Mineral resource assessment: Compliance between Emergy and Exergy respecting Odum's hierarchy concept

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HAL Id: hal-00879451

https://hal.archives-ouvertes.fr/hal-00879451

Submitted on 4 Nov 2013

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Emergy and Exergy respecting Odum’s hierarchy

concept

Nadia Jamali-Zghal, Olivier Le Corre, Bruno Lacarrière

To cite this version:

Nadia Jamali-Zghal, Olivier Le Corre, Bruno Lacarrière. Mineral resource assessment: Compliance between Emergy and Exergy respecting Odum’s hierarchy concept. Ecological Modelling, Elsevier, 2014, 272, pp.208-219. �10.1016/j.ecolmodel.2013.09.017�. �hal-00879451�

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(9)

!" #$% &'(#$)* +(,*# !"#- .!"%*/ 0**,.!"12 #$'# '" -(% 3%4-*!# !* '" !3%'5 .!6#,(% !! -7 ' .!"%('5 M '"3 (-+82 #$% *4%+!9+ +-"+%"#('#!-" %6%(1: -7 #$% .!"%('5 excM +'" !" ;% %<,'#%3 #- #$% .!"!.,. '.-,"# -7 =-(8 #$'# *$-,53 ;% 7,("!*$%3 #- *%4'('#% #$% !# .!"%('5 >+-"+%"#('#%3 '# xM? 7(-. #$% (-+82 *%% &<,'#!-" @/ A# +'" ;% *%%" #$'# #$% "$ $!1$%( #$% -(% 1('3%2 #$% 5-=%( #$% %B-(# "%%3%3 #- %6#('+# #$% .!"%('5 7(-. #$% .!"%2 " 4-!"#!"1 #- #$% 7'+# #$'# +-"+%"#('#%3 .!"%('5* (%4(%*%"# ' "'#,('5 %"%(1: +-"#(!;,#!-"2 "% %'*!"1 !#* %6#('+#!-"/ "& excM(xM) = −RT ◦ ln(λMxM) + (1 − λMxM) λMxM ln (1 − λMxM) >@? "' C$%(% R (%4(%*%"#* #$% ,"!D%(*'5 1'* +-"*#'"#2 T◦ #$% *#'"3'(3 '.;!%"# #%.4%(E "( '#,(% '"3 λM #$% .-5'( .'** ('#!- ;%#=%%" #$% .!"%('5 '"3 #$% .!6#,(% -7 (-+8 '"3 ") .!"%('5/ "! F'5%(- !3%"#!9%* #$(%% #$%-(%#!+'5 *#'#%* -7 .!"%('5 %D-5,#!-" #- +'5+,5'#% #$% %6%(1: "" #$'# *$-,53 ;% !"D%*#%3 #- 7-(. .!"%('5 3%4-*!#* 7(-. #$% (%7%(%"+% %"D!(-".%"#2 *%% "# G!1,(% HI #$ J/ K#'#% L >M%7%(%"+% %"D!(-".%"#?I 055 #$% (%7%(%"+% *,;*#'"+%* '(% 3!*4%(*%32 # .!6%3 '"3 !" #$%(.-3:"'.!+ %<,!5!;(!,. ⇒ N$% %6%(1: !* O%(-/ #% #& H/ K#'#% J >P!*4%(*%3 .!"%('5*?I Q$%.!+'5 (%'+#!-"* #--8 45'+%2 #$% .!"%('5* '(% #' 7-(.%3 '"3 3!*4%(*%3 '# #$%!( +(,*#'5 +-"+%"#('#!-" xMcr -" #$% &'(#$ +(,*#2 #$% #( *4%+!9+ %6%(1: -7 #$% 3!*4%(*%3 .!"%('5 exM(xMcr) -7 #$% .!"%('5 !*I #) exM(xMcr) = exchM >R? @/ K#'#% H >Q-"+%"#('#%3 .!"%('5*?I N$% 3!*4%(*%3 .!"%('5 !* +-"+%"#('#%3 '# xM #! !"#- .!"%*2 #$% *4%+!9+ %6%(1: -7 #$% +-"+%"#('#%3 .!"%('5 exM(xM) !* #$%"I #"

(10)

exM(xM) = exchM + excM(xM) !"#

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" ( exR(xMcr) = ex r M(xMcr) = kchM ·exchM ExR(xMcr) → 0 !J#

(11)

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◦ M)7 ?# % #%2$ $#-4+ #%+ $#&0+) +3+04. ExR(x ◦ M)0+-/%+$ 2#$ 1-321'17 & exR(x ◦ M) = ex r M(x ◦ M) = kchM ·exchM + kcM ·excM(x ◦ M) ExR = m ◦ M·MM·exR(x ◦ M) @AB m◦

M 2$ #%+ 2,2#2-( #&#-( 1-$$ &; 12,+0-( 2, #%+ 12,+8 MM 2$ #%+ 1&(-0 1-$$ &; #%+

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(12)

lim xM→_x_McrexR(xM) = kchM ·exchM lim xM→_x_McrExR(xM) = 0 ! "# $%&'() * ()+(),)-., ./) ,+)0%10 )2)(&3 45 ./) 6%-)(78 (),)(9) 7. :%;)()-. !"

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(13)

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0B+0 .0, ,@">.A> "#"$%& emM(xMcr) ., @$'@'$0.'/+) 0' 0B" ,@">.A> "#"$%& '( 0B" %" 2+$0BC, >$1,0 emcr +/- 0' .0, >$1,0+) >'/>"/0$+0.'/ xMcrE ,"" 2N1+0.'/ O %# !!"#$%&'( ) * !"#$% &' ()*+"#*"( !),"#-.* + %$ "*+,- .,/01 /2 345+,67 48 68803+9 :/ ;+ 6 </=1,/90<: /2 :>+ "6,:>?8 <,08:@ #>+ 6;8/70:+ +3+,.- /2 65- 345+,67 .,/01 EmM(xMcr)48 +A067 :/ :>+ 6;8/70:+ +3+,.- /2 :>+ "6,:>?8 <,08: EmcrB EmM(xMcr) def = Emcr %! %% mMcr $"@$","/0, 0B" #+,, '( 0B" #./"$+) ./ 0B" 2+$0BC, >$1,0 +/- mcr $"@$","/0, %& 0B" 0'0+) #+,, '( 0B" 2+$0BC, >$1,07 P"/>" 0B" ,@">.A> "#"$%& '( + %' #./"$+) ME emM(xMcr)>+/ *" "J@$",,"- +, ('))'9,O &( mMcr = xMcr ·mcr emM(xMcr) = Emcr mMcr = emcr xMcr : =

Q-1# :,"" R.0+0.'/ != "#@B+,.K"- 0B+0 0B" "#"$%& @"$ #+,, :'$ ,@">.A> "#"$%&= &)

'( + #./"$+) F+$& 9.0B .0, +*1/-+/>" ./ 0B" 2+$0BC, >$1,07 MB" #'$" 0B" #./"$+) &

(14)

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* !"# $"%&'!(' $" %)"%'"*&!*$)" )+ ,!*'&$!- &'./$&'( !" $"%&'!(' $" *0' '"'&1# 2'& ,!(( 30'" %)"%'"*&!*$)" $"%&'!(' $" (),' 2!&* )+ ! 4$)1')%0',$%!- %#%-'5 *0' ','&1# 2'& ,!(( $"%&'!('( 6 < :(&0 <>BBB$??

'( ')

>4 7)"%'"*&!*$)" )+ ,$"'&!-( $"*) )&' 8'2)($*(C :(&0 <=")$)"-' D? #)/+##+( )*$) 3-0E ' ;$/+( )- ")# ("#;+/#+( #)$)+ <$) 3/&#)$, 3-'3+')/$)"-' xMcr?@ 3-'3+')/$)+( 0"'+/$,# '" /+8&"/+( 0-/+ .-/F )- %+ 6-/0+( $'( )*&#@ )*+ 0-/+ )*+ 0"'+/$, "# 3-'3+')/$)+( '# )*+ 0-/+ $G$",$%,+ +'+/12 *$( %++' (+1/$(+( $'( )*+ *"1*+/ )*+ +0+/12 ;+/ 0$## '$ .",, %+ <"4+4 )*+ *"1*+/ ")# ;-#")"-' "' )*+ 0$)+/"$, *"+/$/3*2?4 H3)&$,,2@ )*+ 8&$,E '% ")2 -6 $ 0"'+/$, /+#+/G+ (+;+'(# -' )*+ 8&$')")2 m◦ M $'( -' )*+ 3-'3+')/$)"-' '& -6 )*+ #)-/+( 0"'+/$, x◦ M4 A' )*"# .-/F@ ") "# 3-'#"(+/+( )*$) )*+ #$0+ #-&/3+# '! -6 +'+/12 <τF "# )*+ $G+/$1+ )/$'#6-/0")2 -6 )*+#+ #-&/3+#? )*$) 3/+$)+( ("#;+/#+( ''

0"'+/$,#@ 3-'3+')/$)+( 6&/)*+/ $ ;$/) -6 )*+0 "')- 0"'+#@ #++ I"1&/+ J4 5*&#@ "(( )*+ #;+3"K3 +0+/12 -6 )*+ 0"'+/$, /+#+/G+ emR 3$' %+ 3$,3&,$)+( %2 &#"'1 )*+ "() 0"'+/$,L# +9+/12@ )*+ #;+3"K3 +0+/12 -6 )*+ ("#;+/#+( 0"'+/$, emM(xMcr)$'( )*+ "( -/+ 1/$(+ -6 )*+ (+;-#") x◦ M4 "(" )**+,-"!$% . '1/"#$% -, /'!"#(* #"0"#2"0 ( "(# #+, -./01,- .2 ,3,045 6+76 10,76,8 89-:,0-,8 ;93,07<- 93 6+, "706+=- 10/-6 70, "($

(15)

*+,*-.*/ .0 1* .2* 345* 3067-* .24. -08-*8.74.*/ 4 ,47. 09 .2*5 :8.0 5:8*74; 7*3*7<*3= τF def = EmM(xMcr) ExM(xMcr) def = EmR(x ◦ M) ExR(x ◦ M) " #$ !" !# %&' ()(*(+, -.'/(0/ '1'234 emR(x ◦ M) 56 +) 52' 2'-'27' 56 1()'2+, M 8(*& +) 52' !$ 32+9' x◦ M /+) :' /+,/;,+*'9< *&')< :4 ;-()3 =>;+*(5) ? @ !% emR(x ◦ M) = emM(xMcr) · exR(x ◦ M) exM(xMcr) = emcr xMcr · kchM ·exchM + kcM ·excM(x ◦ M) kchM ·exchM = emcr xMcr · 1 +kcM ·excM(x ◦ M) kchM ·exchM " ?$ &! %&' *5*+, +15;)* 56 '1'234 EmR(x ◦ M) -*52'9 ()(*(+,,4 () *&' 1()' /+) :' 'AB && .2'--'9 +- 65,,58-@ &' EmR(x ◦ M) = m ◦ M ·emR(x ◦ M) " !$ C* -&5;,9 :' ;)9'2,()'9 *&+* emR(x ◦ M) +)9 EmR(x ◦

M) +2' *&' &(3&'-* 7+,;'- *&'

&

1()' /+) '7'2 2'+/& 9;2()3 (*- ,(6' /4/,'D &(

C* /+) :' )5*'9 *&+* *&' 2'-;,*- 56 =>;+*(5) ? +2' /5)-(-*')* 8(*& *&' 1+*'2(+, &)

&('2+2/&4D %&' -.'/(0/ '1'234 56 *&' 1()'2+, 2'-'27' ()/2'+-'- 8(*& (*- 52' 32+9' &"

x◦

M +)9 /5)7'2-',4 8&') (*- 52' /5)*')* 9'/,()' *5 (*- ,58'-* 7+,;' "xMcr$< *&' &#

-.'/(0/ '1'234 56 *&' 9'.5-(* *')9- *5 (*- ,58'-* ,'7', () *&' 1+*'2(+,E- &('2+2/&4< &$

8&(/& /522'-.5)9- *5 *&' /2;-*+, -.'/(0/ '1'234 56 *&' 1()'2+,@ &%

(16)

lim

x◦ M→xMcr

emR = emM(xMcr)

"#$%&#$' ()#* +)# ,-.*&,*/# 01 +)# 2%*#3,4 xMcr +#*&$ +0 5()%/) 2#,*$ +),+ !" +)# ()04# 6,3+)7$ /3.$+ %$ /0280$#& 01 2%*#3,4 9:' +)# $8#/%;/ #2#3<= 01 +)# !# 2%*#3,4 3#$#3># +#*&$ +0 +)# $8#/%;/ #2#3<= 01 +)# 6,3+)7$ /3.$+? !! lim xMcr→₁ emR = emcr

@* 03&#3 +),+ 6A.,+%0* 83080$#& -= 5B0)#* #+ ,4C' DEEF: ,*& 5"30(*' DEEF: (%44 !

-# /0)#3#*+ (%+) +)# 2,+#3%,4 )%#3,3/)= /0*/#8+' +)# #*3%/)2#*+ 1,/+03 %$ #G8#/+#& !$

+0 #G83#$$ +)# 3#,4 (03H +),+ I,+.3# ),& %*>#$+#& +0 /0*/#*+3,+# +)# 2%*#3,4 !%

,*& +)# -,$#4%*# $)0.4& -# +)# 2%*%2.2 $8#/%;/ #2#3<= +)# 2%*#3,4 /0.4& 3#,/) !&

*,+.3,44= ()#* %+ %$ &%$8#3$#&C J)#3#103#' +0 #*),*/# +)%$ 2#+)0&040<= +)# 3#$.4+$ !'

01 6A.,+%0* K 2,= -# .$#&C @*$+#,& 01 ,* .*%>#3$,4 -,$#4%*# BR 103 ,*= 2%*#3,4

!(

3#$#3>#' +)# /3.$+,4 $8#/%;/ #2#3<= 01 +)# /033#$80*&%*< 2%*#3,4 emM(xMcr) %$ !)

83080$#&' $%*/# %+$ 20$+ *,+.3,4 &#<3,&#& $+,+# %$ ()#* %+ %$ &%$8#3$#&C J)# " #*3%/)2#*+ 1,/+03 ERR /,* -# &#;*#& ,$ +)# 3,+%0 -#+(##* +)# $8#/%;/ #G#3<= 01 # +)# /0*/#*+3,+#& 03# 3#$#3># exR(x ◦ M),*& +)# /3.$+,4 $8#/%;/ #G#3<= 01 +)# 2%*#3,4 !

exM(xMcr)C J)%$ 8#32%+$ +0 %*+#<3,+# +)# #L03+ %*>#$+#& -= I,+.3# +0 .8<3,&#

+)# 2%*#3,4 1302 %+$ &%$8#3$#& $+,+# +0 %+$ /0*/#*+3,+#& $+,+# %* +)# 2%*#C $

!"#"$%&%"' ( ) !"#$%&# !&' #&(%)*+#&, -!),.( .- +%&#(!$ (#"#(/#" * %

#*+ ,-.+/01+ 0. -..23+4 56 ,+ 5*+ 782.5-/ .9+70:7 +3+8;< 6= 5*+ 301+8-/>

(17)

BR(xMcr) def = emM(xMcr) " #$ #*+ +,-./*0+,1 23/14- .5 6+7,+6 35 1*+ -31.4 8+19++, 1*+ 5:+/.7/ +;+-<= 42 1*+ /4,/+,1-31+6 4-+ -+5+->+ exR(x ◦ M) 3,6 1*+ /-?513@ 5:+/.7/ +;+-<= 42 1*+ 0.,+-3@ exM(xMcr)A EFR(x ◦ M) def = exR(x ◦ M) exM(xMcr) " !$ ! "

%& '()*+, -. */,.0+1/., &(2& 3)0 2/ 2440)4012&. .5.067 .82+*2&1)/9 :0'& &(. ')*0;.'

#

2/, 42&(<27' '()*+, -. 1,./&1:.,= >(*'9 1& '..5' */0.2+ &) .?*2&.9 2' ,)/. 1/

$%

@?*2&1)/ 9 &(. '4.;1:; .5.067 )3 51/.02+ 0.'.08.' <1&()*& ;)/'1,.01/6 &(.10

$&

3)052&1)/ 40);.''= >(.0.3)0.9 A0)4)'1&1)/ <2' 52,. 23&.0 ,.:/1/6 2 &(.)0.&1;2+

$'

.8)+*&1)/ 40);.'' )3 51/.02+ 0.'.08.'9 1++*'&02&., 1/ B16*0. C= >(. 0.'*+&' 0.D.;&'9

$

&(./9 &(. 42&(<27 )3 51/.02+ 0.'.08. 3)052&1)/ 2' &(. '4.;1:; .5.067 )3 &(.

$$

0.'.08. ,.4./,' )/ &(. ;(.51;2+ 40)4.0&1.' 2/, 2-*/,2/;. )3 &(. 51/.02+ 2/, &(.

$(

1/1&12+ ;)/,1&1)/' )3 &(. 0.'.08.9 '.. @?*2&1)/ E=

$)

!"#"!$% & ' ! "!"#$% &'"#(#)&% *(+ ,-# .'-)&"/')(* )%)*"0 ( $!

B !"# $%&"'(%) *(+,"'+"+- &!" +&.'"* "$"'/0 *"1'"%+"+2 3 " F,*5 "GHHH2$$

$" $#

E= !"#$%"&'( ') "*+ ,&(+#$- )#', "*+ ,&(+ !"#$% &' #&( )*+,#)-. -$/ )'$)+$&0-&#'$ !"

1'&+$&#-.2 - ,#$+0-. 0+(+03+ )-$ 4+ )'$(#/+0+/ -( -$ +,+0%5 0+(+03'#0 670$#(*+/ !#

45 8-&70+ &' ,++& *7,-$ $++/(9 :#&* &*+ +;&0-)&+/ ,#$+0-. &*+ ,#$+ .'(+( !$

)*+,#)-. -$/ )'$)+$&0-&#'$ 1'&+$&#-.2 &*+ ,#$+0-. )'$&+$& mM -$/ %0-/+ xM '6

! &*+ '0+ 4'/5 /#,#$#(*+( <mM < m ◦ M= xM < x ◦ M> "*#)* .+-/( &' - 0+/7)&#'$ !%

'6 #&( (&'0+/ +,+0%59 ?& #( +;1+)&+/ &*-&2 /70#$% &*+ ,#$#$%2 $'& '$.5 &*+ &'&-. !!

(18)

"#"$%& '( )*" '$" +',& EmR ,-#-.-/*"/ +0) 12/' -)/ /3"4-54 "#"$%& em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

)*" #-."$12 41. +" /-#021)", +& 1 .1)0$12 ,-20)-'. '( )*" '$" +',& %"."$1)", +& "! )*" /1#" /'0$4"/ '( "."$%& )*1) 4'.4".)$1)", -) +"('$"6 /"" A-%0$" B8 >*-/ #"1./ "" )*1) )*" '$" +',& '( )*" $"/"$;" 7-22 +" ,-20)", ($'# -)/ -.-)-12 '$" %$1," x◦ M )' 1 "# 2'7"$ 4'.4".)$1)-'. xM 1()"$ #-.-.%8 "$

!!"#$%&'( ) * !"#$% &"'#"()" *+ &"',-.-.$ !-."#(, #")"#/") + "% #*+ +,-./0-123 24 -*+ 513+./6 4.25 -*+ 513+ 0/3 7+ /881516/-+9 -2 / 3/-:./6 916:-123 24 -*+ 2.+ 729; 4.25 1-8 131-1/6 2.+ <./9+ x◦ M -2 / 62=+. 0230+3-./-123 xM> <+3+./-+9 7; -*+ 8/5+ 82:.0+8 24 +3+.<; -*/- ?.+@12:86; 0230+3-./-+9 -*+ 513+./6 13-2 -*+ 513+A τF def = EmR(x ◦ M) ExR(x ◦ M) def = EmR(xM) ExR(xM) < != #& #' >*0/ )*" /3"4-54 "#"$%& '( )*" ,"42-.-.% '$" $"/"$;" emR(xM) 41. +" ":3$"//", #( 1/ ('22'7/ C #

(19)

emR(xM) = emR(x ◦ M) · exR(xM) exR(x ◦ M) "##$%&'() *$ +,-.*'$( /0 emR(xM) = emcr xMcr · kchM ·exchM + kcM ·excM(xM) kchM ·exchM = emcr xMcr · 1 +kcM ·excM(xM) kchM ·exchM 1 !2 !" 345 %56.'('() 7*$%5& 565%)8 .9*5% 6'('() EmR(xM):'*4 .( $%5 )%.&5 xM #$(; !# *.'('() . ,-.(*'*8 mM $9 5<*%.#*.=>5 6'(5%.>? '7 *45(0 !$ EmR(xM) = mM ·emR(xM) 1 @2

A(#5 *45 $%5 %575%B5 4.7 =55( 5<4.-7*5&? '*7 $%5 )%.&5 .(& 7C5#'D# 565%)8 !! .CC%$.#457 %57C5#*'B5>8 *45 #%-7*.> #$(#5(*%.*'$( xMcr .(& *45 #%-7*.> 7C5#'D# !% 565%)8 $9 *45 6'(5%.> emR(xMcr)E 345$%5*'#.>>8? *45 #$(7-65& %575%B5 '7 .* *45 !& >$:57* C$7'*'$( '( *45 6.*5%'.> 4'5%.%#48 *4.* '* #.( 5B5% .**.'( (.*-%.>>8 E 345 %'

7*$%5& 565%)8 $9 *45 %575%B5 *5(&7 *$ F5%$E 345 #456'#.> C$*5(*'.> $9 *45 6'(5%.> %( '( *45 6'(5 '7 ($ >$()5% 5<C>$'*.=>5E %) lim xM→_x_Mcr emR(xM) = emM(xMcr) lim xM→x_Mcr EmR(xM) = 0 1GH2 "(.>$)$-7 *$ +,-.*'$( ? *45 7C5#'D# 565%)8 $9 . &5#>'('() 6'(5%.> %575%B5 %

emR(xM) 6.8 =5 5<C%5775&? .7 . 9-(#*'$( $9 '*7 =.75>'(5 BR .(& '*7 &5C>5*'$(

%" 9.#*$% DFRE 3$ =5 '( #$(9$%6'*8 :'*4 *45 6.*5%'.> 4'5%.%#48? *45 =.75>'(5 74$->& %# %5C%575(* *45 4')457* 7C5#'D# 565%)8 $9 *45 6'(5 &-%'() '*7 >'95 #8#>5? .(& *45 %$ &5C>5*'$( 9.#*$% '7 5<C5#*5& *$ ,-.(*'98 *45 #$(#5(*%.*'$( .='>'*8 $9 *45 6'(5 .* %! .(8 7*.)5 $9 &5)%.&.*'$(E "##$%&'() *$ +,-.*'$( !? emR(x ◦ M)#.( =5 &5D(5& .7 %%

(20)

"#$ %&'$()*$ +, "#$ -$'$-.$/ "#$ -&")+ %$"0$$* "#$ '1$2)32 $4$-56 +, "#$ +-$ %+76 !"

&,"$- 8)*)*5 exR(xM)&*7 "#$ )*)")&( '1$2)32 $4$-56 +, "#$ 8)*$ exR(x

◦

M) 2&* %$

"#

2+*')7$-$7 &' "#$ 7$1($")+* ,&2"+- +, "#$ -$'$-.$9 "$

!"#"$%&%"' ( ) !"#$%&# !&' '#($#)%*& +!,)*- *+ '#,$%&%&. /%&#-!$ -#"#-0#" * "% #*+ ,-,.,/0 12+3,43 +5+678 9: .*+ 5,-+6/0 ,1 269291+; /1 </1+0,-+= BR(x ◦ M) def = emR(x ◦ M) :; < #*+ ;+20+.,9- :/3.96 ,1 ;+4-+; /1 .*+ 6/.,9 <+.>++- .*+ 12+3,43 +?+678 9: .*+ 96+ <9;8 /:.+6 5,-,-7 exR(xM) /-; .*+ ,-,.,/0 12+3,43 +?+678 9: .*+ 5,-+ exR(x ◦ M)= DFR(xM) def = exR(xM) exR(x ◦ M) :;;< " "& !"! ! #$%&'( %)*+,*-./0 /1 +*02 2%'&*2*-./0 2,% -/ $.0.0' "'

= 8)*$ 2&* %$ 2+*')7$-$7 &' & 1&-" +, "#$ >&-"#?' 2-@'" 2+*"&)*)*5 #)5# 2+*2$*"-&"$7 "(

8)*$-&( +-$'9 A#$ $4"-&2")+* +, "#$'$ +-$' *+" +*(6 2&@'$' "#$ (+'' +, 8)*$?' 2#$8)2&( ")

&*7 2+*2$*"-&")+* 1+"$*")&( %@" &('+ "#$ 7$.&'"&")+* +, "#$ (&*7'2&1$9 B+*"-&-)(6 "+ "!

"#$ 1-$.)+@' '$2")+* 0#$-$ +*(6 "#$ 7$1($")+* +, "#$ 8)*$-&( -$'$-.$ #&' %$$* '"@7)$7/ ""

"#$ 1-$'$*" '$2")+* 7$&(' 0)"# "#$ 7$5-&7&")+* +, "#$ 0#+($ 8)*$ 7@$ "+ )"' $41(+)"&")+* &##

&*7 "#$ $C+-"' "#&" '#+@(7 %$ &11()$7 "+ -$8$76 "#$ 8)*)*5 )81&2"'9 &#$

A#$ 2+*2$*"-&"$7 8)*$-&( 2+*"$*" +, "#$ 8)*$ 7)C$-$*")&"$' )" ,-+8 )"' $*.)-+*8$*"9 &#%

D" 2&* %$ &''@8$7/ "#$*/ "#&" "#$ "+"&( $8$-56 +, "#$ 8)*$ EmM ine)' $E@&( "+ "#$ "+"&(

&# $8$-56 +, )"' 8)*$-&( -$'$-.$ EmR9 &#& +$$,-#&%"' . )1/#-.2 *+ /%&#"* &#' #*+ .9./0 +5+678 9: .*+ >*90+ 5,-+ EmM ine ,1 +@A/0 .9 .*+ .9./0 +5+678 9: &#(

(21)

*+, -*./012 0/,/03/ EmR4 EmM ine(xM) = EmR(xM) mM = xM ·mM ine " #$ !" !#

%&&'()*+, -' ./01-*'+ 23 -45 675&*8& 595(,: '; -45 9*+5 emM ine &1+ <5 5=7(5665)

!$ 16 ;'>>'?6@ %! emM ine= xM ·emR(xM) emM ine(xM) = emcr· xM xMcr · 1 + kc·exc(xM) kch·exch " A$

B45+ -45 9*+5 *6 &'97>5-5>: 5=4106-5) *-6 '(5 ,(1)5 -5+)6 -' -45 &(06-1> &'+&5+-(1C %%

-*'+ '; -45 9*+5(1> xMcr 1+) -45 6-'(5) 595(,: *+ ;'(9 '; 9*+5(1> (565(D56 177('1&456 %&

E5(' <5&1065 -45 /01+-*-: '; 5=-(1&-1<>5 9*+5(1> <5&'956 +5,>*,*<>5F G165) '+ ./01C %' -*'+ A -45 675&*8& 595(,: '; -45 )57>5-5) 9*+5 -5+)6 -' -45 675&*8& 595(,: '; -45 % .1(-4H6 &(06- emcr@ %( lim xM→_x_McremM ine(xM) = emcr " I$ %) J4*6 64'?6 -41- -45 )57>5-5) 9*+5 >'6- 1>> 6 675&*8& 7'-5+-*1>6 -41- )*K5(5+-*1-5 %" ;('9 -45 (56- '; -45 60(('0+)*+, >1+) "?45(5 -45 1D5(1,5 675&*8& 595(,: *6 emcr$F L+ %# -5(96 '; 91-5(*1> 4*5(1(&4:M -4*6 951+6 -41- -45 9*+5 *6 1- -45 >'?56- 7'6*'+ -41- %$

&1+ 5D5( (51&4 *+ -45 4*5(1(&4:F G56*)56M )0(*+, 9*+*+,M -45 .1(-4 *+ -45 9*+5 416 <55+ &!

(59'D5) -' '<-1*+ -45 9*+5(1>F N5+&5M -45 -'-1> 595(,: '; -45 597-: 9*+5 -5+)6 -' &%

E5('M 1+) -45 7'6-C9*+*+, 5&'6:6-59 *6 '0- '; <1>1+&5 <5&1065 1 71(- '; -45 .1(-4H6 && *6 9*66*+,F %))*-*'+1> ?'(O *6 (5/0*(5) -' (56-'(5 -45 )5D16-1-5) >1+)F L- *6 7'66*<>5 &' -' <0>>)'E5 -45 51(-4 *+-' -45 9*+5M <0- P)09 675&*85) "655 Q*-1-*'+ 3$ -41- +1-0(1> & (56-'(5) 7'6-C9*+*+, >1+) *6 9'(5 D1>01<>5 -41+ 1(-*8&*1>>: (5&'D5(5) >1+)F J406M -45 &(

(22)

"#$%&$"'()* "%")* $+$&,- "#)" .)"/&$ 0#%/*1 '23$0" "% &$0"%&$ 4%& '"0$*4 "#$ 5)*)2($ %4 !"

"#$ $(%0-0"$+ EmRest ()2 5$ ()*(/*)"$1 5- +/*"'6*-'2, "#$ &$+%3$1 +)00 %4 "#$ +'2$

!#

mrem 7'"# "#$ 06$('8( $+$&,- %4 "#$ 9)&"#:0 (&/0" emcr;

!$

EmRest = mrem·emcr < =>

?" 0#%/*1 5$ %/"*'2$1 "#)" EmRest &$6&$0$2"0 "#$ "#$%&$"'()**- +'2'+/+ 7%&@ "#)"

!%

0#%/*1 5$ 1%2$ "% &$$0")5*'0# "#$ 6%0"A+'2'2, *)21B 5$()/0$ )4"$& &$0"%&)"'%2 "#$ 4%&+$& &'

+'2$ 5$(%+$0 )2 %&1'2)&- 6)&" %4 "#$ 9)&"#:0 (&/0" 7'"#%/" )2- 6)&"'(/*)&*- (#$+'()* &(

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