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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 23 "Complex Disc Arithmetic
" }}{PARA 257 "" 0 "" {TEXT 256 234 "written by  Grimmer, Markus, Depa
rtment of Mathematics, University of Wuppertal, Germany, http://www.ma
th.uni-wuppertal.de/wrswt \n<\251 1999-2002 Scientific Computing/Softw
are Engineering Research Group, University of Wuppertal, Germany>" }
{TEXT 257 1 "\n" }}{PARA 258 "" 0 "" {TEXT -1 63 "NOTE: This worksheet
 demonstrates the use of the Maple package " }{TEXT 258 12 "intpakX v1
.0" }{TEXT -1 25 " for interval arithmetic." }}}{EXCHG {PARA 259 "" 0 
"" {TEXT -1 102 "This document is not the package.  It only shows how \+
to work with the functions and types provided by " }{TEXT 260 12 "intp
akX v1.0" }{TEXT -1 156 ".  You must create the package in an empty di
rectory before loading the package ( i.e., /usr/maple/intpakX/lib)  On
ce created, load the package as follows:\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "restart;\nlibname:=\"C:/mylib/interval\", libname;
\nwith(intpakX);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q2C:/
mylib/interval6\"Q=C:\\Program~Files\\Maple~8/libF'" }}{PARA 7 "" 1 "
" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
{PARA 7 "" 1 "" {TEXT -1 61 "Warning, the assigned name midpoint now h
as a global binding\n" }}{PARA 7 "" 1 "" {TEXT -1 85 "Warning, the pro
tected names ilog10, max and min have been redefined and unprotected\n
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ar%#&*G%$&**G%#&+G%#&-G%#&/G%-&Co
nvex_HullG%(&arccosG%(&arcsinG%(&arctanG%&&cabsG%&&caddG%&&cdivG%*&cdi
v_optG%'&cmultG%+&cmult_optG%%&cosG%&&coshG%&&csubG%%&expG%+&intersect
G%*&intpowerG%$&lnG%%&sinG%&&sinhG%%&sqrG%&&sqrtG%%&tanG%&&tanhG%'&uni
onG%&EvalfG%6Interval_IntegerpowerG%4Interval_Round_DownG%2Interval_Ro
und_UpG%-Interval_addG%0Interval_arccosG%0Interval_arcsinG%0Interval_a
rctanG%-Interval_cosG%.Interval_coshG%0Interval_divideG%-Interval_expG
%0Interval_hyp_rdG%0Interval_hyp_ruG%3Interval_intersectG%,Interval_ln
G%2Interval_midpointG%5Interval_option_zeroG%/Interval_powerG%6Interva
l_range_valuesG%4Interval_reciprocalG%/Interval_scaleG%-Interval_sinG%
.Interval_sinhG%-Interval_sqrG%.Interval_sqrtG%2Interval_subtractG%-In
terval_tanG%.Interval_tanhG%/Interval_timesG%1Interval_trig_rdG%1Inter
val_trig_ruG%-Interval_ulpG%/Interval_unionG%/Interval_widthG%,addinfi
nityG%2centred_form_evalG%%cexpG%,cni_range3dG%2complex_disc_plotG%5co
mplex_polynom_plotG%2compute_all_zerosG%<compute_all_zeros_with_plotG%
7compute_combined_rangeG%9compute_mean_value_rangeG%8compute_monotonic
_rangeG%=compute_naive_interval_rangeG%.compute_rangeG%0compute_range3
dG%:compute_taylor_form_rangeG%*constructG%,expinfinityG%,ext_int_divG
%1horner_eval_centG%0horner_eval_optG%'ilog10G%+infinitylnG%%initG%3in
terval_list_plotG%5interval_list_plot3dG%0intpakX_greaterG%/intpakX_il
og10G%,intpakX_maxG%,intpakX_minG%$invG%&is_inG%$maxG%.max_abs_errorG%
$midG%)midpointG%$minG%'newtonG%,newton_plotG%1newton_with_plotG%.powe
rinfinityG%#rdG%)rel_diamG%#ruG%,sqrinfinityG%-sqrtinfinityG%3subdivid
e_adaptiveG%1subdivide_equi3dG%6subdivide_equidistantG%1subtractinfini
tyG%.timesinfinityG%$ulpG%&widthG%)x0_startG" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 23 "Complex Disc Arithmetic" }}{PARA 0 "" 0 "" {TEXT -1 
99 "In addition to real intervals, there are also types and procedures
 for complex interval arithmetic." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
32 "Data Types and basic operations " }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 287 "\nData Types: type/complex_disc and type/complex_interval\nA c
omplex disc is given by a complex number (center of the disc) and a va
lue for the radius. It is defined as a list of three entries, [z_re,z_
im,r].\nA complex interval is given by two real intervals specifying a
 rectangular area." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A1:=[
1,0,1];\ntype(A1,complex_disc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
A1G7%\"\"\"\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "B1:=[1.,2.]; \nB2:=[3.,4.];
\nB3:=[B1,B2];\ntype(B3,complex_interval);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B1G7$$\"\"\"\"\"!$\"\"#F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B2G7$$\"\"$\"\"!$\"\"%F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#B3G7$7$$\"\"\"\"\"!$\"\"#F)7$$\"\"$F)$\"\"%F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 32 "\nDisplay complex disc intervals:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "complex_disc_plot(A1);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 486 486 486 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"#\"\"!$F*F*7
$$\"3G4hRPij!*>!#<$\"3Ikwb#=y_O\"!#=7$$\"3W6B))4-Qn>F/$\"3%[#\\ff*)GLD
F27$$\"3R.h)yke[#>F/$\"3gLj&[K5J!QF27$$\"37'oz=42`'=F/$\"3j9NXR4U7]F27
$$\"3U0vG._U!z\"F/$\"3t_H-qceDhF27$$\"31NR'oTd#3<F/$\"3!GO#*3qU&fqF27$
$\"3#H$>jubk6;F/$\"3!\\@@&\\!>8\"zF27$$\"3&HuIc:x5]\"F/$\"3-$>p(Qh-a')
F27$$\"3O#3nW'R,#Q\"F/$\"3aK+16bcT#*F27$$\"3kY@iF'QDD\"F/$\"3s&Q\"[M\"
oen*F27$$\"3G7:E#o8X8\"F/$\"33)fVPO<\"4**F27$$\"3sdfAI*GS***F2$\"2%HhJ
<#)******F/7$$\"3#>]e2k<vj)F2$\"3ap%>wGZn!**F27$$\"39WGk\\y\"QN(F2$\"3
PA>=\\D`V'*F27$$\"3]?c$*o5*oA'F2$\"3IzF#e`m3E*F27$$\"3e-vPEZod\\F2$\"3
vU'yI2&oN')F27$$\"3_PRkB0^gRF2$\"3!G;)RQ+BqzF27$$\"3&\\vNKl$o5HF2$\"3A
fvOg?x_qF27$$\"39=uW%z&z&4#F2$\"3u3<J%QZc7'F27$$\"3C>(*=gKpS8F2$\"3D%H
$\\4/k,]F27$$\"3gpajosj!p(!#>$\"3In'[*>HvXQF27$$\"3#4$4'f!\\wjLF_r$\"3
o$p9m#Q%=d#F27$$\"3?3BIe!)*pH*!#?$\"31V?00]Ug8F27$$\"3Z!e%*Rnr!4[!#B$
\"3vCA\\O%485$Fjr7$$\"3'zu%pF\")>c\"*Fjr$!3q#4*>c=8]8F27$$\"39![kuKX$f
KF_r$!3UNbFGIGKDF27$$\"3')y%o'pyPntF_r$!3q4&=j`Bsw$F27$$\"31D<!z'Q\"eK
\"F2$!3kENX')3zv\\F27$$\"3uw+sGA$R0#F2$!37W(H!pUCrgF27$$\"3au'3H1M%*)G
F2$!3Y?#o?\"yMJqF27$$\"3?c@))pNT^RF2$!3)4j2yeGL'zF27$$\"3&*ydu\"e6$4]F
2$!3I_Mh6Mil')F27$$\"3K*Rz#=&=sA'F2$!3#\\IN,%***4E*F27$$\"3]Avq=WL\"R(
F2$!3=V>(fp[Pl*F27$$\"37)yeZAL%3()F2$!3;$RoI*>C;**F27$$\"3-@jqlhUp**F2
$!3W1O#>E`*****F27$$\"39BKODBxG6F/$!3lYZ3X=u;**F27$$\"3C$RqX8/bD\"F/$!
3U$[o%H'z!o'*F27$$\"3[)yb&Q)zNQ\"F/$!36.2<#>x]B*F27$$\"3Nh(4w0I.]\"F/$
!3wHgb5wMe')F27$$\"3#\\#)[*R]$4h\"F/$!3/a*[=H2o\"zF27$$\"3TgnkQ+$*4<F/
$!3)4d)eg?sUqF27$$\"3%f=s$Ry,!z\"F/$!3D1$H<bQ38'F27$$\"3+95i^PVn=F/$!3
e>)zD(p_v\\F27$$\"3R$e@`4+D#>F/$!3i&>2ndo*fQF27$$\"3s'GAp))oa'>F/$!3%)
f]nRQ=0EF27$$\"3/elf`bp!*>F/$!3/up91t'4O\"F27$F($\"36YKhSr8/#)!#F-%'CO
LOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q!6\"Fd[l-%%VIEWG6$%(DEFAUL
TGFi[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "\nArithmetical Operations
:\n&cadd, &csub, &cmult, &cdiv\nAs for real intervals, you can also do
 the basic arithmetics with disc intervals. " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 235 "\nArea optimal multiplication and division:\nYou can a
lso use area optimal multiplication and division instead of their cent
ered counterparts (where you get the new center i.e. by  \nmultiplying
 the centers of the discs to be multiplied)." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "A2:=[-1,1,1];\nA3:=A1 &cmult A2;\nA4:=A1 &cmult_
opt A2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G7%!\"\"\"\"\"F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G7%$!+++++5!\"*$\"+++++5F($\"+zN@
9MF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G7%$!+/#)Q!R\"!\"*$\"+/#)
Q!R\"F($\"+cAcpHF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 39 "Range Enclosure for Complex Polynomials" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "De
fine and display complex polynomials:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "p1:=(0.1+0.1*I)*z^5+0.2*I*z^4-0.1*I*z^3+(-0.2-0.1*I)
*z+2.0+1.0*I;\nZ:=[-0.2,0.4,1];\ntype(Z,complex_disc);\ntype(p1,polyno
m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G,,*&^$$\"\"\"!\"\"F(F))%
\"zG\"\"&F)F)*&^#$\"\"#F*F))F,\"\"%F)F)*&^#$F*F*F))F,\"\"$F)F)*&^$$!\"
#F*F6F)F,F)F)^$$\"#?F*$\"#5F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"ZG7%$!\"#!\"\"$\"\"%F(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr
ueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "complex_polynom_plot(p1,Z);" }}{PARA 13 "" 1 "
" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6%-%'CURVESG6$7iq7$$\"(Cmu\"!\"'$
\"'KWzF*7$$\"+eoI]<!\"*$\"+>svJs!#57$$\"+Tc!\\x\"F0$\"++>'QX'F37$$\"+A
v9;=F0$\"+<-vtdF37$$\"+ys1x=F0$\"+zvNW^F37$$\"+'GN'o>F0$\"+9hQnXF37$$
\"+NFz\"3#F0$\"+!3=r?%F37$$\"+WY'>@#F0$\"+UWvQTF37$$\"+ETS]BF0$\"+KN@E
WF37$$\"+?cQ'[#F0$\"+8Qu-^F37$$\"+PZH4EF0$\"+F'3l<'F37$$\"+(\\'Q,FF0$
\"+fi80vF37$$\"+d&>Lw#F0$\"+(zel5*F37$$\"+\"=Quy#F0$\"+wcd(4\"F07$$\"+
wKVkFF0$\"+=(>dH\"F07$$\"+6(Rro#F0$\"+5zX*\\\"F07$$\"++_2bDF0$\"+O@Z(o
\"F07$$\"+zZesBF0$\"+2a_W=F07$$\"+'Q;q9#F0$\"+Cxjd>F07$$\"+Zx/#)=F0$\"
+dk5;?F07$$\"+)HW#*f\"F0$\"+/Y/2?F07$$\"+>!e'\\8F0$\"+NQWR>F07$$\"+K=:
86F0$\"+mV0;=F07$$\"*$R;1**F0$\"+xz;B<F07$$\"*nVYy)F0$\"+SZg9;F07$$\"*
#yD(y(F0$\"+c*p;\\\"F07$$\"*L2D$pF0$\"+e\"RfN\"F07$$\"*+:UB'F0$\"+$G4'
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$\"+HaO;sF37$$\"*(=ou_F0$\"+-sU$e&F37$$\"*mN&3bF0$\"+lV-bRF37$$\"*1on#
fF0$\"+S[p`BF37$$\"*Qr]_'F0$\"*kbh,)F37$$\"*Wz%=sF0$!)=JraF07$$\"*l<t/
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6F0$!*@%*)R_F07$$\"+QQ^,8F0$!*REu?'F07$$\"+r`&[Y\"F0$!*<UZ)pF07$$\"+m:
mN;F0$!*JN;c(F07$$\"+k<@%y\"F0$!*J'e))yF07$$\"+VpaM>F0$!*c9^1)F07$$\"+
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\"+f1=mDF0$!*'4XNrF07$$\"+YTL?FF0$!*B#z]kF07$$\"+'GAQ'GF0$!*mghg&F07$$
\"+u]\\'3$F0$!*;e<x$F07$$\"+\\&=HE$F0$!*;lpf\"F07$$\"+CcEQLF0$!)hK<IF0
7$$\"+@A(yR$F0$\"+(*>HS5F37$$\"+P1jTMF0$\"+Sc.4CF37$$\"+>]ppMF0$\"+FQ(
[y$F37$$\"+xDP\"[$F0$\"+zo&yN'F37$$\"+(R]FW$F0$\"+.\\Xr()F37$$\"+Et%oN
$F0$\"+Z1b,6F07$$\"+V$)eMKF0$\"+*[Y+H\"F07$$\"+#HT+5$F0$\"+3*ptU\"F07$
$\"+_jL`HF0$\"+Q`2H:F07$$\"+r-5\"z#F0$\"+dY()*f\"F07$$\"+@ipLEF0$\"+(G
yOj\"F07$$\"+,<A$[#F0$\"+z;HN;F07$$\"+3-y^BF0$\"+d8$*4;F07$$\"+\"z&\\b
AF0$\"+!3`>d\"F07$$\"+`O)o<#F0$\"+;[WC:F07$$\"+kJD6@F0$\"+%))ywY\"F07$
$\"+n*QQ1#F0$\"+U@&*49F07$$\"+(eK6.#F0$\"+vaA`8F07$$\"+A'e:,#F0$\"+Bj9
-8F07$$\"+`CA)*>F0$\"+Kb+B7F07$$\"+r(\\0+#F0$\"+Go]r6F07$$\"+7\\J2?F0$
\"+*o0^8\"F07$$\"+6df8?F0$\"+\"4MT6\"F07$$\"+TQ/<?F0$\"+())4k5\"F07$$
\"+F8`??F0$\"+1d5,6F07$$\"+yD\\B?F0$\"+t,b)4\"F07$$\"+g&\\e-#F0$\"+KJE
)4\"F07$$\"+cU1F?F0$\"+p#Q05\"F07$$\"+r`,E?F0$\"+h]B06F07$$\"+H*)>A?F0
$\"+!*QH66F07$$\"+&)*f],#F0$\"+795=6F07$$\"+&G@P+#F0$\"+5,([7\"F07$$\"
+(G#\\))>F0$\"+[;&*H6F07$$\"+QrOq>F0$\"+P$3?8\"F07$$\"+Sx**\\>F0$\"+M%
=,8\"F07$$\"+1SpF>F0$\"+;G9B6F07$$\"+o871>F0$\"+l$436\"F07$$\"+AOr()=F
0$\"+;fB%4\"F07$$\"+Y&pI(=F0$\"+%[+S2\"F07$$\"+W80j=F0$\"+/X+^5F07$$\"
+&=A&e=F0$\"+^5aF5F07$$\"+r1?f=F0$\"+.\\w05F07$$\"+AA.k=F0$\"+>G[s)*F3
7$$\"+-3sq=F0$\"+1D]U(*F37$$\"+<'oy(=F0$\"+zYRc'*F37$$\"+SEF%)=F0$\"+5
_/5'*F37$$\"+nOB')=F0$\"+otQ2'*F37$$\"+(\\sC)=F0$\"+_)))Hi*F37$$\"+'ew
D(=F0$\"+.:sJ'*F37$$\"+y'Q_&=F0$\"+R1C.'*F37$$\"+M_0K=F0$\"+r>$H]*F37$
$\"+7fl1=F0$\"+uds6$*F37$$\"+5>N\"y\"F0$\"+sdR2!*F37$$\"+g\\%)e<F0$\"+
]'=Qa)F37$$\"++SiY<F0$\"+#**>V%zF3-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F]
^mF\\^m-%+AXESLABELSG6$Q!6\"Fa^m-%%VIEWG6$%(DEFAULTGFf^m" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 281 "\nThere are three methods for enclosing \+
the range of a complex polynomial, two based on a Horner-like evaluati
on with centered or area optimal multiplication, the third based on ce
ntered forms (similar to the mean value form for real numbers). Study \+
the example for the differences:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "pH:=horner_eval_cent(p1,Z);\npO:=horner_eval_opt(p1,Z
);\npC:=centred_form_eval(p1,Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#pHG7%$\"++Siq?!\"*$\"+++Kk#*!#5$\"+tX7X?F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#pOG7%$\"+[[c:?!\"*$\"+;hYeg!#5$\"+Z(>Rt\"F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pCG7%$\"++Siq?!\"*$\"+++Kk#*!#5$\"+
]G!4&=F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "c1:=complex_di
sc_plot(pH,color=blue,thickness=2,linestyle=4):\nc2:=complex_disc_plot
(pO,color=green,thickness=2,linestyle=3):\nc3:=complex_disc_plot(pC,co
lor=red,thickness=2,linestyle=2):\nc4:=complexplot(subs(z=Z[1]+I*Z[2]+
Z[3]*(cos(t)+I*sin(t)),p1),t=0..2*Pi,color=black,thickness=2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display([c1,c2,c3,c4],scalin
g=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 
"6)-%'CURVESG6&7S7$$\"3T*****Hd[d6%!#<$\"3M+++++Kk#*!#=7$$\"3eWI+7&)f'
4%F*$\"3/YH,'R[c?\"F*7$$\"3]7UmQq.\\SF*$\"3l=x<]6_W9F*7$$\"3eCMMZ^2iRF
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#**>V%zFj[n-Fhz6&FjzF\\[lF\\[lF\\[lFd[l-%+AXESLABELSG6%Q!6\"Fg]p-%%FON
TG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$F\\^pF\\^p" 1 2 0 
1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "\nThe dis
play command can be found in the Maple plots package. For the options \+
cf. the plot command options." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
32 "The Complex Exponential Function" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 61 "\nFinally, there's the exponential function for complex discs:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "cexp(Z);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%$\"+7'45a(!#5$\"+ExG)=$F&$\"+$=5oS\"!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 261 "" 0 
"" {TEXT 259 11 "Disclaimer:" }{TEXT -1 240 " While every effort has b
een made to validate the solutions in this worksheet, Waterloo Maple I
nc. and the contributors are not responsible for any errors contained \+
and are not liable for any damages resulting from the use of this mate
rial." }}}}{MARK "1 0 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }