Damião Cruz
2, Antonio Teixeira do Amaral Júnior
3and Carlos Alberto Scapim
1*1
Universidade Estadual de Maringá, Av. Colombo, 5790, 87020-900, Maringá, Paraná, Brazil. 2Universidade Federal de Viçosa, Viçosa, Minas Gerais, Brazil. 3Universidade Estadual do Norte Fluminense Darcy Ribeiro, Campos dos Goytacazes, Rio de Janeiro, Brazil. *Author for correspondence. E-mail: cascapim@uem.br
!! ! " # $ %
" # " " & ! ' " & " ' &
!! (" " # ! ' #! )
" " ! * + * , - .
- ' ! " ! ! !
(" " "! & && & " & ! " " & " " " & !! &
! & ! ! & " !
" " " && & ! " (" # # & ! %
" ! & & " !!
! & " &! & # ! " !
& ! # # # ! !
" ! & ! ! #
!"# $ % !! ! " # ! /
& '( ) *" + ,- ." " / 0 " 1 %" *0 2 " 3*
"* * 4. 0 0 !" & ' 0 !1!
# +2 $ "3# % 4 ! ' +2 " 5
3 6 7 "3# & !
! #! 8 +9 * + * ,
-: . - ! 8 1 ! % ! 3
! 2 2 ; & ! &
" 1 ! ! 0
3 & & & +2
! 2 0
0 & " ! !" # +9
2 &
< 2 " ! &
" 3& # ! & &! 8 =
# ! 2 3 . " ! = 2
2 2 # ! ' +2
! 0 2 ! +2
3
) 4 5 . " ! ! " 2 = # ! /
(" ' ) " > ! &
! - ' ! " $? @
; ! ! AB1!
C # " " " # D E F &
! " # D , -, G.:/
H%* - 2 * + " !! I
& " !! & ! #!
! " A F <E ! @ D %,:E(
CJGE7, ! @D G(7 ! >D <:,:E,
! KD , G.:/ ! KD *E/:/ ! KI
F " ! &
- ' ! " && " !
" ! & " # # #!
& "
& "
& G " " # A& ! <@ K
<@ K ;( <@ K <@ <@ $I - ' !
# ) L )
! " " " #
" M A N B OE
! D P ,7 ! D %,:E(
CJGE7, ! @D , G.:/ ! >I
/ " " " & &
" " )
0 #! " " # & !
" # " ! & " - ' !
A GP, P:
! D F <E ! @#D E, GP !
KD G(7 ! KI "
& . - ' ! )
! # " & " & !
! ! ; " & "
. && ! &
" ) & !
E " " !
" # & ! & &
) ! - ' ! A,EFFE !
> #D E/* ! >I 0 ! &
A 7(:,/: ! KI &
A*:GP,H F7/7 ! D *E:E, !
I A<:,:E, ! @D KD
P GP7/EGE ! KD *E/:/ ! KD
E/* ! I " & #
AP ,7 ! #D %,:E( CJGE7,
! @D *E G ! >D , G.:/ ! KD
,7P7* /;7 ! KD G(7 ! KD
,G;7/P ! D %,:E( CJGE7, !
I
E & " " " & # ! & ) !
!! & " ! ) &
" # " #
! # ! & " ! &
" # & ! ! !
#! & ! ! % "
" !! ! ! ! !& #!
! " " # &
! " " AF,HB ! D %,:E(
CHGE7, ! @D F <E ! @ D
E, GP ! KI
- " & !! ! !
& - ' ! % ! B
A K I ! ' !! ! & !
& " & " &
/ ! A I
# ! & " & "
E " !! ! A F <E ! I
!! ! !
< " " ! #
" , ! & !
! - " !
!
& " # ! & !
! & " " ! # # !
! ! A @ I #
! & ! "
! !! !
" # !! #! #
" G " & < N " "
#Q & " # &
! #
& " " "
" ! & ! # -,
G.:/ F !! " !
& ! !
" "
E " % CR ! A @I
! " # # ! &
& & " # & ! !
!! ! ,
C - ' ! (" ! "
HG- H F -, G.:/ HG- H F
* + H%* & ! # " !
& ! # " # #
<, : ! -, G.:/ HG- H F
-, G.:/
! A KI ) " ! &
!! ! " &
& " # AE F B1! I
" !! A,
-I ! S I - ' ! "
" # ! &
! ! ! D I
" # ! - ' !
! " !! " ! &
&& & ! D I
! # && ! # ! " #
! & ! ! # " "
& &
" " Q &
&& & A<:,:E, D , /
CJGE7, D E 7G ! D %,:E(
CJGE7, ! @D , G.:/ ! >I #
" # ! && ! #
" & !
" ! ! - ' ! E " *
A I % CR ! A @I , !
! A KI ) " - ' ! !
& " " & # # !
# " ! &
! " && & "
# ! !! # #
N " " & #
" !! ! " "
- ' ! ! "
!
" $
" # % !! !
" & !
!! # ! &
" '
& A
I
!! ! # $ % " # #
" ! % #! " " " #
# " @
!! # !
& # #
! &! !
& #! " && 0 &
" # " & "
O & # !! " !!
& " !!
; # !
" &
. S * , - * + D
" : "
A " & & " # I
' #! ) " "
! <! & &
! A #
I " " & ! ' "
$ ) & K AG < 7$ O 7I ! ) &
! G ! & ("
" ! #
(" & !! ! S ! " "
A<;I D " " A:;I D ! A.LI
) " D " & AN:ID & "
A I D A<:I <:
! " T " &
" !! &
> UF " ! # : #
E +2 " ) !
& $ ! # "
/ " ! &
& ) ! " ! &
" !
(" ! & & "
& " ! T & # # ! # "
% (" O T &
# # ! "
" ! & " ! &
" & ! ! & .
:# " A @@I & "
(" ! Q ! &
& .:G: & AF,HB
@I
F " " % A V $I
! " & 0
& " ! ! & !!
#! " & " Q !
" "
& && " 0 &
" & # % A V I
Q ! !! " & " 0
# " && A
" I ! "
! " && "
" # & !! A V I E & " "
&& !
! & #!
& #
(" & " " ' "
0 & " && & !
" " " ! " ! "
0 & " " &
A V I E " !
" & 0 " # #!
# & ! " &
" ! " - "
& && & ! & "
&& "
& 0 " " ! &
" & 0 " ! ! &
! " "
! A I ) " !! ! &
& & &
! E <
- ' ! ! # & &
" && & ! " "
" ! & ! & "
- " ! ! # #!
" ! " " #! #
" " & "
# & " # & " E ! ! &
" ) " ! &&
! ! - C ! A @I " ! &
! ' # !! ! ! & " !
" " & !
' # % "
" !! " " "
" !) ! E & " "
! ) ! # #!
!) ! (" " & "
! # ! & ! "
" ' - !! "
" " ! # "
AF7E -, ! I "
# " & " & !
(" ! !! " " #
& " & && & &
" A V I " " ! "
A V $I && " ! ! &
" & 0 !
(" & " " &
&& & " ! & "
A V I !
& 0 " " & "
&& # " " &
& 0 !! && "
! ! & & 0 ! #
" ! !
F "
& & " 0 V & !
& " # " " " "
A V $I # ! & "
* ) F ' A K I &
0 & " &
! &
! & " ! " "
0 & " " ! &
(" ! & " ! &
" " #! ! #
" ! # "
& ! " !! !
F " 0 & " "
& " ! " " &
A V I & " 0 &
! & " ) #!
" " & & & " "
! EE A. ,PG:,D :-:,; ,( @@D F,HB
! I 7 " " " " ! E*
#! & " " &
. :# " A @@I ! A I
F ' ! A I
: & " &&
A( #! I ! " " " ! &
v
j &! " " & " !
E " " !
!! & ("
&& & " " " M
& (" " "
" " & " & " "
j
v
ˆ
! " >K " "
("
v
j & ! " !! " ! S
" " " " & ("
! ! " & " "
& # &
F " A.L
<:I ! " !
! &
v
j " "! A( #! I
(" & " ! &
! # # "
! !! & " !
0 & " && "
" " &! &
/ " - # ) A I <
! CR A I ! A I
! A I % CR ! A @I
, ! ! A >I #
!! ! &
v
ˆ
j & !74" : & " " A
v
jI &&" A P I # ! &
! !! ! "
(
* <; :; .L N: G:
$> $ $ $ > >>
K > K $ @ >
@ K $ $$ K>>
@ K K$ > K>
@K K $@$ KK >
K K$ K
P Avi−vjI @ K $@ K
K > >K $@$ $K K >
<;S ! " " :;S " " .LS ! N:S " & G:S # & S & "
74" : & " " A
v
jI&& " A P I #
! & A<:I "
" %M" #
.
* E EE EEE
$@$ @@ >
$ $ >
$ >> @
K@$ @ >
@$ $> K
@$ > $
P Avi−vjI > @ K
$ @$ @>
ES " & ! * + EES "
! * , - EEES "
! * +
(" "
v
j ! & !! "! E
! E
" "
" " ! & "
v
j ! & !!! !! ! "
& ! & " & &
* + * , - "
& " " " " &! !
" & # ! !
& ! ! & !
" !
" F" A K$I "
& " #
&& " " !
" "
-( #! " # ! "
! & !! "
!!
v
j && S <; AK @ I :; A $ I .LA $ I N: A KKI G: A$ @I A@ @ I <: & "
" & ! * +
A I <: & " "
! * , - A >I
<: & " "
! * + A I
: & " # "
" ! ! "
" # " !
! " ( #!
; # " ! & " 0 &&
& " & 0 # !
" ! ! & (" "
" # !
! ! & : "
" #! "
! # &
# " & "
# ' "
& ! &
A*:GF7* OLD F,HB K D F <E !
D F,HB ! D F,HB $I
<
Q "
W ! & ! " "
" " ! " &
" & " " " "
&! " & & " 7 !
" # A I " ! & "
Q
"W & # & &
! # "
&! & " && 7 " " "
ij
hˆ ! Q !
& & " "
# ! " & "
" & " #
" & " " ! " & "
& " A( #! I
! ! & " & "
"
! & "WQ !! !
" # !
&! " & # " A( #! I
74" : & " ! " # "
" ! ! " A P I
# " # " ! ! &
! & & %M &
!! ! "
(
; # <; :; .L N: G:
K$ $ >$ $ K$ K
$ >$ $ $ >>$
$$ $ $ $ >$
$ $ $ $ $ > K $
@ $ K $@$ @ $
$ $ $ @ $ @@ $
>$ K $ @ > $
$ $ $ $ $ @ $ >$ K
@ $ $ $ $ > $ > $
$ $ > K@ $>$ $
$ K @$ @ $ > $ $
@ @ $ $ @@$ K$
$ $ @$ @ $@$
@ K$ >@$ K>$
$ @ >$ $@$ $
P Ah h
ij− klI $ K > @
P Ah h
ij− jkI @ $ > $ >
S S S S $S @S <;S !
" " :;S " " .LS ! N:S " & G:S # & S & "
74" : & " ! " # " "
A P I # " # " ! !
& A<:I "
& & %M & !! ! "
. ; #
E EE EEE
@ $
K $
K$ $@$ $
$ > $$ >
@ @$ $
$ $ K$
K$ K$
$ @@$ >@$
@ $ @$ $
$ @@$
$ @ $ KK
@ $ @ $
$ K $ @$
@ $ K >
$ @ >$ K $
P Ah h
ij− klI @K
P Ah h
ij− ikI @ @>
S S S S $S @S ES "
& ! * + EES " ! *
, - EEES " ! * +
(" ! " ! # #
! A I ! A I *
A I E& " ! ) # '
& & #!
" " !! #!
' " & ! "
# E & " "
& ! #
' A N B OE !
D P ,7 ! #I
" !
# ! ' & E & " " "
# & ' #
! E "
! " " #
! & ! # ! !
" # #!
& " ! & ' # & "
#! & " " " && #
" " ! ! &
" ! ! ("
# " " !
# " ! & &
& " /
E ! ! " " " &
" ! # " " #
" " #
" " ! (" " "
! & "WQX " 0 !! #
#
A( #! I ! " " # "
& ! !
# #
! # ! & "
!!
" ' " "
&& " !!
" & !
, " A( #! I "
" #
" ! & "
& !! ! A( #! I
! " !! " # &
! # " &
v
j ! A( #! I " " & "& & " !
& ! " " #
# " " ("
! # A I "
" !! !
A( #! I
(" # " # & !
" # 0 ! & &
- " " !
# & # &
# & # " #
" " & ! ! " "
!! " " ! " ) &
&& & ! " A( #! I " & &
" # "
" " ! &
" & A( #! I
" ! & " & " #
# & " & "
" !
" " ("
" " " Sii
(" " & ! " &
$K ± @ ) " > ± K )
" " " " " & " !
A( #! $I (" ! "
" "
" # " " " # ! & " # " #
! " & #
" " && & ! "
E & & & " 0 &
" ! " & # &
A V I "
" ! & ! " AhjI
& "
& " E ! " &
" " ! #
& hj KK
! "
) #! " " ! !
& !! S I " & ! "
" ! & "WQ " "
D I "
& # "
7 " " "
" !
" (" " & ' "
! && ! " " 0
& " # & "
& & ' & #
" " "
74" : & " && & AhI ! AhjI
" A P I & & !
!! ! "
(
* .L N: G:
$ K K > K
K $$$
>
$K $ $ $
>K @
$K>
P Ah h
i− jI >
h $K > $K
P AhI @ K $
.LS ! N:S " & G:S # & S &
"
F " "
& "
A& !
* + I - " "
A ! * , - I
" " & "
! A( #! @I E
" & &&
" " " # # !!
" && & !
M & "
#! F " "
&& " &
" " ! " &
A/ ,E ;D -,:N- O:, D <:,:E, D
, / CJGE7, D F <E ! D
%,:E( CJGE7, ! @D , G.:/ !
KI " "
& #
! " " && ! "
! &
74" : & " && & AhI ! Ah
jI
" A P I #
! & A<:I "
" %M" #
.
* E EE EEE
> @
K >
@ $ @
$ K @ @>
> $ $$
P Ah h
i− jI @ $
h > @ @>
P AhI $ > $ K
ES " & ! * + EES "
! * , - EEES "
! * +
(" " &
! " !! ! "
" " " " "
" A( #! @I 7 "
" " ! " ! "
A( #! @I " # "
! ! # ( #!
& " & " #
" & " A( #!
>I & ! " " #
" " " " ! &
s
ˆ
ij ! " "" " " # "
" " " & #! && &
s
ij & " & & " " " # AI &
# !! " # &
, " & " " #
"
! # & #
! " ' # "
"
s
ij !"
& sij ! !!
! & A( #! KI
" ! # !
& " # " " " " "
& " & !
74" 8 & & & " AsijI
A P I & & ! & &
" # & !! ! "
(
; # .L N: G:
K$ K $ K@
K K@ @
@ @ > >
$ @ $ >$ @
@ K > $
@ $ >K
K KK
$ > K $
@ > > @@
$ > @
$ $ $> @$@
@ $ @@ $> K
$ $$ > >
@ @ @ >@
$ @ $ @ > @$
P As s
ij− ikI >
P As s
ij− klI @ K
S S S S $S @S .LS !
N:S " & G:S # & S & "
74" : & & " AsijI
A P I # " # & & &
" # & !! ! "
. ; #
E EE EEE
K$ > @
K $ $
@ K
$ >
@ >$K K K
@ $ @ @
K >>> K
$ K > K
@ @ $$
$ $
$ K @
@ @ @
$ >>K K
@ @ > K
$ @ K >$ $
P As s
ij− ikI @ $
P As s
ij− klI K@$ > > K
S S S S $S @S ES "
& ! * + EES " ! *
, - EEES " ! * +
(" & " " # "
! # ! ! &
# & !
! & "
(" " & - ' ! " Q !
& !
& ! " " # ! ! "
! !
" !
! ! !
F " ! & "
# ; !! :# " A > I
# " & "
! # # & !! # "
#
# ! " # ! #
# ! !
& " " #
% " & - ' ! " Q ! &
! &
! #
! !
(" " "
# && ! " "
(" & " & !
"
" & # !
" & & "
& & "
! & !
# # #
! !
GP, P: , D F,HB F P D F <E F D
E/*Y,E7 / D <EG(7 , C - D (7G:( !
!1! # 5
!" " 9* *#
>
,G;7/P : D *E G C D E/* , . D 7,
% :& 8 ! 1 ! +2 & 3!
= !" " 9*
*# >
F7E -, , , D E, GP . * D F,HB F P D
*E G C F ! +9
! +2 !" P%( , # 2 "
" " K >K > $
F,HB F P D ,:. BBE C D F ,G:E,7 < F
( %"4 7 *: 04 % *"4. *"
" : * + S H%*
F,HB F P ) 10 %" " : ;9
* + S H%* $
F,HB F P ) * " "S 2 S !
! 1 3 * + S H%* @
P GP7/EGE ( D F <E F D , /
CJGE7, ( D G.7/EG F D F; P7
% < D 7(( D /7<: P .
! # !! ) 0
""% % 004 "% " . 4 # K
K
P ,7 D , / CJGE7, ( D <:,:E,
. D G(7 % D . -,E:/ < F D F <E F D
%,:E( CJGE7, < D E/*Y,E7 / ,
! # & ! " 4
@ @ @ @
P ,7 D , / CJGE7, ( D <:,:E,
. D G(7 % D F <E F D %,:E(
CJGE7, < D P ;:, , % D Z*E/ ,
F ! +9 6 !
! +2 !" < 9 4
$ K #
%,:E( CJGE7, < D , / CJGE7, ( D
<:,:E, . D F,HB F P D F <E F
F # 5 !"
! ! ! )" ;9 0" 9= 4"
$ @ > @
%,:E( CJGE7, < D , / CJGE7, ( D
, G.:/ , D *E G < .
# & !! # ! 0 ""% %
004 "% " . 4 # >
. ,PG:, F 7 D :-:,; ,( !
& " !! ! !
! *" $ @@
; // H:, , D :-:,; ,( , ! & !!
# ! 0 " " $ @ >
CE ; F D F;7 C N D L O N ( P !! ! !
& ! " " ! ' A / I
> 9 4 ? ." @ 94 # ? 94 9 " $
K @
/ ,E ; / / - D -,:N- O:, C / P !! ! ! &
! ( #%
> K
E, GP . * D 7HB / * D . /*[7 C F F D
.HE ,[: / C D :/7 * D G(7 E F
. # ! " & - ' !
! &90.# @ K
7(:,/: / D F <E F D -, FFEGE /
,7P7* /;7 D - ,,:(7 , , E &! 8
"3 # " "3#
!" < " " 4 @
K K > K
<:,:E, . D , / CJGE7, ( :
& # "
0 ""% % 004 "% " . 4 #
<:,:E, / O D G.7/EG F D F <E F D
< F;:F7 F < D -7G (7 F D F; P7
% < ! " ' &
! & " !& # ! & F
A / I ( #% $ $@ $@@ @
<:,:E, / O D F <E F D G.7/EG F D
F; P7 % < D < F;:F7 F < D 7,
% ; ' & !! " !& # !
? " . 4 # K $ K
, G.:/ , D , / CJGE7, ( D *E G
< D %,:E( CJGE7, < < &
" # 0 ""% %
004 "% " . 4 # > K> $ >
, G.:/ , D , / CJGE7, ( D F <E
F D %,:E( CJGE7, < D <:,:E, .
. " # & !
!! ! A" " % ( 4" 94 " " .
> K
,EFFE . F / D E/* G D < ./E ,EGE D
F <E F - " A / I
A" " % ( 4" 94 " " . @ >
>
,EFFE . F / D E/* G D < ./E ,EGE D
F <E F # ! &
A / I A" " % ( 4" 94
" " . @ K >#
,7P7* /;7 D 7, % D G(7 : D
F <E F D ,G;7/P : ! " # ! @
& ! & " S - "
" " " / $ $$ @ K
G(7 % D , / CJGE7, ( D %,:E(
CJGE7, < D , G.:/ , D <:,:E, .
< +2 " 1 3 ! +2
! +2 !" HG- H # ! +2
@@ K >
G(7 % D , / CJGE7, (D %,:E(
CJGE7, < D , G.:/ , D F <E F D
7, % . & " "
! ! !
" 9* *# $ @$ @$K K
N B OE : D F (,7 C / D . //7 < - D
< (:,GE G : . B D E/* , D /HP:, ,
, < ! "3# !"
' ! E F "
4" %" ( 4. " @ >
F <E F D < F;:F7 F < D (7G:( D
-, FFEGE / D <EG(7 , C - ! !1!
" ! +9 !"
@
F <E F D <EG(7 , C - D , / CJGE7, ( D P GP7/EGE ( F # # ! & "
! 0 ""% % 004 "%
" . 4 # @ @ @
F <E F D -, FFEGE / D <EG(7 , C - D
, / CJGE7, ( D ,7P7* /;7 D E/*
, D 7(:,/: / F 1 1
2 ! +9 !"
< 9 4 @ @ @#
E/* % < ( - D /7<: : % D < ./E ,EGE
D F <E F :&& & "
0 ""% %
004 "% " . 4 # > @ >
E/* ( D <EG(7 , - C D F <E F D
G.7/EG F D F; P7 % < D
F ,* /;7 G .
!! # !) PG
0 ""% % 004 "% " . 4 #
@
E 7G . D F <E F D < F;:F7 F < D
<EG(7 , C - D -, FFEGE / D (7G:(
P 2 ! +9 !"
@ $$ @
EG.; , O D F; HP; ,L - P *" 4
*" . % ;9 " " " 4# G P !" S
O ! < #! " K$
*:GF7* OL , D F,HB F P F +2 !
1 ! !1! " 4" %"
A" : $ K K
*:GP,H F7/7 : F . D F <E F D
< F;:F7 F < D 7/E*:E, * , D -, FFEGE
/ D .7GF /*: *EPE. / F # !
# ! ! !" 2
! - ! )" ;9 0" 9= 4" @
*E G C D (( % < ! & !
& # # ! & !
! !& A" " % ( 4" 94
4 # @ @$ >
*E G C D (:E\:E, - D / :EP , * D
F <E F D * /:G(EGE / , ! &
& " ! &
& ! 0 ""% % 004 "% " . 4 # > > K >
*E:E, , D ,7P7* /;7 D F <E F D
(: GG P C D , / CJGE7, ( D
-E.G7((7 / P " 6
"3# !" G :
< - ! " 9* *#
@
*E/:/ % 7 D , / CJGE7, ( D <:,:E, . D F <E F D *E G < D %,:E( CJGE7, < :&& & ! " # ! & "
HG- H ! , <P )
" 9* *# $ K
B G:((: * ! # ! 1
! +9 !" A / IS "
2 2 *
94 %" " $ > K K
! " #$$%
& '( #$$(
/ & S (" ! # " & "
F F # / " " #