikdacaGGSaGaaGymaaqabaaa
keaacqqHJoWudaWgaaWcbaGaaGOmaiaacYcacaaIYaaabeaaaaaaki
aawIcacaGLPaaaaiaawUfacaGLDbaaaaa@5E20@
|
(
β
^
1
β
^
2
)=
[
(
Χ
′
1
0
0
Χ
′
2
)(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,1
)(
Χ
1
0
0
Χ
2
) ]
−1
×(
Χ
′
1
0
0
Χ
′
2
)(
Σ
1,1
Σ
1,2
Σ
2,1
Σ
2,1
)(
Υ
1
Υ
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada
qaauaabeqaceaaaeaacuaHYoGygaqcamaaBaaaleaacaaIXaaabeaa
aOqaaiqbek7aIzaajaWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkai
aawMcaaiabg2da9maadmaabaWaaeWaaeaafaqabeGacaaabaGafu4P
dmKbauaadaWgaaWcbaGaaGymaaqabaaakeaacaaIWaaabaGaaGimaa
qaaiqbfE6adzaafaWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaa
wMcaamaabmaabaqbaeqabiGaaaqaaiabfo6atnaaCaaaleqabaGaaG
ymaiaacYcacaaIXaaaaaGcbaGaeu4Odm1aaWbaaSqabeaacaaIXaGa
aiilaiaaikdaaaaakeaacqqHJoWudaahaaWcbeqaaiaaikdacaGGSa
GaaGymaaaaaOqaaiabfo6atnaaCaaaleqabaGaaGOmaiaacYcacaaI
XaaaaaaaaOGaayjkaiaawMcaamaabmaabaqbaeqabiGaaaqaaiabfE
6adnaaBaaaleaacaaIXaaabeaaaOqaaiaaicdaaeaacaaIWaaabaGa
eu4Pdm0aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaGaay
5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOqaaiabgEna
0oaabmaabaqbaeqabiGaaaqaaiqbfE6adzaafaWaaSbaaSqaaiaaig
daaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacuqHNoWqgaqbamaaBaaa
leaacaaIYaaabeaaaaaakiaawIcacaGLPaaadaqadaqaauaabeqaci
aaaeaacqqHJoWudaahaaWcbeqaaiaaigdacaGGSaGaaGymaaaaaOqa
aiabfo6atnaaCaaaleqabaGaaGymaiaacYcacaaIYaaaaaGcbaGaeu
4Odm1aaWbaaSqabeaacaaIYaGaaiilaiaaigdaaaaakeaacqqHJoWu
daahaaWcbeqaaiaaikdacaGGSaGaaGymaaaaaaaakiaawIcacaGLPa
aadaqadaqaauaabeqaceaaaeaacqqHLoqvdaWgaaWcbaGaaGymaaqa
baaakeaacqqHLoqvdaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaay
zkaaaaaaa@8460@
|
[
(
Χ
1
0
−
Σ
2,1
Σ
1,1
−1
Χ
1
Χ
2
)(
β
1
β
2
),(
Σ
1,1
0
0
Σ
22.1
) ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada
qadaqaauaabeqaciaaaeaacqqHNoWqdaWgaaWcbaGaaGymaaqabaaa
keaacaaIWaaabaGaeyOeI0Iaeu4Odm1aaSbaaSqaaiaaikdacaGGSa
GaaGymaaqabaGccqqHJoWudaqhaaWcbaGaaGymaiaacYcacaaIXaaa
baGaeyOeI0IaaGymaaaakiabfE6adnaaBaaaleaacaaIXaaabeaaaO
qaaiabfE6adnaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaa
daqadaqaauaabeqaceaaaeaacqaHYoGydaWgaaWcbaGaaGymaaqaba
aakeaacqaHYoGydaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzk
aaGaaiilamaabmaabaqbaeqabiGaaaqaaiabfo6atnaaBaaaleaaca
aIXaGaaiilaiaaigdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacqqH
JoWudaWgaaWcbaGaaGOmaiaaikdacaGGUaGaaGymaaqabaaaaaGcca
GLOaGaayzkaaaacaGLBbGaayzxaaaaaa@5EC6@
|
(
Ι,0
)(
Χ
′
1
Σ
1,1
Χ
1
Χ
′
1
Σ
1,2
Χ
2
Χ
′
2
Σ
2,1
Χ
1
Χ
′
2
Σ
2,2
Χ
2
)(
0
Ι
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
qHzoqscaGGSaGaaGimaaGaayjkaiaawMcaamaabmaabaqbaeqabiGa
aaqaaiqbfE6adzaafaWaaSbaaSqaaiaaigdaaeqaaOGaeu4Odm1aaW
baaSqabeaacaaIXaGaaiilaiaaigdaaaGccqqHNoWqdaWgaaWcbaGa
aGymaaqabaaakeaacuqHNoWqgaqbamaaBaaaleaacaaIXaaabeaaki
abfo6atnaaCaaaleqabaGaaGymaiaacYcacaaIYaaaaOGaeu4Pdm0a
aSbaaSqaaiaaikdaaeqaaaGcbaGafu4PdmKbauaadaWgaaWcbaGaaG
OmaaqabaGccqqHJoWudaahaaWcbeqaaiaaikdacaGGSaGaaGymaaaa
kiabfE6adnaaBaaaleaacaaIXaaabeaaaOqaaiqbfE6adzaafaWaaS
baaSqaaiaaikdaaeqaaOGaeu4Odm1aaWbaaSqabeaacaaIYaGaaiil
aiaaikdaaaGccqqHNoWqdaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOa
GaayzkaaWaaeWaaeaafaqabeGabaaabaGaaGimaaqaaiabfM5ajbaa
aiaawIcacaGLPaaaaaa@62B7@
|
(
β
˜
1
β
˜
2
)∼
Ν
2
κ
[
(
β
1
β
2
),(
Var(
β
˜
1
)
cov(
β
˜
1
,
β
˜
2
)
cov(
β
˜
2
,
β
˜
1
)
Var(
β
˜
2
)
) ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGabaaabaGafqOSdiMbaGaadaWgaaWcbaGaaGymaaqabaaakeaa
cuaHYoGygaacamaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPa
aacqWI8iIocqqHDoGtdaWgaaWcbaGaaGOmamaaCaaameqabaGaeqOU
dSgaaaWcbeaakmaadmaabaWaaeWaaeaafaqabeGabaaabaGaeqOSdi
2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOSdi2aaSbaaSqaaiaaikda
aeqaaaaaaOGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaciaaae
aacaWGwbGaamyyaiaadkhadaqadaqaaiqbek7aIzaaiaWaaSbaaSqa
aiaaigdaaeqaaaGccaGLOaGaayzkaaaabaGaci4yaiaac+gacaGG2b
WaaeWaaeaacuaHYoGygaacamaaBaaaleaacaaIXaaabeaakiaacYca
cuaHYoGygaacamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaa
qaaiGacogacaGGVbGaaiODamaabmaabaGafqOSdiMbaGaadaWgaaWc
baGaaGOmaaqabaGccaGGSaGafqOSdiMbaGaadaWgaaWcbaGaaGymaa
qabaaakiaawIcacaGLPaaaaeaacaWGwbGaamyyaiaadkhadaqadaqa
aiqbek7aIzaaiaWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaa
aaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@6FAE@
|
(
Υ
1
Υ
2
)∼
Ν
2ν
[
(
Ι
2
⊗Χ
)(
β
1
β
2
),(
σ
1
2
γ
γ
σ
2
2
)⊗
Ι
ν
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP
du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiablYJi6i
abf25aonaaBaaaleaacaaIYaGaeqyVd4gabeaakmaadmaabaWaaeWa
aeaacqqHzoqsdaWgaaWcbaGaaGOmaaqabaGccqGHxkcXcqqHNoWqai
aawIcacaGLPaaadaqadaqaauaabeqaceaaaeaacqaHYoGydaWgaaWc
baGaaGymaaqabaaakeaacqaHYoGydaWgaaWcbaGaaGOmaaqabaaaaa
GccaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiGaaaqaaiabeo8a
ZnaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiabeo7aNbqaaiabeo
7aNbqaaiabeo8aZnaaDaaaleaacaaIYaaabaGaaGOmaaaaaaaakiaa
wIcacaGLPaaacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaO
Gaay5waiaaw2faaaaa@63C6@
|
(
σ
2,0
4
γ
0
2
−2
γ
0
σ
2,0
2
γ
0
2
σ
1,0
4
−2
γ
0
σ
1,0
2
−2
γ
0
σ
2,0
2
−2
γ
0
σ
1,0
2
2(
σ
1,0
2
,
σ
2,0
2
+
γ
0
2
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeWadaaabaGaeq4Wdm3aa0baaSqaaiaaikdacaGGSaGaaGimaaqa
aiaaisdaaaaakeaacqaHZoWzdaqhaaWcbaGaaGimaaqaaiaaikdaaa
aakeaacqGHsislcaaIYaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGa
eq4Wdm3aa0baaSqaaiaaikdacaGGSaGaaGimaaqaaiaaikdaaaaake
aacqaHZoWzdaqhaaWcbaGaaGimaaqaaiaaikdaaaaakeaacqaHdpWC
daqhaaWcbaGaaGymaiaacYcacaaIWaaabaGaaGinaaaaaOqaaiabgk
HiTiaaikdacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqh
aaWcbaGaaGymaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiabgkHiTi
aaikdacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqhaaWc
baGaaGOmaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiabgkHiTiaaik
dacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqhaaWcbaGa
aGymaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiaaikdadaqadaqaai
abeo8aZnaaDaaaleaacaaIXaGaaiilaiaaicdaaeaacaaIYaaaaOGa
aiilaiabeo8aZnaaDaaaleaacaaIYaGaaiilaiaaicdaaeaacaaIYa
aaaOGaey4kaSIaeq4SdC2aa0baaSqaaiaaicdaaeaacaaIYaaaaaGc
caGLOaGaayzkaaaaaaGaayjkaiaawMcaaaaa@7D4A@
|
{
Μ
(
Χ
0
0
Χ
)
[
(
σ
1,0
2
γ
0
γ
0
σ
2,0
2
)⊗
Ι
ν
]
Μ
(
Χ
0
0
Χ
)
}
+
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq
qHCoqtdaWgaaWcbaWaaeWaaeaafaqabeGacaaabaGaeu4PdmeabaGa
aGimaaqaaiaaicdaaeaacqqHNoWqaaaacaGLOaGaayzkaaaabeaakm
aadmaabaWaaeWaaeaafaqabeGacaaabaGaeq4Wdm3aa0baaSqaaiaa
igdacaGGSaGaaGimaaqaaiaaikdaaaaakeaacqaHZoWzdaWgaaWcba
GaaGimaaqabaaakeaacqaHZoWzdaWgaaWcbaGaaGimaaqabaaakeaa
cqaHdpWCdaqhaaWcbaGaaGOmaiaacYcacaaIWaaabaGaaGOmaaaaaa
aakiaawIcacaGLPaaacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4ga
beaaaOGaay5waiaaw2faaiabfY5annaaBaaaleaadaqadaqaauaabe
qaciaaaeaacqqHNoWqaeaacaaIWaaabaGaaGimaaqaaiabfE6adbaa
aiaawIcacaGLPaaaaeqaaaGccaGL7bGaayzFaaWaaWbaaSqabeaacq
GHRaWkaaaaaa@6062@
|
(
η
′
(
Μ
Α
Σ
0
Μ
Α
)
+
Β
1
(
Μ
Α
Σ
0
Μ
Α
)
+
η
⋮
η
′
(
Μ
Α
Σ
0
Μ
Α
)
+
Β
π
(
Μ
Α
Σ
0
Μ
Α
)
+
η
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaqaabe
qaaiqbeE7aOzaafaWaaeWaaeaacqqHCoqtdaWgaaWcbaGaeuyKdeea
beaakiabfo6atnaaBaaaleaacaaIWaaabeaakiabfY5annaaBaaale
aacqqHroqqaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHRaWk
aaGccqqHsoGqdaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabfY5ann
aaBaaaleaacqqHroqqaeqaaOGaeu4Odm1aaSbaaSqaaiaaicdaaeqa
aOGaeuiNd00aaSbaaSqaaiabfg5abbqabaaakiaawIcacaGLPaaada
ahaaWcbeqaaiabgUcaRaaakiabeE7aObqaaiabl6UinbqaaiqbeE7a
OzaafaWaaeWaaeaacqqHCoqtdaWgaaWcbaGaeuyKdeeabeaakiabfo
6atnaaBaaaleaacaaIWaaabeaakiabfY5annaaBaaaleaacqqHroqq
aeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHRaWkaaGccqqHso
GqdaWgaaWcbaGaeqiWdahabeaakmaabmaabaGaeuiNd00aaSbaaSqa
aiabfg5abbqabaGccqqHJoWudaWgaaWcbaGaaGimaaqabaGccqqHCo
qtdaWgaaWcbaGaeuyKdeeabeaaaOGaayjkaiaawMcaamaaCaaaleqa
baGaey4kaScaaOGaeq4TdGgaaiaawIcacaGLPaaaaaa@72A6@
|
Κ=(
φ
1
,…,
φ
ρ(
Τ
)
)(
1
λ
1
…
0
⋮
⋱
⋮
0
⋯
1
λ
ρ(
Τ
)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdSKaey
ypa0ZaaeWaaeaacqaHgpGAdaWgaaWcbaGaaGymaaqabaGccaGGSaGa
eSOjGSKaaiilaiabeA8aQnaaBaaaleaacqaHbpGCdaqadaqaaiabfs
6aubGaayjkaiaawMcaaaqabaaakiaawIcacaGLPaaadaqadaqaauaa
beqadmaaaeaadaWcaaqaaiaaigdaaeaadaGcaaqaaiabeU7aSnaaBa
aaleaacaaIXaaabeaaaeqaaaaaaOqaaiablAcilbqaaiaaicdaaeaa
cqWIUlstaeaacqWIXlYtaeaacqWIUlstaeaacaaIWaaabaGaeS47IW
eabaWaaSaaaeaacaaIXaaabaWaaOaaaeaacqaH7oaBdaWgaaWcbaGa
eqyWdi3aaeWaaeaacqqHKoavaiaawIcacaGLPaaaaeqaaaqabaaaaa
aaaOGaayjkaiaawMcaaaaa@5CF2@
|
(
Σ
γ
τ(γ)
)=(
1
1
1
1
ζ
2
ζ
1
ζ
ζ
2
)×(
θ
σ(θ)
σ
2
(θ)
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa
qabeWabaaabaGaeu4OdmfabaGaeq4SdCgabaGaeqiXdqNaaiikaiab
eo7aNjaacMcaaaaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaafaqabe
WadaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiab
eA7a6naaCaaaleqabaGaaGOmaaaaaOqaaiabeA7a6bqaaiaaigdaae
aacqaH2oGEaeaacqaH2oGEdaahaaWcbeqaaiaaikdaaaaaaaGccaGL
OaGaayzkaaGaey41aq7aaeWaaeaafaqabeWabaaabaGaeqiUdehaba
Gaeq4WdmNaaiikaiabeI7aXjaacMcaaeaacqaHdpWCdaahaaWcbeqa
aiaaikdaaaGccaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaa
a@5EA2@
|
Τ∈(
0,∞
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiPdqLaey
icI48aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaaaa
@3D65@
|
ω=(
ω
1
,…,
ω
ν−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey
ypa0ZaaeWaaeaacqaHjpWDdaWgaaWcbaGaaGymaaqabaGccaGGSaGa
eSOjGSKaaiilaiabeM8a3naaBaaaleaacqaH9oGBcqGHsislcaaIXa
aabeaaaOGaayjkaiaawMcaaaaa@44F6@
|
Α
∞
=Α×(
0,∞
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaS
baaSqaaiabg6HiLcqabaGccqGH9aqpcqqHroqqcqGHxdaTdaqadaqa
aiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaaaaa@41DF@
|
Β
χ
(
0,∞;ℂ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaW
baaSqabeaacqaHhpWyaaGcdaqadaqaaiaaicdacaGGSaGaeyOhIuQa
ai4oaiablkqiJcGaayjkaiaawMcaaaaa@3FC1@
|
ℝ
δ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW
baaSqabeaacqaH0oazaaaaaa@3939@
|
δ∈ℕ\{ 0 }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaey
icI4SaeSyfHuQaaiixamaacmaabaGaaGimaaGaay5Eaiaaw2haaaaa
@3E57@
|
σ⊂∂Ω
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey
OGIWSaeyOaIyRaeyyQdCfaaa@3CAB@
|
[
χ
κ
,χ ]⊂Κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq
aHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacYcacqaHhpWyaiaawUfa
caGLDbaacqGHckcZcqqHAoWsaaa@415E@
|
χ∈
Ω
¯
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
icI4SafuyQdCLbaebaaaa@3AD8@
|
Μ
σ
={
Κ,Λ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiNd00aaS
baaSqaaiabeo8aZbqabaGccqGH9aqpdaGadaqaaiabfQ5aljaacYca
cqqHBoataiaawUhacaGL9baaaaa@4036@
|
Γ
∞,α
(
∂Ω×[
0,∞ ]
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaW
baaSqabeaacqGHEisPcaGGSaGaeqySdegaaOWaaeWaaeaacqGHciIT
cqqHPoWvcqGHxdaTdaWadaqaaiaaicdacaGGSaGaeyOhIukacaGLBb
GaayzxaaaacaGLOaGaayzkaaaaaa@46B7@
|
χ∈∂Ω
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
icI4SaeyOaIyRaeuyQdCfaaa@3C26@
|
{
β
σ
Κ
:σ∈Β,Κ∈Μ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq
aHYoGydaqhaaWcbaGaeq4WdmhabaGaeuOMdSeaaOGaaiOoaiabeo8a
ZjabgIGiolabfk5acjaacYcacqqHAoWscqGHiiIZcqqHCoqtaiaawU
hacaGL9baaaaa@47BB@
|
{
Κ
Κ
ν
:ν∈〚
0,Ν+1〛,Κ∈Μ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq
qHAoWsdaqhaaWcbaGaeuOMdSeabaGaeqyVd4gaaOGaaiOoaiabe27a
UjabgIGiopaaimaabaGaaGimaiaacYcacqqHDoGtcqGHRaWkcaaIXa
aacaGLAaJaay5gWaGaaiilaiabfQ5aljabgIGiolabfY5anbGaay5E
aiaaw2haaaaa@4C69@
|
τ
0
∈[
0,∞ ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS
baaSqaaiaaicdaaeqaaOGaeyicI48aamWaaeaacaaIWaGaaiilaiab
g6HiLcGaay5waiaaw2faaaaa@3EFD@
|
υ
0
∈
Λ
2
(
Ω
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS
baaSqaaiaaicdaaeqaaOGaeyicI4Saeu4MdW0aaWbaaSqabeaacaaI
YaaaaOWaaeWaaeaacqGHPoWvaiaawIcacaGLPaaaaaa@3FB2@
|
φ
0
∈
Γ
˙
∞
(
Υ
0
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aaS
baaSqaaiaaicdaaeqaaOGaeyicI4Safu4KdCKbaiaadaahaaWcbeqa
aiabg6HiLcaakmaabmaabaGaeuyPdu1aaSbaaSqaaiaaicdaaeqaaa
GccaGLOaGaayzkaaaaaa@4142@
|
Η
2
(
ε
−γ×τ
,
Ω
∞
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4LdG0aaW
baaSqabeaacaaIYaaaaOWaaeWaaeaacqaH1oqzdaahaaWcbeqaaiab
gkHiTiabeo7aNjabgEna0kabes8a0baakiaacYcacqqHPoWvdaWgaa
WcbaGaeyOhIukabeaaaOGaayjkaiaawMcaaaaa@4613@
|
Ω
ρ
:={
χ∈Ω:
ρ
2
<| χ |<2ρ,ρ>0 }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdC1aaW
baaSqabeaacqaHbpGCaaGccaGG6aGaeyypa0ZaaiWaaeaacqaHhpWy
cqGHiiIZcqqHPoWvcaGG6aWaaSaaaeaacqaHbpGCaeaacaaIYaaaai
abgYda8maaemaabaGaeq4XdmgacaGLhWUaayjcSdGaeyipaWJaaGOm
aiabeg8aYjaacYcacqaHbpGCcqGH+aGpcaaIWaaacaGL7bGaayzFaa
aaaa@5313@
|
Κ={
χ=(
χ
1
,
χ
2
)∈
ℝ
2
:ρ>0,0<ω<
ω
0
}
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdSKaey
ypa0ZaaiWaaeaacqaHhpWycqGH9aqpdaqadaqaaiabeE8aJnaaBaaa
leaacaaIXaaabeaakiaacYcacqaHhpWydaWgaaWcbaGaaGOmaaqaba
aakiaawIcacaGLPaaacqGHiiIZcqWIDesOdaahaaWcbeqaaiaaikda
aaGccaGG6aGaeqyWdiNaeyOpa4JaaGimaiaacYcacaaIWaGaeyipaW
JaeqyYdCNaeyipaWJaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaGccaGL
7bGaayzFaaaaaa@550B@
|
Α={
τ∈[
0,Τ ]:|
υ
˙
(
τ
) |≥1 }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey
ypa0ZaaiWaaeaacqaHepaDcqGHiiIZdaWadaqaaiaaicdacaGGSaGa
euiPdqfacaGLBbGaayzxaaGaaiOoamaaemaabaGafqyXduNbaiaada
qadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSlaawIa7aiabgwMi
ZkaaigdaaiaawUhacaGL9baaaaa@4E38@
|
η:(
0,1
)→[
0,∞ )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGMaai
OoamaabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaGaeyOK
H46aaKGeaeaacaaIWaGaaiilaiabg6HiLcGaay5waiaawMcaaaaa@42AA@
|
φ:(
0,1
)×(
0,∞
)×(
0,∞
)×(
−∞,0
)→[
0,∞ )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai
OoamaabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaGaey41
aq7aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaiabgE
na0oaabmaabaGaaGimaiaacYcacqGHEisPaiaawIcacaGLPaaacqGH
xdaTdaqadaqaaiabgkHiTiabg6HiLkaacYcacaaIWaaacaGLOaGaay
zkaaGaeyOKH46aaKGeaeaacaaIWaGaaiilaiabg6HiLcGaay5waiaa
wMcaaaaa@5719@
|
Γ
2
[
0,1 ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaW
baaSqabeaacaaIYaaaaOWaamWaaeaacaaIWaGaaiilaiaaigdaaiaa
wUfacaGLDbaaaaa@3C69@
|
Κ(
ρ
)={
υ∈Κ:|
‖ υ ‖ |<ρ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdS0aae
WaaeaacqaHbpGCaiaawIcacaGLPaaacqGH9aqpdaGadaqaaiabew8a
1jabgIGiolabfQ5aljaacQdadaabdaqaamaafmaabaGaeqyXduhaca
GLjWUaayPcSdaacaGLhWUaayjcSdGaeyipaWJaeqyWdihacaGL7bGa
ayzFaaaaaa@4E3A@
|
χ∈
ℝ
ν
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey
icI4SaeSyhHe6aaWbaaSqabeaacqaH9oGBaaaaaa@3C87@
|
α∈[
0,
π
−
−1 )
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey
icI48aaKGeaeaacaaIWaGaaiilaiabec8aWnaaCaaaleqabaGaeyOe
I0caaOGaeyOeI0IaaGymaaGaay5waiaawMcaaaaa@40E0@
|
υ∈
Κ(
ρ
2
)
¯
\Κ(
ρ
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXduNaey
icI48aa0aaaeaacqqHAoWsdaqadaqaaiabeg8aYnaaBaaaleaacaaI
YaaabeaaaOGaayjkaiaawMcaaaaacaGGCbGaeuOMdS0aaeWaaeaacq
aHbpGCdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaaaaa@458D@
|
Κ
β
(
ρ
)={
υ∈
Κ
β
:‖ υ ‖<ρ }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdS0aaW
baaSqabeaacqaHYoGyaaGcdaqadaqaaiabeg8aYbGaayjkaiaawMca
aiabg2da9maacmaabaGaeqyXduNaeyicI4SaeuOMdS0aaWbaaSqabe
aacqaHYoGyaaGccaGG6aWaauWaaeaacqaHfpqDaiaawMa7caGLkWoa
cqGH8aapcqaHbpGCaiaawUhacaGL9baaaaa@4EC8@
|
0∈
Ω
1
⊂
Ω
¯
1
⊂
Ω
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgI
GiolabfM6axnaaBaaaleaacaaIXaaabeaakiabgkOimlqbfM6axzaa
raWaaSbaaSqaaiaaigdaaeqaaOGaeyOGIWSaeuyQdC1aaSbaaSqaai
aaikdaaeqaaaaa@43B9@
|
υ
*
∈Κ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW
baaSqabeaacaGGQaaaaOGaeyicI4SaeuOMdSeaaa@3B9A@
|
Γ(
0,σ
)≡0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aae
WaaeaacaaIWaGaaiilaiabeo8aZbGaayjkaiaawMcaaiabggMi6kaa
icdaaaa@3E98@
|
Υ
1
∼(
Χ
β1
,
Σ
1,1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyPdu1aaS
baaSqaaiaaigdaaeqaaOGaeSipIOZaaeWaaeaacqqHNoWqdaWgaaWc
baGaeqOSdiMaaGymaaqabaGccaGGSaGaeu4Odm1aaSbaaSqaaiaaig
dacaGGSaGaaGymaaqabaaakiaawIcacaGLPaaaaaa@43BC@
|
(
Υ
1
,
Υ
2
)=
Σ
1,2
≠0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq
qHLoqvdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeuyPdu1aaSbaaSqa
aiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyypa0Jaeu4Odm1aaSbaaS
qaaiaaigdacaGGSaGaaGOmaaqabaGccqGHGjsUcaaIWaaaaa@448B@
|
Ω
¯
=
∪
Κ∈Μ
Κ
¯
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuyQdCLbae
bacqGH9aqpdaWeqbqaaiqbfQ5alzaaraaaleaacqqHAoWscqGHiiIZ
cqqHCoqtaeqaniablQIivbaaaa@4050@
|
max{
1,2,3 }
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacg
gacaGG4bWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa
ay5Eaiaaw2haaaaa@3E90@
|
sup{
1,2,3 }=3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw
hacaGGWbWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa
ay5Eaiaaw2haaiabg2da9iaaiodaaaa@4065@
|
Α⊂Β
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey
OGIWSaeuOKdieaaa@3AB5@
|
inf{
1,2,3 }=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyAaiaac6
gacaGGMbWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa
ay5Eaiaaw2haaiabg2da9iaaigdaaaa@4048@
|
∪
α
Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaambuaeaacq
qHroqqaSqaaiabeg7aHbqab0GaeSOkIufaaaa@3AAA@
|
∏
α
Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebuaeaacq
qHroqqaSqaaiabeg7aHbqab0Gaey4dIunaaaa@3B08@
|
α∪β
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeS
OkIuLaeqOSdigaaa@3A69@
|
Α=∅
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey
ypa0JaeyybIymaaa@39D6@
|
α∪β
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey
OkIGSaeqOSdigaaa@3AD7@
|
cov(
β
1
,
β
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGG2bWaaeWaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaGccaGG
SaGaeqOSdi2aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@402B@
|
Var(
β
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg
gacaWGYbWaaeWaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaaakiaa
wIcacaGLPaaaaaa@3CCA@
|
∀χ∈Χ:∃ψ∈Ψ:χ=ψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaeq
4XdmMaeyicI4Saeu4PdmKaaiOoaiabgoGiKiabeI8a5jabgIGiolab
fI6azjaacQdacqaHhpWycqGH9aqpcqaHipqEaaa@4737@
|
tanΑ=
sinΑ
cosΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg
gacaGGUbGaeuyKdeKaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6ga
cqqHroqqaeaaciGGJbGaai4BaiaacohacqqHroqqaaaaaa@43A9@
|
cotΑ=
1
tanΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGG0bGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaciiDaiaa
cggacaGGUbGaeuyKdeeaaaaa@402D@
|
secΑ=
1
cosΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw
gacaGGJbGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaci4yaiaa
c+gacaGGZbGaeuyKdeeaaaaa@4024@
|
cscΑ=
1
sinΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaaco
hacaGGJbGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaci4Caiaa
cMgacaGGUbGaeuyKdeeaaaaa@4027@
|
sec
2
Α−
tan
2
Α=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw
gacaGGJbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0IaciiD
aiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyypa0
JaaGymaaaa@42E5@
|
tan(−Α)=−tanΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg
gacaGGUbGaaiikaiabgkHiTiabfg5abjaacMcacqGH9aqpcqGHsisl
ciGG0bGaaiyyaiaac6gacqqHroqqaaa@4292@
|
cos(−Α)=cosΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaaiikaiabgkHiTiabfg5abjaacMcacqGH9aqpciGGJbGa
ai4BaiaacohacqqHroqqaaa@41A9@
|
sin2Α=2sinΑcosΑ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbGaaGOmaiabfg5abjabg2da9iaaikdaciGGZbGaaiyAaiaa
c6gacqqHroqqciGGJbGaai4BaiaacohacqqHroqqaaa@4518@
|
cos2Α=
cos
2
Α−
sin
2
Α
=1−2
sin
2
Α=2
cos
2
Α−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGJb
Gaai4BaiaacohacaaIYaGaeuyKdeKaeyypa0Jaci4yaiaac+gacaGG
ZbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0Iaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeeabaGaeyypa0Ja
aGymaiabgkHiTiaaikdaciGGZbGaaiyAaiaac6gadaahaaWcbeqaai
aaikdaaaGccqqHroqqcqGH9aqpcaaIYaGaci4yaiaac+gacaGGZbWa
aWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0IaaGymaaaaaa@5856@
|
sin3Α=3sinΑ−4
sin
3
Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbGaaG4maiabfg5abjabg2da9iaaiodaciGGZbGaaiyAaiaa
c6gacqqHroqqcqGHsislcaaI0aGaci4CaiaacMgacaGGUbWaaWbaaS
qabeaacaaIZaaaaOGaeuyKdeeaaa@47BE@
|
sin
2
Α=
1
2
−
1
2
cos2Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyypa0ZaaSaa
aeaacaaIXaaabaGaaGOmaaaacqGHsisldaWcaaqaaiaaigdaaeaaca
aIYaaaaiGacogacaGGVbGaai4CaiaaikdacqqHroqqaaa@4512@
|
sin
4
Α=
3
8
−
1
2
cos2Α+
1
8
cos4Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacaaI0aaaaOGaeuyKdeKaeyypa0ZaaSaa
aeaacaaIZaaabaGaaGioaaaacqGHsisldaWcaaqaaiaaigdaaeaaca
aIYaaaaiGacogacaGGVbGaai4CaiaaikdacqqHroqqcqGHRaWkdaWc
aaqaaiaaigdaaeaacaaI4aaaaiGacogacaGGVbGaai4Caiaaisdacq
qHroqqaaa@4C7C@
|
cos
5
Α=
5
8
cosΑ+
5
16
cos3Α+
1
16
cos5Α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaWbaaSqabeaacaaI1aaaaOGaeuyKdeKaeyypa0ZaaSaa
aeaacaaI1aaabaGaaGioaaaaciGGJbGaai4BaiaacohacqqHroqqcq
GHRaWkdaWcaaqaaiaaiwdaaeaacaaIXaGaaGOnaaaaciGGJbGaai4B
aiaacohacaaIZaGaeuyKdeKaey4kaSYaaSaaaeaacaaIXaaabaGaaG
ymaiaaiAdaaaGaci4yaiaac+gacaGGZbGaaGynaiabfg5abbaa@5220@
|
sinΑ+sinΒ=2sin
1
2
(
Α+Β
)cos
1
2
(
Α−Β
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbGaeuyKdeKaey4kaSIaci4CaiaacMgacaGGUbGaeuOKdiKa
eyypa0JaaGOmaiGacohacaGGPbGaaiOBamaalaaabaGaaGymaaqaai
aaikdaaaWaaeWaaeaacqqHroqqcqGHRaWkcqqHsoGqaiaawIcacaGL
PaaaciGGJbGaai4BaiaacohadaWcaaqaaiaaigdaaeaacaaIYaaaam
aabmaabaGaeuyKdeKaeyOeI0IaeuOKdieacaGLOaGaayzkaaaaaa@542B@
|
cos
−1
(
−χ
)=π−
cos
−1
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWaaeaacqGH
sislcqaHhpWyaiaawIcacaGLPaaacqGH9aqpcqaHapaCcqGHsislci
GGJbGaai4BaiaacohadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqaH
hpWyaaa@48EF@
|
cos(
χ×ψ
)+
tan
2
χ+ψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaeWaaeaacqaHhpWycqGHxdaTcqaHipqEaiaawIcacaGL
PaaacqGHRaWkciGG0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaa
GccqaHhpWycqGHRaWkcqaHipqEaaa@48FC@
|
cos(
χ×ψ
)+
tan
2
(
χ+ψ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaeWaaeaacqaHhpWycqGHxdaTcqaHipqEaiaawIcacaGL
PaaacqGHRaWkciGG0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaa
GcdaqadaqaaiabeE8aJjabgUcaRiabeI8a5bGaayjkaiaawMcaaaaa
@4A85@
|
cosχ×ψ+
tan
2
χ+ψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaeq4XdmMaey41aqRaeqiYdKNaey4kaSIaciiDaiaacgga
caGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaey4kaSIaeqiYdK
haaa@4773@
|
cosχ×ψ+
tan
2
(
χ+ψ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaeq4XdmMaey41aqRaeqiYdKNaey4kaSIaciiDaiaacgga
caGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacqaHhpWycqGHRa
WkcqaHipqEaiaawIcacaGLPaaaaaa@48FC@
|
tan
(
χ+2×ψ
)
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg
gacaGGUbWaaeWaaeaacqaHhpWycqGHRaWkcaaIYaGaey41aqRaeqiY
dKhacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@4274@
|
sin
2
(χ+ψ)+cosχ+
ψ
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiabeE8aJjabgUca
RiabeI8a5jaacMcacqGHRaWkciGGJbGaai4BaiaacohacqaHhpWycq
GHRaWkcqaHipqEdaahaaWcbeqaaiaaikdaaaaaaa@4887@
|
cos(
xψ
)+
tan
2
x+ψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k
aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEai
abgUcaRiabeI8a5baa@4571@
|
cos(
xψ
)+
tan
2
(
χ+y
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k
aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaae
aacqaHhpWycqGHRaWkcaWG5baacaGLOaGaayzkaaaaaa@46E4@
|
cosχy+
tan
2
χ+y
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa
CaaaleqabaGaaGOmaaaakiabeE8aJjabgUcaRiaadMhaaaa@43BC@
|
cosχy+
tan
2
(
x+ψ
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa
CaaaleqabaGaaGOmaaaakmaabmaabaGaamiEaiabgUcaRiabeI8a5b
GaayjkaiaawMcaaaaa@455B@
|
cosh
2
χ−
sinh
2
χ=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaakiabeE8aJjabgkHi
TiGacohacaGGPbGaaiOBaiaacIgadaahaaWcbeqaaiaaikdaaaGccq
aHhpWycqGH9aqpcaaIXaaaaa@457C@
|
sin
2
χ+
cos
2
χ=1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaey4kaSIaci4y
aiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaeyypa0
JaaGymaaaa@4399@
|
sin
−1
χ≠sin
χ
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM
gacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeq4XdmMaeyiy
IKRaci4CaiaacMgacaGGUbGaeq4Xdm2aaWbaaSqabeaacqGHsislca
aIXaaaaaaa@4490@
|
cos(
χ+ψ
)=cosχ⋅cosψ−sinχ⋅sinψ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbWaaeWaaeaacqaHhpWycqGHRaWkcqaHipqEaiaawIcacaGL
PaaacqGH9aqpciGGJbGaai4BaiaacohacqaHhpWycqGHflY1ciGGJb
Gaai4BaiaacohacqaHipqEcqGHsislciGGZbGaaiyAaiaac6gacqaH
hpWycqGHflY1ciGGZbGaaiyAaiaac6gacqaHipqEaaa@57A1@
|
cos2⋅χ=
cos
2
χ−
sin
2
χ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+
gacaGGZbGaaGOmaiabgwSixlabeE8aJjabg2da9iGacogacaGGVbGa
ai4CamaaCaaaleqabaGaaGOmaaaakiabeE8aJjabgkHiTiGacohaca
GGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeE8aJbaa@4A79@
|