Content-Type: multipart/mixed; boundary="-------------1106030305746" This is a multi-part message in MIME format. ---------------1106030305746 Content-Type: text/plain; name="11-84.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-84.keywords" impedance tomography, inverse source problem, nonlinear integral equation, Fourier method, simplified regularized Gauss-Newton. ---------------1106030305746 Content-Type: application/postscript; name="BMMS_ADG_ConstCond.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BMMS_ADG_ConstCond.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95b Copyright 2005 Radical Eye Software %%Title: C:/ADGWorkingDirectory/ConstCondInclusionArticles/BMMS_ADG_ConstCond.dvi %%CreationDate: Fri Jun 03 09:57:20 2011 %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: Helvetica-Bold CMR8 Helvetica Helvetica-Oblique CMR10 %%+ CMMI10 CMMI8 CMSY10 CMMI6 CMEX10 CMR6 MSBM10 CMSY8 MSBM7 CMSY6 %%+ CMMI12 CMR12 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -R0 %+ C:/ADGWorkingDirectory/ConstCondInclusionArticles/BMMS_ADG_ConstCond.dvi %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2011.06.03:0957 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 68 /D put dup 70 /F put dup 113 /q put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67 3C15C2153C061EB541F66C11E7EE77D5D77C0B11E1AC55101DA976CCACAB6993 EED1406FBB7FF30EAC9E90B90B2AF4EC7C273CA32F11A5C1426FF641B4A2FB2F 4E68635C93DB835737567FAF8471CBC05078DCD4E40E25A2F4E5AF46C234CF59 2A1CE8F39E1BA1B2A594355637E474167EAD4D97D51AF0A899B44387E1FD933A 323AFDA6BA740534A510B4705C0A15647AFBF3E53A82BF320DD96753639BE49C 2F79A1988863EF977B800C9DB5B42039C23EB86953713F730E03EA22FF7BB2C1 D97D33FD77B1BDCC2A60B12CF7805CFC90C5B914C0F30A673DF9587F93E47CEA 5932DD1930560C4F0D97547BCD805D6D854455B13A4D7382A22F562D7C55041F 0FD294BDAA1834820F894265A667E5C97D95FF152531EF97258F56374502865D A1E7C0C5FB7C6FB7D3C43FEB3431095A59FBF6F61CEC6D6DEE09F4EB0FD70D77 2A8B0A4984C6120293F6B947944BE23259F6EB64303D627353163B6505FC8A60 00681F7A3968B6CBB49E0420A691258F5E7B07B417157803FCBE9B9FB1F80FD8 CA0DA1186446DD565542BCCC7D339A1EB34C7F49246E8D72E987EB477C6DB757 99AF86CEBCD7605C487A00CD2CD093098182DC57B20D78ECE0BECF3A0BF88EBA C866DB19F34BBBED6634AFC0F08D2AFB2A92578A6F8B4ADCD6594737FF6EED7D 5B536DA9E3E2CADB40DB7C600EA4D100D33C3B92B1CF857E012C4EB370BA8295 55B50047CA2D7522A6966F9A3A389BD74D6191B95467B973F1BC4B393F76BA14 EC03D9CD831F164CE780E5BC8AC29D00EC3E90ED7CF8AD01CEFB18483ACD688B 2285516598609CE41194E7C81E17C10C8115F73491EDA18681A38475A08817C4 6F49FC5ED79CFD7841A379A96512477213F9F781618A6712B531F6B49BEDCE63 74EF6B01918593BDE87200FDF1E13864058256E8D92E770C8CCF789786C3EE29 9956E595AF4FA49C6C2D54E9E98812B175B8ADCE7D0B046FCB4F3A332AEEC576 556FCFD7303E5B8CE68DF6B91AE6E31BBA9673E592C6078D84CBCE80A82AAAA2 94B08A7036D06FD83B5B5A8E49E1C4A80697DFE48042BA7D021E487DA565B0C1 711F0FACE363F5A1D34F62ACCBA1CE281F9E1397E6BAEFEE85B11FC05A066DA6 68617B9DD43C121ED90D119B1DD8376CA2522F2A711365C4996DFE012F2EF46C AADCC43999D203D936EFE68C5811BB47C7692E4C9EB9411FC5854E1945230EEC 599212CEEC7B783433C85E50A0C679A6DC0F90AC3DF77299282B16165628A983 87B624839319DCA89D82BE76C7B5F8E7182162FA2ABF1E4D8F3C78B8E9DA6690 60E104B6B3AE2D8C25DC38E70594D06BECFA636D7EF3C3897C954E834C6E81DE FC4D24DDB01ACECF3B5DC51618E0C20E89F8DAFE595B1E78C9D3016F53A0E3D4 1731C7D680B2A26E64145C49D044E24AD4334983AAB74E4F99369E68A4650401 906E0FD623035A03B3BB54CB8774C7BC541E4CF2335AF6B4691561AE8CD0FFEE E3EE5BBF1B9FB671AB44CC73E8BCB440133819B26B431C02A7F6C2BF31BE8FE8 4B0A71927048DD284DCA73E4A4669969D0E80154AB3BFEA72AB3FCEEDA661FAD 7569E98A8630D65519EE7673289C216FAAE00D16DEBA0479717CBCE26A2F4159 3897767099E7877CA59088B0577A9D17992BB6A6254B1464E2AC45E164D0F016 1BFDA542E2D568157A62400A110F265BFF66744A913958F9838AE36BE2B9C452 5B1974029463E8E2CF5578972ACEEAC7CF03D1CF3FF8F5348AC695116F0A8A2E D112B3706C5EE2F930A1A59EAAD2ABD35AC2B2E80754E3FBF33AB42999112022 935D7F6912331DCD3B3CF2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 5 /diamondmath put dup 14 /openbullet put dup 26 /propersubset put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 76 /L put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 100 /hatwidest put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 61 /slash put dup 83 /S put dup 105 /i put dup 107 /k put dup 108 /l put dup 112 /p put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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%%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 15 /bullet put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 67 /C put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 72 /H put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 93 /sharp put dup 99 /c put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 15 /epsilon1 put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 73 /I put dup 78 /N put dup 82 /R put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 99 /c put dup 101 /e put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put dup 120 /x put dup 126 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fu(Z)26679 72774 y Fo(\()p Fn(r)34 b Fo(\))28202 72274 y Ff(j)p Fm(n)p Ff(j)29352 72774 y Fn(e)29916 72274 y Fm(i)157 b(ns)31455 72774 y Fn(;)15669 b Fp(\(2.7\))-189 76663 y(which)337 b(is)g(the)g(Green')-61 b(s)337 b(function)g(f)-36 b(or)336 b(the)h(unit)f(circular)h(domain)f Fo(\012)31038 76845 y Fu(1)31901 76663 y Fp(in)h(polar)g(coordinates)f Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))p Fn(:)p eop end %%Page: 4 4 TeXDict begin 4 3 bop -189 715 a Fp(Without)382 b(loss)f(of)h(gener)-12 b(ality)-121 b(,)391 b(w)-12 b(e)382 b(restr)18 b(ict)381 b(ourselv)-30 b(es)381 b(b)-24 b(y)381 b(choosing)h Fj(the)f(har)30 b(monic)381 b(ref)-36 b(erence)380 b(potential)-189 2537 y Fn(u)505 2719 y Fu(0)1368 2537 y Fp(of)337 b(the)f(f)-36 b(or)30 b(m)16126 4704 y Fn(u)16820 4886 y Fu(0)17683 4704 y Fo(:=)336 b Fn(U)20127 4901 y Fm(k)20695 4704 y Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))338 b(=)25007 3884 y(\()p Fn(r)34 b Fo(\))26530 3444 y Fm(k)p 25007 4425 2093 49 v 25382 5535 a Fl(j)p Fn(k)k Fl(j)27232 4704 y Fn(e)27796 4203 y Fm(i)158 b(k)24 b(s)29278 4704 y Fn(;)202 b(k)374 b Fl(2)337 b Fg(Z)32775 4886 y Fu(0)33301 4704 y Fn(;)-189 7374 y Fp(which)g(is)g(the)g(solution)g(of)g(boundar) 36 b(y)336 b(v)-30 b(alue)337 b(prob)-24 b(lem)336 b(\(1.3\))g(under)g (scaled)h(tr)18 b(igonometr)g(ic)336 b(function)20427 10486 y Fn(g)381 b Fo(=)22798 9666 y Fn(e)23362 9226 y Fm(i)158 b(k)24 b(s)p 22798 10207 2046 49 v 23150 11317 a Fl(j)p Fn(k)38 b Fl(j)24976 10486 y Fn(;)202 b(k)375 b Fl(2)337 b Fg(Z)28474 10668 y Fu(0)29000 10486 y Fn(;)-189 13463 y Fp(as)g(a)g(Dir)18 b(ichlet)337 b(boundar)36 b(y)336 b(v)-30 b(alue)337 b(at)g Fn(@)67 b Fo(\012)18735 13645 y Fu(1)19261 13463 y Fn(:)-189 15285 y Fj(F)-36 b(orw)-18 b(ard)347 b(Map)h Fn(F)168 b Fj(.)452 b Fp(The)348 b(de\002ned)g(oper)-12 b(ator)348 b Fn(F)517 b Fp(mapping)348 b Fn(q)392 b Fp(to)29335 14808 y Fm(@)52 b(u)p 29322 15006 1119 49 v 29322 15703 a(@)g(n)30923 15285 y Fp(is)348 b(b)-24 b(y)348 b(taking)h(the)f(nor)30 b(mal)348 b(der)18 b(iv)-30 b(ativ)g(e)348 b(of)-189 17107 y(\(2.6\))5697 19988 y Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\)\()p Fn(t)p Fo(\))1109 b(=)13087 19168 y Fn(@)67 b(u)p 12842 19710 1895 49 v 12842 20820 a(@)g(\027)14152 21002 y Fm(x)15206 19988 y Fo(=)17210 19168 y Fn(@)p 16618 19710 V 16618 20820 a(@)g(\027)17928 21002 y Fm(x)18848 18339 y Fi(Z)19521 21088 y Fu(\012)20201 21211 y Fh(1)20919 19988 y Fn(G)p Fo(\()p Fn(x;)202 b(\030)56 b Fo(\))p Fn(S)70 b Fo(\()p Fn(\030)56 b Fo(\))p Fn(d\030)g(;)10659 23315 y Fo(=)12709 21665 y Fi(Z)13382 24414 y Fu(\012)14062 24537 y Fh(1)15158 22495 y Fn(@)67 b(u)p 14913 23036 V 14913 24147 a(@)g(\027)16223 24329 y Fm(x)16941 23315 y Fn(G)p Fo(\()p Fn(x;)202 b(\030)56 b Fo(\))p Fl(r)21664 23512 y Fm(\030)22168 23315 y Fn(a)p Fl(r)23819 23512 y Fm(\030)24322 23315 y Fn(U)25150 23512 y Fm(k)25719 23315 y Fo(\()p Fn(\030)g Fo(\))p Fn(d\030)g(;)12709 25873 y Fp(b)-24 b(y)337 b(applying)f(Green')-61 b(s)337 b(second)g(f)-36 b(or)30 b(m)-12 b(ula)13046 28027 y(and)337 b(use)f(the)h(f)-36 b(act)337 b(that)1212 b Fn(a)p Fl(j)26184 28224 y Fm(@)52 b Fu(\012)27412 28347 y Fh(1)28264 28027 y Fo(=)336 b(0)10659 30452 y(=)12709 28802 y Fi(Z)13382 31551 y Fu(\012)14062 31700 y Fk(S)14912 30452 y Fn(a)p Fl(r)16563 30649 y Fm(\030)17444 29632 y Fn(@)67 b(u)p 17200 30173 V 17200 31284 a(@)g(\027)18510 31466 y Fm(x)19227 30452 y Fn(G)p Fo(\()p Fn(x;)202 b(\030)56 b Fo(\))p Fl(r)23950 30649 y Fm(\030)24454 30452 y Fn(U)25282 30649 y Fm(k)25850 30452 y Fo(\()p Fn(\030)g Fo(\))p Fn(d\030)g(;)12709 33028 y Fp(due)337 b(to)f(change)h(of)g(v)-30 b(ar)18 b(iab)-24 b(les)336 b(from)g(car)48 b(tesian)337 b(to)g(polar)13046 35182 y(and)g(the)f(use)h(of)g(P)-61 b(oisson)338 b(K)-48 b(er)30 b(nel)337 b(f)-36 b(or)336 b(Green')-61 b(s)337 b(function)13046 37336 y(also)g(inser)48 b(ting)337 b Fn(a)f Fo(=)h Fn(\037)23587 37524 y Fu(\012)24267 37673 y Fk(S)10659 40078 y Fo(=)12709 38429 y Fi(Z)13921 38779 y Fu(2)p Fm(\031)13382 41178 y Fu(0)15221 38429 y Fi(Z)16433 38779 y Fm(q)32 b Fu(\()p Fm(s)p Fu(\))15894 41178 y(0)18309 40078 y Fl(r)19319 40275 y Fm(\030)19823 40078 y Fn(P)168 b Fo(\()p Fn(r)-34 b(;)202 b(t)270 b Fl(\000)f Fn(s)p Fo(\))h Fl(\001)f(r)27137 40275 y Fm(\030)27640 40078 y Fn(U)28468 40275 y Fm(k)29037 40078 y Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))p Fn(r)34 b(dr)g(ds)p Fo(;)1414 b Fn(\030)56 b Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))13046 42506 y Fp(after)336 b(inser)48 b(ting)337 b(an)g(e)-36 b(xplicit)337 b(e)-36 b(xpression)336 b(of)h(the)g(P)-61 b(oisson)337 b(K)-48 b(er)30 b(nel)10659 45182 y Fo(=)13209 44362 y(1)p 12842 44903 1341 49 v 12842 46013 a(2)p Fn(\031)14517 43532 y Fi(Z)15729 43882 y Fu(2)p Fm(\031)15190 46281 y Fu(0)17029 43532 y Fi(Z)18241 43882 y Fm(q)i Fu(\()p Fm(s)p Fu(\))17703 46281 y(0)21146 44030 y Fi(X)20117 46665 y Fm(n)p Ff(6)p Fu(=0)p Fm(;n)p Ff(2)p Fu(Z)24127 45182 y Fl(j)p Fn(n)p Fl(j)p Fn(e)26093 44681 y Fm(i)157 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))29435 45182 y Fn(e)29999 44681 y Fm(i)g(k)24 b(s)31480 45182 y Fn(r)32061 44681 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)g Ff(j\000)p Fu(2)36180 45182 y Fn(r)34 b(dr)g(ds)-189 49018 y Fp(W)-36 b(e)337 b(arr)18 b(iv)-30 b(e)336 b(at)h(the)g(f)-36 b(orw)-18 b(ard)335 b(map)i Fn(F)505 b Fp(mapping)337 b Fn(q)380 b Fp(to)24358 48541 y Fm(@)52 b(u)p 24345 48739 1119 49 v 24345 49436 a(@)g(n)25597 49018 y Fn(;)9485 52274 y(F)168 b Fo(\()p Fn(q)43 b Fo(\)\()p Fn(t)p Fo(\))340 b(=)15457 51454 y(1)p 15090 51995 1341 49 v 15090 53105 a(2)p Fn(\031)16765 50624 y Fi(Z)17978 50975 y Fu(2)p Fm(\031)17439 53373 y Fu(0)20306 51122 y Fi(X)19278 53758 y Fm(n)p Ff(6)p Fu(=0)p Fm(;n)p Ff(2)p Fu(Z)24832 51454 y Fl(j)p Fn(n)p Fl(j)p 23420 51995 4226 49 v 23420 53105 a(j)p Fn(n)p Fl(j)268 b Fo(+)i Fl(j)p Fn(k)38 b Fl(j)27778 52274 y Fn(q)43 b Fo(\()p Fn(s)p Fo(\))29872 51773 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)32791 52274 y Fn(e)33355 51773 y Fm(i)157 b(k)24 b(s)34836 52274 y Fn(e)35400 51773 y Fm(i)158 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p 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b(or)473 b(e)-36 b(v)-30 b(er)36 b(y)474 b Fn(l)589 b Fl(2)566 b Fo(I)-404 b(R)p Fn(;)474 b Fp(with)g(Lipschitz) g(constant)g(giv)-30 b(en)474 b(b)-24 b(y)500 b Fo(~)-633 b Fn(c;)474 b Fp(where)500 b Fo(~)-633 b Fn(c)474 b Fp(is)2841 31375 y(depending)337 b(on)g Fn(U)132 b Fo(\()p Fn(q)43 b Fo(\))338 b Fp(and)e Fn(l)24 b(:)-189 34193 y Fj(Proof.)606 b Fp(W)-36 b(e)337 b(f)-36 b(ollo)-18 b(w)336 b(in)h(par)-12 b(allel)337 b(to)g(the)f(proof)h(of)g(Proposition)f(4.1.)417 b(in)337 b(Ring)g(\(Ring,)g(1995\).)1224 37343 y(1.)606 b(Obser)36 b(v)-30 b(e)337 b(that,)9399 40773 y Fl(k)p Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\))p Fl(k)13084 40273 y Fu(2)13084 41180 y Fm(H)13925 40928 y Fk(l)14250 41180 y Fu(\()p Fe(T)p Fu(\))16734 40773 y Fo(=)1107 b(\()19755 39953 y(1)p 19388 40494 1341 49 v 19388 41605 a(2)p Fn(\031)20861 40773 y Fo(\))21332 40273 y Fu(2)22302 39622 y Fi(X)22061 42247 y Fm(n)p Ff(2)p Fu(Z)23834 42370 y Fh(0)24295 40773 y Fo(\(1)270 b(+)f Fn(n)27582 40273 y Fu(2)28107 40773 y Fo(\))28578 40273 y Fm(l)28925 40773 y Fl(j)29464 39123 y Fi(Z)30676 39474 y Fu(2)p Fm(\031)30138 41872 y Fu(0)32109 39953 y Fl(j)p Fn(n)p Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))35605 39513 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p 32109 40494 6414 49 v 33203 41605 a Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)38655 40773 y Fn(e)39219 40273 y Ff(\000)p Fm(ins)41333 40773 y Fn(ds)p Fl(j)42869 40273 y Fu(2)16734 44809 y Fl(\024)19026 43657 y Fi(X)18784 46283 y Fm(n)p Ff(2)p Fu(Z)20557 46406 y Fh(0)21353 43989 y Fo(\(1)270 b(+)f Fn(n)24640 43549 y Fu(2)25165 43989 y Fo(\))25636 43549 y Fm(l)11 b Fu(+1)p 21353 44530 5833 49 v 22157 45640 a Fl(j)p Fn(n)p Fl(j)268 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)27656 44809 y Fn(:)2841 48910 y Fp(Since)321 b Fo(\(1)211 b(+)h Fn(n)9364 48470 y Fu(2)9889 48910 y Fo(\))10360 48470 y Fm(l)11 b Fu(+1)11909 48910 y Fn(=)p Fo(\()p Fl(j)p Fn(n)p Fl(j)211 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)p Fo(\))336 b Fl(\024)h Fo(\(1)212 b(+)f Fn(n)22355 48470 y 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y(F)-36 b(our)18 b(ier)336 b(T)-145 b(r)-12 b(ansf)-36 b(or)30 b(m)335 b(is)13001 58644 y Fl(j)p Fn(n)p Fl(jj)p Fo(\()p Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\)\))34 b(^)-302 b(\()p Fn(n)p Fo(\))p Fl(j)337 b Fo(=)g Fl(j)p Fn(n)p Fl(j)23519 58143 y Fm(k)24587 57824 y Fo(1)p 24219 58365 1341 49 v 24219 59475 a(2)p Fn(\031)25693 58644 y Fl(j)26232 56994 y Fi(Z)27443 57344 y Fu(2)p Fm(\031)26905 59743 y Fu(0)28876 57824 y Fl(j)p Fn(n)p Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))32372 57384 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p 28876 58365 6414 49 v 29971 59475 a Fl(j)p Fn(n)p Fl(j)268 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)35423 58644 y Fn(e)35987 58143 y Ff(\000)p Fm(in)157 b(s)38257 58644 y Fn(ds)p Fl(j)2841 62117 y Fp(Since)338 b Fn(H)7316 61677 y Fu(1)7842 62117 y Fo(\()p Fg(T)p Fo(\))f Fl(\032)g(C)71 b Fo(\()p Fg(T)p Fo(\))p Fn(;)337 b Fp(clear)18 b(ly)337 b Fo(sup)20024 62408 y Fm(s)p Ff(2)p Fe(T)21934 62117 y Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))p Fl(j)338 b Fo(=)e Fl(k)p Fn(q)43 b 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Fo(\()p Fg(T)p Fo(\))h Fl(\032)e Fn(H)34102 76267 y Fm(l)34449 76767 y Fo(\()p Fg(T)p Fo(\))p eop end %%Page: 6 6 TeXDict begin 6 5 bop 2841 715 a Fp(with)337 b Fn(l)361 b Fl(2)336 b Fo(I)-404 b(R)p Fn(:)337 b Fp(Theref)-36 b(ore)2841 4833 y Fl(k)p Fn(F)168 b Fo(\()p Fn(q)5406 5015 y Fu(1)5934 4833 y Fo(\))269 b Fl(\000)g Fn(F)168 b Fo(\()p Fn(q)9845 5015 y Fu(2)10372 4833 y Fo(\))p Fl(k)11449 5129 y Fm(H)12290 4877 y Fk(l)12613 5129 y Fu(\()p Fe(T)p Fu(\))15098 4833 y Fo(=)17147 2397 y Fi(0)17147 4579 y(@)18450 3682 y(X)18208 6307 y Fm(n)p Ff(2)p Fu(Z)19981 6430 y Fh(0)20442 4833 y Fo(\(1)270 b(+)f Fn(n)23729 4333 y Fu(2)24255 4833 y Fo(\))24726 4333 y Fm(l)25073 4833 y Fl(j)25910 4013 y Fo(1)p 25543 4554 1341 49 v 25543 5665 a(2)p Fn(\031)27218 3183 y Fi(Z)28430 3534 y Fu(2)p Fm(\031)27891 5932 y Fu(0)29863 4013 y Fl(j)p Fn(n)p Fl(j)p Fo(\()p Fn(q)32277 4195 y Fu(1)32802 4013 y Fo(\()p Fn(s)p Fo(\))34312 3573 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)37499 4013 y 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b(uity)336 b(of)h Fn(F)505 b Fp(with)337 b(respect)f(to)-189 47778 y Fn(C)87 b Fo(\()p Fg(T)p Fo(\))p Fl(\000)p Fp(nor)30 b(m)337 b(on)g Fo(\()p Fp(dom)p Fn(F)168 b Fo(\))12518 47338 y Ff(\016)13045 47778 y Fn(:)-189 49600 y Fj(Der)18 b(iv)-30 b(ativ)g(e)336 b(of)h Fn(F)34 b(:)337 b Fp(No)-18 b(w)337 b(w)-12 b(e)337 b(tur)30 b(n)337 b(to)f(der)18 b(iving)337 b(F)-55 b(r)22572 49578 y(\264)22437 49600 y(echet)337 b(der)18 b(iv)-30 b(ativ)g(e)336 b(of)h Fn(F)34 b(:)-189 51896 y Fj(Proposition)387 b Fp(3.2)p Fj(.)570 b Fp(The)367 b(f)-36 b(orw)-18 b(ard)366 b(map)i Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\))389 b(:)f(\()p Fp(dom)p Fn(F)168 b Fo(\))26223 51456 y Ff(\016)27138 51896 y Fl(\032)387 b Fn(H)29574 51456 y Fu(1)30100 51896 y Fo(\()p Fg(T)p Fo(\))i Fl(!)e Fn(H)34944 51456 y Fm(l)35291 51896 y Fo(\()p Fg(T)p Fo(\))368 b Fp(is)g(F)-55 b(r)39877 51874 y(\264)39742 51896 y(echet)367 b(diff)-36 b(erentiab)-24 b(le)-189 53718 y(at)337 b(e)-36 b(v)-30 b(er)36 b(y)336 b(element)h Fn(q)380 b Fl(2)337 b Fo(\()p Fp(dom)p Fn(F)168 b Fo(\))15391 53278 y Ff(\016)16255 53718 y Fp(with)337 b(der)18 b(iv)-30 b(ativ)g(e)336 b(giv)-30 b(en)336 b(b)-24 b(y)9422 57163 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)p Fo(\)\()p Fn(t)p Fo(\))338 b(=)17803 56343 y(1)p 17435 56884 1341 49 v 17435 57995 a(2)p Fn(\031)19111 55513 y Fi(Z)20323 55864 y Fu(2)p Fm(\031)19784 58262 y Fu(0)21872 56012 y Fi(X)21623 58642 y Fm(n)p Ff(2)p Fe(Z)23411 58765 y Fh(0)24075 57163 y Fl(j)p Fn(n)p Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))27571 56663 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)30488 57163 y Fn(e)31052 56663 y Fm(i)158 b(k)180 b(s)32691 57163 y Fn(h)p Fo(\()p Fn(s)p Fo(\))p Fn(e)35463 56663 y Fm(i)158 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))38806 57163 y Fn(ds;)7119 b Fp(\(3.4\))-189 61169 y(f)-36 b(or)336 b(all)h Fn(h)g Fl(2)g Fn(H)6361 60729 y Fu(1)6887 61169 y Fo(\()p Fg(T)p Fo(\))p Fn(:)g Fp(The)g(F)-36 b(our)18 b(ier)335 b(tr)-12 b(ansf)-36 b(or)30 b(m)336 b(of)h 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b(diff)-36 b(erentiability)337 b(of)g Fn(F)34 b(;)336 b Fp(it)h(is)g(suf\002cient) g(to)g(sho)-18 b(ws)337 b(the)g(f)-36 b(ollo)-18 b(wing)417 12759 y(\(1.\))605 b Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])337 b(:)f Fn(H)8200 12319 y Fu(1)8726 12759 y Fo(\()p Fg(T)p Fo(\))i Fl(!)f Fn(H)13469 12319 y Fm(l)13815 12759 y Fo(\()p Fg(T)p Fo(\))h Fp(is)f(a)g(bounded)g(linear)f(oper)-12 b(ator)-61 b(.)417 15577 y(\(2.\))605 b Fn(D)34 b(F)482 b Fp(is)314 b(a)f(locally)h(Lipschitz)g(contin)-12 b(uous)313 b(dependent)g(on)g Fn(q)43 b(;)314 b Fp(i.e)-18 b(.,)318 b(there)313 b(e)-36 b(xists)314 b Fn(L)p Fo(\()p Fn(q)43 b Fo(\))338 b Fn(>)e Fo(0)p Fn(;)313 b Fp(such)h(that)14819 18616 y Fl(k)p Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\))272 b Fl(\000)d Fn(F)168 b Fo(\()90 b(~)-696 b Fn(q)45 b Fo(\))p Fl(k)22463 18912 y Ff(L)p Fu(\()p Fm(H)24317 18660 y Fk(l)24640 18912 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)27064 18660 y Fk(l)27386 18912 y Fu(\()p Fe(T)p Fu(\)\))29467 18616 y Fl(\024)336 b Fn(L)p Fl(k)p Fn(q)313 b Fl(\000)359 b Fo(~)-696 b Fn(q)44 b Fl(k)35434 18912 y Fm(H)36275 18660 y Fk(l)36597 18912 y Fu(\()p Fe(T)p Fu(\))417 22154 y Fp(\(3.\))605 b Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])337 b Fp(is)g(Gateaux)g(der)18 b(iv)-30 b(ativ)g(e)336 b(of)h Fn(F)506 b Fp(at)337 b Fn(q)43 b(;)337 b Fp(i.e)-18 b(.)14614 25193 y Fo(lim)14490 25966 y Fm(\034)102 b Ff(!)p Fu(0)16623 25193 y Fl(k)p Fn(F)168 b Fo(\()p Fn(q)315 b Fo(+)269 b Fn(\034)137 b(h)p Fo(\))270 b Fl(\000)f Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\))272 b Fl(\000)d Fn(\034)137 b(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)p Fl(k)33204 25489 y Fm(H)34045 25237 y Fk(l)34367 25489 y Fu(\()p Fe(T)p Fu(\))36082 25193 y Fo(=)336 b(0)p Fn(;)2841 28470 y Fp(where)h Fn(\034)474 b Fl(2)336 b Fo([0)p Fn(;)202 b Fl(1)p Fo(\))337 b Fp(f)-36 b(or)336 b(e)-36 b(v)-30 b(er)36 b(y)337 b Fn(h)g Fl(2)g Fn(H)20401 28030 y Fm(l)20747 28470 y Fo(\()p Fg(T)p Fo(\))p Fn(:)-189 31288 y Fp(Let)410 b(us)f(sho)-18 b(w)410 b(\002rst)f(that)h Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])409 b Fp(as)h(de\002ned)g(in)f(\(3.4\))g(is)h (a)g(bounded)f(linear)g(oper)-12 b(ator)409 b(from)f Fn(H)45312 30848 y Fu(1)45838 31288 y Fo(\()p Fg(T)p Fo(\))j Fp(into)-189 33110 y Fn(H)917 32670 y Fm(l)1264 33110 y Fo(\()p Fg(T)p Fo(\))338 b Fp(f)-36 b(or)336 b(some)g(arbitr)-12 b(ar)36 b(y)336 b(b)-24 b(ut)337 b(\002x)-36 b(ed)336 b Fn(l)361 b Fl(2)336 b Fo(I)-404 b(R)p Fn(:)337 b Fp(Using)3127 37540 y Fl(k)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)p Fl(k)8280 37836 y Fm(H)9121 37583 y Fk(l)9444 37836 y Fu(\()p Fe(T)p Fu(\))11928 37540 y Fo(=)13978 35103 y Fi(0)13978 37285 y(@)15288 36388 y(X)15039 39018 y Fm(n)p Ff(2)p Fe(Z)16827 39141 y Fh(0)17288 37540 y Fo(\(1)270 b(+)f Fn(n)20575 37039 y Fu(2)21101 37540 y Fo(\))21572 37039 y Fm(l)21919 37540 y Fl(j)22755 36720 y Fo(1)p 22389 37261 V 22389 38371 a(2)p Fn(\031)24064 35890 y Fi(Z)25276 36240 y Fu(2)p Fm(\031)24737 38639 y Fu(0)26576 37540 y Fl(j)p Fn(n)p Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))30072 37039 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)32989 37540 y Fn(e)33553 37039 y Fm(i)158 b(k)180 b(s)35192 37540 y Fn(h)p Fo(\()p Fn(s)p Fo(\))p Fn(e)37964 37039 y Fm(i)158 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))41307 37540 y Fn(ds)p Fl(j)42843 37039 y Fu(2)43369 35103 y Fi(1)43369 37285 y(A)44429 35405 y Fu(1)p Fm(=)p Fu(2)46098 37540 y Fn(;)11928 42660 y Fl(\024)13978 40223 y Fi(0)13978 42405 y(@)15288 41508 y(X)15039 44139 y Fm(n)p Ff(2)p Fe(Z)16827 44262 y Fh(0)17288 42660 y Fo(\(1)270 b(+)f Fn(n)20575 42159 y Fu(2)21101 42660 y Fo(\))21572 42159 y Fm(l)21919 42660 y Fl(j)p Fn(n)p Fl(j)23321 42159 y Fu(2)23845 42660 y Fl(k)p Fn(q)43 b Fl(k)25641 42159 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))25641 42959 y Ff(1)29762 42660 y Fl(j)30598 41840 y Fo(1)p 30232 42381 V 30232 43491 a(2)p Fn(\031)31906 41010 y Fi(Z)33119 41360 y Fu(2)p Fm(\031)32580 43759 y Fu(0)34418 42660 y Fn(h)p Fo(\()p Fn(s)p Fo(\))p Fn(e)37190 42159 y Fm(i)159 b(n)p Fu(\()p Ff(\000)p Fm(s)p Fu(\))40194 42660 y Fn(ds)p Fl(j)41730 42159 y Fu(2)42256 40223 y Fi(1)42256 42405 y(A)43317 40525 y Fu(1)p Fm(=)p Fu(2)44985 42660 y Fn(;)11928 47780 y Fl(\024)13978 45343 y Fi(0)13978 47525 y(@)15039 47780 y Fo(\()15759 46628 y Fi(X)15510 49259 y Fm(n)p Ff(2)p Fe(Z)17298 49382 y Fh(0)17760 47780 y Fo(\(1)269 b(+)h Fn(n)21047 47279 y Fu(2)21572 47780 y Fo(\))22043 47279 y Fm(l)22390 47780 y Fo(\(1)g(+)f Fn(n)25677 47279 y Fu(2)26202 47780 y Fo(\))p Fl(k)p Fn(q)43 b Fl(k)28469 47279 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))28469 48080 y Ff(1)32590 47780 y Fl(j)32939 47460 y Fo(^)32927 47780 y Fn(h)p Fo(\()p Fn(n)p Fo(\))p Fl(j)35632 47279 y Fu(2)36158 47780 y Fo(\))36629 45343 y Fi(1)36629 47525 y(A)37690 45646 y Fu(1)p Fm(=)p Fu(2)39358 47780 y Fn(;)11928 52900 y Fl(\024)13978 50464 y Fi(0)13978 52645 y(@)15288 51748 y(X)15039 54379 y Fm(n)p Ff(2)p Fe(Z)16827 54502 y Fh(0)17288 52900 y Fo(\(1)270 b(+)f Fn(n)20575 52399 y Fu(2)21101 52900 y Fo(\))21572 52399 y Fu(\()p Fm(l)11 b Fu(+1\))23853 52900 y Fl(k)p Fn(q)43 b Fl(k)25649 52399 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))25649 53200 y Ff(1)29769 52900 y Fl(j)30118 52580 y Fo(^)30106 52900 y Fn(h)p Fo(\()p Fn(n)p Fo(\))p Fl(j)32811 52399 y Fu(2)33337 50464 y Fi(1)33337 52645 y(A)34397 50766 y Fu(1)p Fm(=)p Fu(2)36066 52900 y Fn(;)11928 57656 y Fl(\024)1107 b Fn(c)14906 55584 y Fi( )15885 56505 y(X)15866 59135 y Fm(n)p Ff(2)p Fe(Z)17857 57656 y Fl(j)18206 57337 y Fo(^)18194 57656 y Fn(h)p Fo(\()p Fn(n)p Fo(\))p Fl(j)20899 57156 y Fu(2)21425 55584 y Fi(!)22384 55886 y Fu(1)p Fm(=)p Fu(2)24188 57656 y Fn(<)336 b(c)p Fl(k)p Fn(h)p Fl(k)27902 57903 y Fm(L)28542 58026 y Fh(2)29003 57903 y Fu(\()p Fe(T)p Fu(\))30381 57656 y Fn(;)-189 62849 y Fp(where)365 b(w)-12 b(e)365 b(chose)g Fn(c)g Fp(suf\002ciently)h (large)f(such)g(that)g Fo(\(1)289 b(+)f Fn(n)27589 62410 y Fu(2)28114 62849 y Fo(\))28585 62410 y Fu(\()p Fm(l)11 b Fu(+1\))p Fm(=)p Fu(2)31807 62849 y Fl(k)p Fn(q)43 b Fl(k)33603 62211 y Fu(\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))33603 63007 y Ff(1)37637 62849 y Fn(<)384 b(c)365 b Fp(f)-36 b(or)365 b(all)g Fn(n)384 b Fl(2)g Fg(Z)p Fn(:)365 b Fp(\(Note)-189 64671 y(that)447 b Fl(k)p Fn(q)43 b Fl(k)4076 64853 y Ff(1)5593 64671 y Fn(<)521 b Fo(1)447 b Fp(since)g Fn(q)564 b Fl(2)521 b Fo(\()p Fp(dom)p Fn(F)168 b Fo(\))18067 64231 y Ff(\016)18594 64671 y Fo(\))p Fn(:)447 b Fp(Hence)g Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])520 b(:)h Fn(H)29526 64231 y Fu(1)30052 64671 y Fo(\()p Fg(T)p Fo(\))g Fl(\032)g Fn(L)34612 64853 y Fu(2)35137 64671 y Fo(\()p Fg(T)p Fo(\))h Fl(!)f Fn(H)40248 64231 y Fm(l)40595 64671 y Fo(\()p Fg(T)p Fo(\))447 b Fp(is)g(a)g(bounded)-189 66493 y(linear)375 b(oper)-12 b(ator)-61 b(.)533 b(Moreo)-18 b(v)-30 b(er)374 b(the)i(results)f(sho)-18 b(ws)376 b(a)g(stronger)f(result)g(than)h(w) -12 b(e)375 b(require)g(and)h(also)g Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])-189 68314 y Fp(can)337 b(be)g(uniquely)g(e)-36 b(xtended)336 b(to)h(a)g(bounded)f(linear)h(oper)-12 b(ator)335 b(from)h Fn(L)33063 68496 y Fu(2)33589 68314 y Fo(\()p Fg(T)p Fo(\))i Fp(into)f Fn(H)39074 67874 y Fu(1)39600 68314 y Fo(\()p Fg(T)p Fo(\))p Fn(:)-189 70136 y Fp(F)-36 b(or)370 b(the)i(proof)f(of)g(\(2.\))521 b(w)-12 b(e)371 b(pic)-24 b(ks)372 b(a)f(neighbourhood)g Fn(U)132 b Fo(\()p Fn(q)43 b Fo(\))373 b Fp(of)e Fn(q)415 b Fp(in)372 b Fn(H)33015 69696 y Fu(1)33541 70136 y Fo(\()p Fg(T)p Fo(\))396 b Fl(\032)e Fn(L)37849 69696 y Fu(2)38375 70136 y Fo(\()p Fg(T)p Fo(\))372 b Fp(such)g(that)f Fl(k)90 b Fo(~)-696 b Fn(q)44 b Fl(k)47619 70318 y Ff(1)49010 70136 y Fl(\024)p eop end %%Page: 8 8 TeXDict begin 8 7 bop -189 715 a Fn(m)337 b(<)f Fo(1)p Fn(;)h Fp(f)-36 b(or)336 b(all)428 b Fo(~)-697 b Fn(q)44 b(:)337 b Fp(Then)f(w)-12 b(e)337 b(ha)-24 b(v)-30 b(e)1540 4572 y Fl(k)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)270 b Fl(\000)f Fn(D)34 b(F)168 b Fo([)90 b(~)-696 b Fn(q)44 b Fo(])p Fn(h)p Fl(k)12117 4868 y Fm(H)12958 4616 y Fk(l)13280 4868 y Fu(\()p Fe(T)p Fu(\))15765 4572 y Fo(=)17814 2136 y Fi(0)17814 4317 y(@)19124 3420 y(X)18875 6051 y Fm(n)p Ff(2)p Fe(Z)20663 6174 y Fh(0)21125 4572 y Fo(\(1)270 b(+)f Fn(n)24412 4072 y Fu(2)24937 4572 y Fo(\))25408 4072 y Fm(l)25755 4572 y Fl(jj)p Fn(n)p Fl(j)401 b(\002)18447 8872 y Fo(1)p 18080 9413 1341 49 v 18080 10524 a(2)p Fn(\031)19755 8042 y Fi(Z)20967 8393 y Fu(2)p Fm(\031)20429 10791 y Fu(0)22065 9692 y Fo(\()p Fn(q)314 b Fl(\000)359 b Fo(~)-696 b Fn(q)43 b Fo(\)[)25995 8063 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)26551 8541 y Fi(X)26609 11146 y Fm(j)51 b Fu(=0)29059 9692 y Fn(q)29643 9192 y Fm(j)30221 9692 y Fo(~)-696 b Fn(q)30715 9192 y Fu(\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j\000)p Fm(j)35162 9692 y Fo(])p Fn(h)p Fo(\()p Fn(s)p Fo(\))p Fn(e)38271 9192 y Ff(\000)p Fm(ns)40065 9692 y Fn(ds)p Fl(j)41601 9192 y Fu(2)42127 7256 y Fi(1)42127 9438 y(A)43188 7558 y Fu(1)p Fm(=)p Fu(2)15765 14812 y Fl(\024)17814 12376 y Fi(0)17814 14558 y(@)19124 13661 y(X)18875 16291 y Fm(n)p Ff(2)p Fe(Z)20663 16414 y Fh(0)21125 14812 y Fo(\(1)270 b(+)f Fn(n)24412 14312 y Fu(2)24937 14812 y Fo(\))25408 14312 y Fm(l)25755 14812 y Fl(j)p Fn(n)p Fl(j)27157 14312 y Fu(2)27682 14812 y Fo(\()p Fl(j)p Fn(n)p Fl(j)g Fo(+)g Fl(j)p Fn(k)38 b Fl(j)269 b Fo(+)g(1\))34937 14312 y Fu(2)35463 14812 y Fn(m)36527 14312 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))40647 14812 y Fl(j)40996 14492 y Fo(^)40984 14812 y Fn(h)p Fo(\()p Fn(n)p Fo(\))p Fl(j)43689 14312 y Fu(2)44215 12376 y Fi(1)44215 14558 y(A)45275 12678 y Fu(1)p Fm(=)p Fu(2)47011 14812 y Fl(\002)22663 18043 y(k)p Fn(q)313 b Fl(\000)359 b Fo(~)-696 b Fn(q)44 b Fl(k)26526 18225 y Ff(1)15765 21587 y Fl(\024)17814 19151 y Fi(0)17814 21333 y(@)19124 20436 y(X)18875 23066 y Fm(n)p Ff(2)p Fe(Z)20663 23189 y Fh(0)21125 21587 y Fo(\(1)270 b(+)f Fn(n)24412 21087 y Fu(2)24937 21587 y Fo(\))25408 21087 y Fm(l)11 b Fu(+1)26957 21587 y Fo(\()p Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)269 b Fo(+)g(1\))34212 21087 y Fu(2)34739 21587 y Fn(m)35803 21087 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))39922 21587 y Fl(j)40271 21267 y Fo(^)40259 21587 y Fn(h)p Fo(\()p Fn(n)p Fo(\))p Fl(j)42964 21087 y Fu(2)43490 19151 y Fi(1)43490 21333 y(A)44551 19453 y Fu(1)p Fm(=)p Fu(2)46287 21587 y Fl(\002)22663 24818 y(k)p Fn(q)313 b Fl(\000)359 b Fo(~)-696 b Fn(q)44 b Fl(k)26526 25000 y Ff(1)-189 27665 y Fp(W)-36 b(e)380 b(choose)g Fn(c)6786 27847 y Fu(1)7692 27665 y Fp(and)f Fn(c)10618 27847 y Fu(2)11524 27665 y Fp(tw)-12 b(o)380 b(constants)g(such)g(that)g Fl(k)p Fn(q)342 b Fl(\000)388 b Fo(~)-696 b Fn(q)44 b Fl(k)28609 27847 y Ff(1)30014 27665 y Fl(\024)409 b Fn(c)31891 27847 y Fu(1)32416 27665 y Fl(k)p Fn(q)342 b Fl(\000)388 b Fo(~)-696 b Fn(q)44 b Fl(k)36337 27928 y Fm(H)37178 27676 y Fh(1)37639 27928 y Fu(\()p Fe(T)p Fu(\))39397 27665 y Fp(f)-36 b(or)379 b(all)471 b Fo(~)-697 b Fn(q)424 b Fp(elements)380 b(in)-189 29487 y Fn(U)132 b Fo(\()p Fn(q)43 b Fo(\))p Fn(;)338 b Fp(the)f(neighbourhood)e(of)i Fn(q)43 b(;)338 b Fp(and)10572 32334 y Fo(\(1)269 b(+)g Fn(n)13858 31834 y Fu(2)14384 32334 y Fo(\))14855 31834 y Fm(l)11 b Fu(+1)16404 32334 y Fo(\()p Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)269 b Fo(+)g(1\))23659 31834 y Fu(2)24185 32334 y Fn(m)25249 31834 y Fu(2\()p Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(\))29706 32334 y Fn(<)336 b(c)31510 31834 y Fu(2)31510 32634 y(2)32036 32334 y Fn(;)202 b Fp(f)-36 b(or)335 b(all)i Fn(n)g Fl(2)f Fg(Z)p Fn(:)-189 35182 y Fp(Then)g(w)-12 b(e)337 b(ha)-24 b(v)-30 b(e)11507 37003 y Fl(k)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)270 b Fl(\000)f Fn(D)34 b(F)168 b Fo([)90 b(~)-696 b Fn(q)43 b Fo(])p Fn(h)p Fl(k)22083 37299 y Fm(H)22924 37047 y Fk(l)23247 37299 y Fu(\()p Fe(T)p Fu(\))24961 37003 y Fl(\024)336 b Fn(L)p Fl(k)p Fn(q)313 b Fl(\000)360 b Fo(~)-697 b Fn(q)44 b Fl(k)30928 37299 y Fm(H)31769 37047 y Fk(l)32091 37299 y Fu(\()p Fe(T)p Fu(\))33469 37003 y Fl(k)p Fn(h)p Fl(k)35379 37299 y Fm(H)36220 37047 y Fk(l)36543 37299 y Fu(\()p Fe(T)p Fu(\))37920 37003 y Fn(;)-189 39448 y Fp(with)337 b Fn(L)g Fo(=)f Fn(c)5269 39630 y Fu(1)5795 39448 y Fn(c)6320 39630 y Fu(2)6845 39448 y Fn(;)h Fp(and)f(consequently)11600 42296 y Fl(k)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)270 b Fl(\000)f Fn(D)34 b(F)168 b Fo([)90 b(~)-696 b Fn(q)43 b Fo(])p Fn(h)p Fl(k)22176 42592 y Ff(L)p Fu(\()p Fm(H)24030 42340 y Fh(1)24492 42592 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)26916 42340 y Fk(l)27239 42592 y Fu(\()p Fe(T)p Fu(\)\))29319 42296 y Fl(\024)336 b Fn(L)p Fl(k)p Fn(q)313 b Fl(\000)360 b Fo(~)-697 b Fn(q)44 b Fl(k)35286 42592 y Fm(H)36127 42340 y Fk(l)36449 42592 y Fu(\()p Fe(T)p Fu(\))37827 42296 y Fn(;)-189 45143 y Fp(f)-36 b(or)336 b(all)427 b Fo(~)-696 b Fn(q)381 b Fl(2)336 b Fn(U)132 b Fo(\()p Fn(q)43 b Fo(\))p Fn(:)-189 46965 y Fp(No)-18 b(w)-73 b(,)337 b(it)g(remains)f(to)h(sho)-18 b(w)337 b(that)g Fn(D)34 b(F)505 b Fp(is)337 b(the)g(Gateaux)f(der)18 b(iv)-30 b(ativ)g(e)337 b(of)f Fn(F)34 b(:)337 b Fp(W)-36 b(e)337 b(\002nd)g(that)13743 49812 y Fl(k)p Fn(F)168 b Fo(\()p Fn(q)314 b Fo(+)269 b Fn(\034)137 b(h)p Fo(\))270 b Fl(\000)f Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\))272 b Fl(\000)d Fn(\034)137 b(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h)p Fl(k)30323 50108 y Fm(H)31164 49856 y Fk(l)31487 50108 y Fu(\()p Fe(T)p Fu(\))33971 49812 y Fo(=)2095 55290 y(=)4145 52854 y Fi(0)4145 55036 y(@)5455 54139 y(X)5205 56769 y Fm(n)p Ff(2)p Fe(Z)6993 56892 y Fh(0)7455 55290 y Fo(\(1)270 b(+)f Fn(n)10742 54790 y Fu(2)11267 55290 y Fo(\))11738 54790 y Fm(l)12085 55290 y Fl(j)12922 54470 y Fo(1)p 12555 55011 V 12555 56122 a(2)p Fn(\031)14230 53640 y Fi(Z)15442 53991 y Fu(2)p Fm(\031)14904 56389 y Fu(0)16540 55290 y Fo(\()17144 54470 y Fl(j)p Fn(n)p Fl(j)p Fo(\(\()p Fn(q)313 b Fo(+)269 b Fn(\034)137 b(h)p Fo(\))23390 54030 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)26577 54470 y Fl(\000)270 b Fn(q)28374 54030 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)31291 54470 y Fo(\))p 17144 55011 14619 49 v 22341 56122 a Fl(j)p Fn(n)p Fl(j)268 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)32165 55290 y(\000)269 b Fn(\034)137 b(q)34628 54790 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)37546 55290 y Fl(j)p Fn(n)p Fl(j)p Fn(h)p Fo(\()p Fn(s)p Fo(\)\))p Fn(e)42191 54790 y Ff(\000)p Fm(ns)43985 55290 y Fn(ds)p Fl(j)45521 54790 y Fu(2)46047 52854 y Fi(1)46047 55036 y(A)47107 53156 y Fu(1)p Fm(=)p Fu(2)2095 60410 y Fo(=)4645 59590 y(1)p 4277 60132 1341 49 v 4277 61242 a(2)p Fn(\031)5953 57974 y Fi(0)5953 60156 y(@)7263 59259 y(X)7013 61889 y Fm(n)p Ff(2)p Fe(Z)8801 62012 y Fh(0)9263 60410 y Fo(\(1)270 b(+)f Fn(n)12550 59910 y Fu(2)13075 60410 y Fo(\))13546 59910 y Fm(l)13893 60410 y Fl(j)p Fn(n)p Fl(j)15295 59910 y Fu(2)15820 60410 y Fl(j)16359 58760 y Fi(Z)17571 59111 y Fu(2)p Fm(\031)17032 61509 y Fu(0)18669 60410 y Fo(\()19273 59590 y(\(\()p Fn(q)314 b Fo(+)269 b Fn(\034)137 b(h)p Fo(\))24118 59150 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)27305 59590 y Fl(\000)269 b Fn(q)29101 59150 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)32019 59590 y Fo(\))p 19273 60132 13218 49 v 23769 61242 a Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)32893 60410 y(\000)269 b Fn(\034)137 b(q)35356 59910 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)38274 60410 y Fn(h)p Fo(\()p Fn(s)p Fo(\)\))p Fn(e)41517 59910 y Ff(\000)p Fm(ns)43312 60410 y Fn(ds)p Fl(j)44848 59910 y Fu(2)45373 57974 y Fi(1)45373 60156 y(A)46434 58276 y Fu(1)p Fm(=)p Fu(2)2095 65530 y Fl(\024)4645 64710 y Fo(1)p 4277 65252 1341 49 v 4277 66362 a(2)p Fn(\031)5953 63094 y Fi(0)5953 65276 y(@)7263 64379 y(X)7013 67009 y Fm(n)p Ff(2)p Fe(Z)8801 67132 y Fh(0)9263 65530 y Fo(\(1)270 b(+)f Fn(n)12550 65030 y Fu(2)13075 65530 y Fo(\))13546 65030 y Fu(\()p Fm(l)11 b Fu(+1\))15828 65530 y Fl(j)16367 63881 y Fi(Z)17578 64231 y Fu(2)p Fm(\031)17040 66630 y Fu(0)20821 64710 y Fo(1)p 19011 65252 4226 49 v 19011 66362 a Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)23369 65530 y Fo([)23706 63902 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)24261 64379 y Fi(X)24320 66985 y Fm(j)51 b Fu(=2)26770 63458 y Fi( )27764 64354 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)28876 65509 y Fn(j)30591 63458 y Fi(!)31752 65530 y Fn(\034)32419 65030 y Fm(j)32906 65530 y Fn(h)p Fo(\()p Fn(s)p Fo(\))35114 65030 y Fm(j)35603 65530 y Fn(q)36187 65030 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)g Ff(j\000)p Fm(j)40268 65530 y Fo(])p Fn(e)41169 65030 y Ff(\000)p Fm(ns)42962 65530 y Fn(ds)p Fl(j)44498 65030 y Fu(2)45024 63094 y Fi(1)45024 65276 y(A)46084 63396 y Fu(1)p Fm(=)p Fu(2)2095 70651 y Fl(\024)4645 69831 y Fo(1)p 4277 70372 1341 49 v 4277 71482 a(2)p Fn(\031)5953 68214 y Fi(0)5953 70396 y(@)7263 69499 y(X)7013 72129 y Fm(n)p Ff(2)p Fe(Z)8801 72252 y Fh(0)9263 70651 y Fo(\(1)270 b(+)f Fn(n)12550 70150 y Fu(2)13075 70651 y Fo(\))13546 70150 y Fu(\()p Fm(l)11 b Fu(+1\))15828 70651 y Fl(j)16367 69001 y Fi(Z)17578 69351 y Fu(2)p Fm(\031)17040 71750 y Fu(0)20821 69831 y Fo(1)p 19011 70372 4226 49 v 19011 71482 a Fl(j)p Fn(n)p Fl(j)269 b Fo(+)g Fl(j)p Fn(k)38 b Fl(j)23369 70651 y Fo([)23706 69022 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)24261 69499 y Fi(X)24320 72105 y Fm(j)51 b Fu(=2)26770 68578 y Fi( )27764 69474 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)28876 70630 y Fn(j)30591 68578 y Fi(!)31752 70651 y Fn(\034)32419 70150 y Fm(j)32906 70651 y Fn(h)p Fo(\()p Fn(s)p Fo(\))35114 70150 y Fm(j)35603 70651 y Fn(q)36187 70150 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)g Ff(j\000)p Fm(j)40268 70651 y Fo(])p Fn(e)41169 70150 y Ff(\000)p Fm(ns)42962 70651 y Fn(ds)p Fl(j)44498 70150 y Fu(2)45024 68214 y Fi(1)45024 70396 y(A)46084 68516 y Fu(1)p Fm(=)p Fu(2)2095 75771 y Fl(\024)1107 b Fn(\034)5014 73334 y Fi(0)5014 75516 y(@)6324 74619 y(X)6074 77249 y Fm(n)p Ff(2)p Fe(Z)7862 77372 y Fh(0)8324 75771 y Fo(\(1)270 b(+)f Fn(n)11611 75270 y Fu(2)12136 75771 y Fo(\))12607 75270 y Fu(\()p Fm(l)11 b Fu(+1\))14889 75771 y Fo(\()17302 74951 y(1)p 15493 75492 V 15493 76602 a Fl(j)p Fn(n)p Fl(j)268 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)19851 75771 y(k)p Fn(q)313 b Fo(+)269 b Fn(\034)137 b Fl(j)p Fn(h)p Fl(jk)25168 75270 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)25168 76070 y(1)28085 75771 y Fo(\))28556 75270 y Fu(2)29082 73334 y Fi(1)29082 75516 y(A)30143 73636 y Fu(1)p Fm(=)p Fu(2)p eop end %%Page: 9 9 TeXDict begin 9 8 bop -189 715 a Fp(where)536 b(w)-12 b(e)537 b(used)g Fn(\034)9554 275 y Fm(k)24 b Ff(\000)p Fu(1)11996 715 y Fl(\024)670 b Fn(\034)14276 275 y Fm(k)24 b(=)p Fu(2)16323 715 y Fp(f)-36 b(or)536 b(all)h Fn(k)709 b Fl(\025)670 b Fo(2)537 b Fp(and)g Fn(\034)808 b Fl(2)671 b Fo(\(0)p Fn(;)202 b Fo(1\))p Fn(:)537 b Fp(The)f(last)h(ser)18 b(ies)536 b(is)h(con)-24 b(v)-30 b(erges)536 b(if)-189 2537 y Fl(k)p Fn(q)313 b Fo(+)269 b Fn(\034)137 b Fl(j)p Fn(h)p Fl(jk)5128 2719 y Ff(1)6461 2537 y Fn(<)337 b Fl(1)g Fp(b)-24 b(y)336 b(the)h(quotient)g(cr)18 b(iter)g(ion,)336 b(hence)h(the)g(whole)g(e)-36 b(xpression)336 b(goes)h(to)g Fo(0)g Fp(as)f Fn(\034)474 b Fl(!)337 b Fo(0)p Fn(:)p 49073 4358 45 819 v 49118 3584 728 45 v 49118 4358 V 49845 4358 45 819 v -189 7172 a Fp(The)f(f)-36 b(ollo)-18 b(wing)337 b(theorem)f(is)h(in)g(par)-12 b(allel)336 b(to)h(the)g(result)g(of)f(Theorem)g(4.5)h(in)g(Ring)g(\(Ring,)g (1995\).)-189 9968 y Fd(Theorem)312 b(3.3.)522 b Fj(The)314 b(oper)-12 b(ator)314 b Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])336 b(:)h Fn(H)20367 9528 y Fu(1)20893 9968 y Fo(\()p Fg(T)p Fo(\))g Fl(!)g Fn(H)25635 9528 y Fm(l)25537 10268 y Ff(\005)26062 9968 y Fo(\()p Fg(T)p Fo(\))316 b Fj(with)e Fn(l)361 b Fl(2)336 b Fo(I)-404 b(R)315 b Fj(is)f(injectiv)-30 b(e)315 b(f)-36 b(or)313 b(all)i Fn(q)380 b Fl(2)337 b Fo(\()p Fj(dom)p Fn(F)168 b Fo(\))49089 9528 y Ff(\016)49616 9968 y Fn(:)-189 12764 y Fj(Proof.)606 b Fp(Let)337 b Fn(h)p Fo(\()p Fn(s)p Fo(\))h Fl(2)f Fp(k)-24 b(er)o Fn(D)34 b(F)168 b Fo([)p Fn(q)43 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y Fp(By)389 b(the)f(help)h(of)f(the)h(results)f(in)g (the)h(proof)f(of)g(Theorem)f(4.5)i(in)f(\(Ring,)g(1995\),)401 b(thus)389 b(w)-12 b(e)388 b(ma)-36 b(y)388 b(conclude)-189 35707 y(that)2311 35387 y Fo(~)2299 35707 y Fn(h)p Fo(\()p Fn(s)p Fo(\))554 b(=)e(0)p Fn(:)466 b Fp(Ho)-18 b(w)-12 b(e)-36 b(v)-30 b(er)-61 b(,)497 b(since)466 b Fn(q)43 b Fo(\()p Fn(s)p Fo(\))18813 35267 y Ff(j)p Fm(k)24 b Ff(j\000)p Fu(1)21109 35707 y Fn(e)21673 35267 y Fm(i)157 b(k)180 b(s)23863 35707 y Fl(6)p Fo(=)553 b(0)p Fn(;)202 b(s)552 b Fl(2)h Fg(T)p Fl(nf)p Fo(0)p Fl(g)p Fn(;)466 b Fp(then)g(leads)f(to)h Fn(h)p Fo(\()p Fn(s)p Fo(\))554 b(=)f(0)p Fn(;)465 b Fp(or)h Fn(h)553 b Fl(2)-189 37529 y Fp(k)-24 b(er)o Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(:)p 49073 37529 45 819 v 49118 36755 728 45 v 49118 37529 V 49845 37529 45 819 v -189 40342 a Fd(Theorem)335 b(3.4.)553 b Fj(The)336 b(e)-36 b(v)-30 b(aluation)336 b(of)h(adjoint)g(oper)-12 b(ator)336 b Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])29400 39903 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Fp(W)-36 b(e)353 b(f)-36 b(ollo)-18 b(w)353 b(in)g(par)-12 b(allel)352 b(to)h(the)g(idea)g(of)g(the)g(proof)g(of)g(Theorem)e(5.1)i(and)g (utiliz)-18 b(e)353 b(some)g(results)g(of)-189 52516 y(the)337 b(appendix)f(in)h(the)g(w)-12 b(or)18 b(ks)337 b(of)g(Ring)g(\(Ring,)g(1995\).)-189 54337 y(\(The)f(f)-36 b(ollo)-18 b(wing)337 b(geometr)18 b(ic)336 b(ser)18 b(ies:)417 b Fn(r)34 b(=)p Fo(\(1)269 b Fl(\000)g Fn(r)34 b Fo(\))22116 53898 y Fu(2)22979 54337 y Fo(=)24258 53428 y Fi(P)25537 54696 y Fm(n)p Ff(2)p Fe(N)27375 54819 y Fh(0)28094 54337 y Fn(nr)29403 53898 y Fm(n)30028 54337 y Fn(;)202 b Fl(j)p Fn(r)34 b Fl(j)336 b Fn(<)g Fo(1)p Fn(;)h Fp(is)g(needed)f(in)h(the)g(proof)-36 b(.\))-189 59946 y Fl(h)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h;)202 b(g)43 b Fl(i)5854 60288 y Fm(H)6695 60027 y Fc(\000)p Fh(1)p Fk(=)p Fh(2)8628 60288 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)11052 60027 y Fh(1)p Fk(=)p Fh(2)12344 60288 y Fu(\()p Fe(T)p Fu(\))14828 59946 y Fo(=)17378 59126 y(1)p 17011 59668 V 17011 60778 a(2)p Fn(\031)18686 58296 y Fi(Z)19898 58647 y Fu(2)p Fm(\031)19360 61045 y Fu(0)21198 57510 y Fi(0)21198 59692 y(@)22759 59126 y Fo(1)p 22392 59668 V 22392 60778 a(2)p Fn(\031)24067 58296 y Fi(Z)25279 58647 y Fu(2)p Fm(\031)24741 61045 y Fu(0)26829 58795 y Fi(X)26579 61425 y Fm(n)p Ff(2)p Fe(Z)28367 61548 y Fh(0)29031 59946 y Fl(j)p Fn(n)p Fl(j)p Fn(q)g Fo(\()p Fn(s)p Fo(\))32527 59446 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)35445 59946 y Fn(e)36009 59446 y Fm(i)157 b(k)180 b(s)37647 59946 y Fn(h)p Fo(\()p Fn(s)p Fo(\))p Fn(e)40419 59446 y Fm(i)159 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))43763 59946 y Fn(ds)44962 57510 y Fi(1)44962 59692 y(A)46224 59946 y Fn(g)43 b Fo(\()p Fn(t)p Fo(\))338 b Fn(dt;)14828 64138 y Fo(=)17378 63318 y(1)p 17011 63859 V 17011 64969 a(2)p Fn(\031)18686 62488 y Fi(Z)19898 62839 y Fu(2)p Fm(\031)19360 65237 y Fu(0)21198 64138 y Fn(q)43 b Fo(\()p Fn(s)p Fo(\))23292 63637 y Ff(j)p Fm(k)24 b Ff(j)24385 64138 y Fn(e)24949 63637 y 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Fo(\))p Fl(j)8590 26372 y Fu(2)9116 26872 y Fl(j)p Fn(n)p Fl(jk)p Fn(q)g Fl(k)12314 26372 y Ff(j)p Fm(n)p Ff(j)12314 27172 y(1)14570 26872 y Fl(\024)16631 25721 y Fi(X)16620 28346 y Fm(n)p Ff(2)p Fu(Z)18595 26872 y Fl(j)e Fo(^)-647 b Fn(g)43 b Fo(\()p Fn(n)p Fo(\))p Fl(j)21560 26372 y Fu(2)22086 26872 y Fl(j)p Fn(n)p Fl(jk)p Fn(q)g Fl(k)25284 26372 y Ff(j)p Fm(n)p Ff(j)25284 27172 y(1)26433 26872 y Fn(;)14570 30311 y Fl(\024)1107 b Fo(\()17102 29159 y Fi(X)17091 31785 y Fm(n)p Ff(2)p Fu(Z)18864 30311 y Fo(\(1)270 b(+)f Fn(n)22151 29810 y Fu(2)22676 30311 y Fo(\))23147 29810 y Fu(1)p Fm(=)p Fu(2)24615 30311 y Fl(j)41 b Fo(^)-647 b Fn(g)43 b Fo(\()p Fn(n)p Fo(\))p Fl(j)27580 29810 y Fu(2)28106 30311 y Fo(\))28577 29810 y Fu(1)p Fm(=)p Fu(2)30044 30311 y Fo(\()30526 29159 y Fi(X)30515 31785 y Fm(n)p Ff(2)p Fu(Z)32288 30311 y Fo(\(1)270 b(+)f Fn(n)35575 29810 y Fu(2)36101 30311 y Fo(\))36572 29810 y Ff(\000)p Fu(1)p Fm(=)p Fu(2)38771 30311 y Fl(j)p Fn(n)p Fl(j)40173 29810 y Fu(2)40698 30311 y Fl(k)p Fn(q)43 b Fl(k)42494 29810 y Fu(2)p Ff(j)p Fm(n)p Ff(j)42494 30610 y(1)44114 30311 y Fo(\))44585 29810 y Fu(1)p Fm(=)p Fu(2)46052 30311 y Fn(;)14570 33677 y Fl(\024)1107 b Fo(\()17102 32526 y Fi(X)17091 35151 y Fm(n)p Ff(2)p Fu(Z)18864 33677 y Fo(\(1)270 b(+)f Fn(n)22151 33177 y Fu(2)22676 33677 y Fo(\))23147 33177 y Fu(1)p Fm(=)p Fu(2)24615 33677 y Fl(j)41 b Fo(^)-647 b Fn(g)43 b Fo(\()p Fn(n)p Fo(\))p Fl(j)27580 33177 y Fu(2)28106 33677 y Fo(\))28577 33177 y Fu(1)p Fm(=)p Fu(2)30044 33677 y Fo(\()30526 32526 y Fi(X)30515 35151 y Fm(n)p Ff(2)p Fu(Z)32288 33677 y Fo(\(1)270 b(+)f Fn(n)35575 33177 y Fu(2)36101 33677 y Fo(\))36572 33177 y Fu(1)p Ff(\000)p Fu(1)p Fm(=)p Fu(2)39241 33677 y Fl(k)p Fn(q)43 b Fl(k)41037 33177 y Fu(2)p Ff(j)p Fm(n)p Ff(j)41037 33977 y(1)42657 33677 y Fo(\))43128 33177 y Fu(1)p Fm(=)p Fu(2)44596 33677 y Fn(;)14570 37044 y Fl(\024)1107 b Fo(\()p Fl(k)p Fn(g)43 b Fl(k)18924 36543 y Fu(2)18924 37497 y Fm(H)19765 37235 y Fh(1)p Fk(=)p Fh(2)21057 37497 y Fu(\()p Fe(T)p Fu(\))22435 37044 y Fo(\)\()23388 35892 y Fi(X)23377 38517 y Fm(n)p Ff(2)p Fu(Z)25151 37044 y Fo(\(1)270 b(+)f Fn(n)28438 36543 y Fu(2)28963 37044 y Fo(\))29434 36543 y Fu(1)p Fm(=)p Fu(2)30901 37044 y Fl(k)p Fn(q)43 b Fl(k)32697 36543 y Fu(2)p Ff(j)p Fm(n)p Ff(j)32697 37343 y(1)34317 37044 y Fo(\))34788 36543 y Fu(1)p Fm(=)p Fu(2)36256 37044 y Fn(;)14570 39791 y(<)1107 b Fl(1)p Fn(;)-189 42536 y Fp(imply)270 b(the)g(unif)-36 b(or)30 b(m)269 b(con)-24 b(v)-30 b(ergence)269 b(of)h(the)g(ser)18 b(ies)270 b(\(3.9\))f(on)h Fg(T)p Fn(:)g Fp(Theref)-36 b(ore)269 b(w)-12 b(e)270 b(conclude)g(that)g Fn(g)44206 42096 y Ff(\003)45069 42536 y Fl(2)337 b(C)71 b Fo(\()p Fg(T)p Fo(\))337 b Fl(\032)-189 44358 y Fn(H)917 43918 y Ff(\000)p Fu(1)2175 44358 y Fo(\()p Fg(T)p Fo(\))p Fn(:)g Fp(Moreo)-18 b(v)-30 b(er)335 b(\(3.7\))h(implies)h(that)12352 47103 y Fl(h)p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])p Fn(h;)202 b(g)43 b Fl(i)18395 47445 y Fm(H)19236 47184 y Fc(\000)p Fh(1)p Fk(=)p Fh(2)21169 47445 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)23593 47184 y Fh(1)p Fk(=)p Fh(2)24885 47445 y Fu(\()p Fe(T)p Fu(\))26599 47103 y Fo(=)337 b Fl(h)p Fn(h;)202 b(g)30208 46603 y Ff(\003)30735 47103 y Fl(i)31206 47366 y Fm(H)32047 47114 y Fh(1)32508 47366 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)34932 47114 y Fc(\000)p Fh(1)36034 47366 y Fu(\()p Fe(T)p Fu(\))-189 49848 y Fp(and)337 b(consequently)8670 53012 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)43 b Fo(])12384 52512 y Ff(\003)12910 53012 y Fn(g)g Fo(\)\()p Fn(s)p Fo(\))339 b(=)d Fn(q)43 b Fo(\()p Fn(s)p Fo(\))19224 52512 y Ff(j)p Fm(k)24 b Ff(j)20317 53012 y Fn(e)20881 52512 y Fm(i)158 b(k)24 b(s)22863 52192 y Fo(1)p 22496 52733 V 22496 53844 a(2)p Fn(\031)24171 51362 y Fi(Z)25383 51713 y Fu(2)p Fm(\031)24844 54111 y Fu(0)26932 51861 y Fi(X)26683 54491 y Fm(n)p Ff(2)p Fe(Z)28471 54614 y Fh(0)29135 53012 y Fl(j)p Fn(n)p Fl(j)p Fn(q)43 b Fo(\()p Fn(s)p Fo(\))32631 52512 y Ff(j)p Fm(n)p Ff(j)33780 53012 y Fn(e)34344 52512 y Fm(i)158 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))37687 53012 y Fn(g)43 b Fo(\()p Fn(t)p Fo(\))p Fn(dt:)5693 b Fp(\(3.10\))p 49073 56588 45 819 v 49118 55814 728 45 v 49118 56588 V 49845 56588 45 819 v -189 60467 a Fv(Singular)368 b(system)g(of)h Fb(D)36 b(F)181 b Fa([)p Fb(q)15142 60666 y Fu(0)15667 60467 y Fa(])370 b Fv(f)-27 b(or)369 b(cir)-27 b(cular)368 b(suppor)27 b(t)367 b(inc)-27 b(lusion.)-189 63240 y Fp(Let)337 b Fn(q)2374 63422 y Fu(0)2900 63240 y Fn(;)202 b Fo(0)336 b Fn(<)h(q)6202 63422 y Fu(0)7064 63240 y Fn(<)g Fo(1)p Fn(;)f Fp(be)h(a)g(constant)g(r)-12 b(adius)336 b(of)h(suppor)48 b(t)337 b(inclusion.)-189 65061 y(Denote)g Fn(\017)4548 65243 y Fm(j)5035 65061 y Fo(\()p Fn(s)p Fo(\))g(=)g Fn(e)8726 64621 y Fm(i)157 b(j)51 b(s)10125 65061 y Fn(;)337 b Fp(w)-12 b(e)337 b(ha)-24 b(v)-30 b(e)8435 68320 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)11769 68502 y Fu(0)12295 68320 y Fo(])p Fn(\017)13124 68502 y Fm(j)13611 68320 y Fo(\)\()p Fn(t)p Fo(\))337 b(=)17328 67168 y Fi(X)17079 69798 y Fm(n)p Ff(2)p Fe(Z)18867 69921 y Fh(0)21996 67500 y Fl(j)p Fn(n)p Fl(j)p 19664 68041 6066 49 v 19664 69202 a Fo(2)p Fn(\031)43 b Fo(1)21610 68850 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)p Fu(+1)25862 68320 y Fn(q)26446 67681 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)g Ff(j)26403 68667 y Fu(0)29364 68320 y Fn(e)29928 67819 y Fm(i)158 b(nt)31574 66670 y Fi(Z)32786 67020 y Fu(2)p Fm(\031)32247 69419 y Fu(0)34086 68320 y Fn(e)34650 67819 y Fm(i)f(j)51 b(s)36049 68320 y Fn(e)36613 67819 y Fm(i)158 b Fu(\()p Fm(k)24 b Ff(\000)p Fm(n)p Fu(\))p Fm(s)40129 68320 y Fn(ds)5459 b Fp(\(3.11\))-189 71986 y(Under)337 b(comple)-36 b(x)336 b(unit)h(monomial)f(basis)h(in)g Fg(T)g Fp(the)g(entr)36 b(y)337 b(of)f Fn(D)34 b(F)168 b Fo([)p Fn(q)31034 72168 y Fu(0)31560 71986 y Fo(])337 b Fp(can)g(be)f(e)-36 b(xpressed)336 b(as)11004 74731 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)14338 74913 y Fu(0)14864 74731 y Fo(]\))15672 74913 y Fm(j)51 b(n)17837 74731 y Fo(=)1106 b Fl(h)p Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)23691 74913 y Fu(0)24218 74731 y Fo(])p Fn(\017)25047 74913 y Fm(j)25533 74731 y Fo(\)\()p Fn(t)p Fo(\))p Fn(;)202 b(\017)28415 74913 y Fm(n)29042 74731 y Fo(\()p Fn(t)p Fo(\))p Fl(i)30893 75073 y Fm(H)31734 74812 y Fc(\000)p Fh(1)p Fk(=)p Fh(2)33666 75073 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)36090 74812 y Fh(1)p Fk(=)p Fh(2)37382 75073 y Fu(\()p Fe(T)p Fu(\))17837 77041 y Fo(=)1106 b Fl(j)p Fn(n)p Fl(j)p Fn(q)21872 76402 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)21829 77388 y Fu(0)24789 77041 y Fn(\016)25374 76540 y Fm(j)51 b Fu(+)p Fm(k)25328 77340 y(n)p eop end %%Page: 11 11 TeXDict begin 11 10 bop -189 715 a Fp(f)-36 b(or)391 b Fn(n)429 b Fl(6)p Fo(=)f(0)p Fn(;)202 b(n;)g(j)497 b Fl(2)429 b Fo(Z)p Fn(;)392 b Fp(and)g Fn(\016)437 b Fp(denoting)392 b(delta)g(Dir)-12 b(ac)391 b(notation.)582 b(W)-36 b(e)392 b(obser)36 b(v)-30 b(e)391 b(here)h(w)-12 b(e)392 b(ha)-24 b(v)-30 b(e)391 b(an)h(in\002nite)-189 2537 y(matr)18 b(ix)337 b(representation)e(of)i Fn(D)34 b(F)168 b Fo([)p Fn(q)15761 2719 y Fu(0)16287 2537 y Fo(])p Fn(;)336 b Fp(which)h(is)g(a)g(non-z)-18 b(ero)336 b(upper-lo)-18 b(w)-12 b(er)335 b(k-diagonal)i(or)f(k-banded.)-189 4358 y(Let)507 b(us)h(mak)-24 b(e)506 b(some)h(notes)g(related)g(to)g (the)h(in\002nite)f(matr)18 b(ix)507 b(representation)f(of)h Fn(D)34 b(F)168 b Fo([)p Fn(q)42575 4540 y Fu(0)43101 4358 y Fo(])507 b Fp(induced)g(b)-24 b(y)-189 6180 y(har)30 b(monic)336 b(ref)-36 b(erence)336 b(potential)h Fn(u)16196 6362 y Fu(0)16721 6180 y Fn(:)-189 8500 y Fj(Note)408 b Fp(3.5)p Fj(.)578 b Fp(Under)382 b('har)30 b(monic)382 b(ref)-36 b(erence)380 b(potential')i Fn(u)26120 8682 y Fu(0)27028 8500 y Fp(of)g(the)g(f)-36 b(or)30 b(m)381 b(of)h(a)g(\002nite)h(linear)e(combination)h(of)-189 10321 y Fj(elementar)36 b(y)336 b(har)30 b(monic)337 b(ref)-36 b(erence)335 b(potential)10323 13360 y Fn(u)11017 13542 y Fu(0)12650 13360 y Fo(:=)15159 12209 y Fi(X)15036 14844 y Fm(k)24 b Ff(2)p Fm(K)17235 13360 y Fn(\013)18010 13557 y Fm(k)18580 13360 y Fn(U)19408 13557 y Fm(k)19976 13360 y Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))15036 16737 y(=)16439 15585 y Fi(X)16316 18220 y Fm(k)24 b Ff(2)p Fm(K)18515 16737 y Fn(\013)19290 16934 y Fm(k)19859 16737 y Fo(\()p Fn(r)34 b Fo(\))21382 16236 y Fm(k)21951 16737 y Fn(e)22515 16236 y Fm(i)158 b(k)24 b(s)23997 16737 y Fn(;)202 b(K)423 b Fo(=)337 b Fl(f)p Fn(k)28505 16919 y Fm(i)29217 16737 y Fl(2)g Fg(Z)31170 16919 y Fu(0)31696 16737 y Fn(;)202 b(i)336 b Fo(=)g(1)p Fn(;)202 b Fl(\001)g(\001)g(\001) 403 b Fn(;)202 b(n)p Fl(g)p Fn(;)-189 20611 y Fp(which)337 b(is)g(the)g(solution)g(of)g(boundar)36 b(y)336 b(v)-30 b(alue)337 b(prob)-24 b(lem)336 b(\(1.3\))g(under)g(boundar)36 b(y)337 b(v)-30 b(alue)13247 23651 y Fn(g)380 b Fo(=)15608 22499 y Fi(X)15485 25134 y Fm(k)24 b Ff(2)p Fm(K)17684 23651 y Fn(\013)18459 23848 y Fm(k)19028 23651 y Fn(e)19592 23150 y Fm(i)157 b(k)24 b(s)21073 23651 y Fn(;)202 b(K)424 b Fo(=)336 b Fl(f)p Fn(k)25581 23833 y Fm(i)26294 23651 y Fl(2)h Fg(Z)28247 23833 y Fu(0)28772 23651 y Fn(;)202 b(i)336 b Fo(=)h(1)p Fn(;)202 b Fl(\001)g(\001)g(\001)403 b Fn(;)202 b(n)p Fl(g)p Fn(:)-189 27549 y Fp(W)-36 b(e)257 b(obser)36 b(v)-30 b(e)256 b(that)h(the)g(in\002nite)g(matr)18 b(ix)256 b Fn(D)34 b(F)168 b Fo([)p Fn(q)20842 27731 y Fu(0)21368 27549 y Fo(])257 b Fp(is)g(a)g(non-z)-18 b(ero)255 b(banded)i(matr)18 b(ix)256 b(with)i(bandwidth)e Fl(j)202 b Fo(min)p Fl(f)p Fn(k)48911 27731 y Fm(i)49287 27549 y Fo(;)g Fn(k)50457 27731 y Fm(i)51169 27549 y Fl(2)-189 29371 y Fn(K)87 b Fl(gj)269 b Fo(+)g(max)p Fl(f)p Fn(k)6844 29553 y Fm(i)7220 29371 y Fo(;)202 b Fn(k)8390 29553 y Fm(i)9102 29371 y Fl(2)337 b Fn(K)87 b Fl(g)p Fp(.)-189 31192 y(In)370 b(the)h(case)f(of)h('har)30 b(monic)370 b(ref)-36 b(erence)369 b(potential')h(is)h Fn(u)25829 31374 y Fu(0)26748 31192 y Fo(:=)392 b Fn(U)29380 30752 y Fm(c)29248 31561 y(k)29843 31192 y Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))394 b(=)e(\()p Fn(r)35186 30752 y Fm(k)35957 31192 y Fo(cos)202 b Fn(k)38 b(s)p Fo(\))p Fn(=k)g(;)202 b(k)432 b Fl(2)393 b Fg(Z)44376 31374 y Fu(0)44902 31192 y Fn(;)370 b Fp(which)h(is)-189 33014 y(the)357 b(solution)h(of)f(boundar)36 b(y)358 b(v)-30 b(alue)357 b(prob)-24 b(lem)357 b(\(1.3\))f(under)h(cosine)h(boundar)36 b(y)357 b(v)-30 b(alue)357 b Fn(g)415 b Fo(=)371 b(\(cos)202 b Fn(k)38 b(s)q Fo(\))p Fn(=k)g(;)202 b(k)410 b Fl(2)-189 34835 y Fg(Z)619 35017 y Fu(0)1145 34835 y Fn(:)317 b Fp(W)-36 b(e)317 b(obser)36 b(v)-30 b(e)316 b(that)h(the)g(in\002nite)g (matr)18 b(ix)317 b Fn(D)34 b(F)168 b Fo([)p Fn(q)23191 35017 y Fu(0)23717 34835 y Fo(])316 b Fp(under)h(tr)18 b(igonometr)g(ical)316 b(polynomial)h(basis)g(is)g(a)g(non-)-189 36657 y(z)-18 b(ero)336 b(symmetr)18 b(ic)337 b(k-bidiagonal.)-189 38479 y(While)449 b(in)g(the)f(case)h(of)g('har)30 b(monic)448 b(ref)-36 b(erence)447 b(potential')i(is)g Fn(u)29866 38661 y Fu(0)30915 38479 y Fo(:=)523 b Fn(U)33678 38039 y Fm(s)33546 38848 y(k)34169 38479 y Fo(\()p Fn(r)-34 b(;)202 b(s)p Fo(\))524 b(=)g(\()p Fn(r)39774 38039 y Fm(k)40544 38479 y Fo(sin)203 b Fn(k)38 b(s)p Fo(\))p Fn(=k)g(;)202 b(k)562 b Fl(2)524 b Fg(Z)49090 38661 y Fu(0)49616 38479 y Fn(;)-189 40300 y Fp(which)309 b(is)f(the)h (solution)f(of)g(boundar)36 b(y)309 b(v)-30 b(alue)308 b(prob)-24 b(lem)307 b(\(1.3\))h(under)f(boundar)36 b(y)309 b(v)-30 b(alue)308 b Fn(g)380 b Fo(=)336 b(\(sin)203 b Fn(k)38 b(s)p Fo(\))p Fn(=k)g(;)202 b(k)376 b Fl(2)-189 42122 y Fg(Z)619 42304 y Fu(0)1145 42122 y Fn(:)389 b Fp(W)-36 b(e)390 b(obtain)f(that)h(the)f(in\002nite)h(matr)18 b(ix)389 b Fn(D)34 b(F)168 b Fo([)p Fn(q)22683 42304 y Fu(0)23209 42122 y Fo(])389 b Fp(under)g(tr)18 b(igonometr)g(ical)389 b(polynomial)g(basis)h(is)g(a)f(non-)-189 43943 y(z)-18 b(ero)336 b(sk)-24 b(e)g(w-symmetr)18 b(ic)336 b(k-bidiagonal.)-189 46263 y Fj(Singular)h(system)g(of)g Fn(D)34 b(F)168 b Fo([)p Fn(q)12981 46445 y Fu(0)13506 46263 y Fo(])p Fn(:)-189 48085 y Fp(Let)337 b Fn(\017)2325 47645 y Fm(s)2325 48424 y(j)2815 48085 y Fo(\()p Fn(t)p Fo(\))g(=)g(\(1)270 b(+)f Fn(j)8939 47645 y Fu(2)9465 48085 y Fo(\))9936 47645 y Ff(\000)p Fm(s=)p Fu(2)12100 48085 y Fn(e)12664 47645 y Fm(i)157 b(j)51 b(t)13968 48085 y Fn(;)337 b Fp(then)10189 51124 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)13523 51306 y Fu(0)14049 51124 y Fo(])p Fn(\017)14878 50624 y Fm(s)14878 51424 y(j)15368 51124 y Fo(\)\()p Fn(t)p Fo(\))1108 b(=)e(\(1)270 b(+)f Fn(j)23503 50624 y Fu(2)24030 51124 y Fo(\))24501 50624 y Ff(\000)p Fm(s=)p Fu(2)26664 51124 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)29998 51306 y Fu(0)30524 51124 y Fo(])p Fn(\017)31353 51306 y Fm(j)31840 51124 y Fo(\)\()p Fn(t)p Fo(\))p Fn(;)18327 53278 y Fo(=)1106 b(\(1)270 b(+)f Fn(j)23503 52777 y Fu(2)24030 53278 y Fo(\))24501 52777 y Ff(\000)p Fm(s=)p Fu(2)27116 52126 y Fi(X)26866 54757 y Fm(n)p Ff(2)p Fe(Z)28654 54880 y Fh(0)29318 53278 y Fl(j)p Fn(n)p Fl(j)p Fn(q)31304 52639 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)31261 53625 y Fu(0)34221 53278 y Fn(\016)34806 52777 y Fm(j)51 b Fu(+)p Fm(k)34760 53578 y(n)36538 53278 y Fn(\017)37030 53460 y Fm(n)37656 53278 y Fo(\()p Fn(t)p Fo(\))p Fn(;)18327 56747 y Fo(=)1106 b(\(1)270 b(+)f Fn(j)23503 56246 y Fu(2)24030 56747 y Fo(\))24501 56246 y Ff(\000)p Fm(s=)p Fu(2)26664 56747 y Fl(j)p Fn(j)69 b Fl(j)p Fn(q)28490 56108 y Ff(j)p Fm(j)51 b Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j)28447 57094 y Fu(0)31269 56747 y Fn(\017)31761 56944 y Fm(j)51 b Fu(+)p Fm(k)33493 56747 y Fo(\()p Fn(t)p Fo(\))p Fn(:)-189 61940 y Fp(Denote)17668 63761 y Fn(\033)18361 63943 y Fm(j)19184 63761 y Fo(=)538 b Fn(q)21249 63122 y Ff(j)p Fm(k)24 b Ff(j)p Fu(+)p Ff(j)p Fm(j)51 b Ff(j)21206 64108 y Fu(0)24230 63761 y Fl(j)p Fn(j)69 b Fl(j)p Fo(\(1)269 b(+)h Fn(j)28599 63261 y Fu(2)29125 63761 y Fo(\))29596 63261 y Ff(\000)p Fm(s=)p Fu(2)31759 63761 y Fn(;)-189 66302 y Fp(then)337 b(from)f(\(3.6\))g(and)h(Theorem)e(\(3.3\),)h(w)-12 b(e)337 b(can)g(sho)-18 b(w)337 b(:)18105 69342 y Fn(D)34 b(F)168 b Fo([)p Fn(q)20968 69524 y Fu(0)21494 69342 y Fo(])21831 68841 y Ff(\003)22356 69342 y Fn(\017)22848 69539 y Fm(j)51 b Fu(+)p Fm(k)24581 69342 y Fo(\()p Fn(t)p Fo(\))337 b(=)f Fn(\033)28270 69524 y Fm(j)28959 69342 y Fn(\017)29451 68841 y Fm(s)29451 69641 y(j)29941 69342 y Fo(\()p Fn(t)p Fo(\))p Fn(:)-189 72381 y Fp(Then)273 b(w)-12 b(e)274 b(demonstr)-12 b(ate)273 b(that)g(the)h(tr)18 b(iple)273 b Fl(f)p Fn(\017)19927 71941 y Fm(s)19927 72720 y(j)20418 72381 y Fo(\()p Fn(t)p Fo(\))p Fn(;)202 b(\017)22829 72578 y Fm(j)51 b Fu(+)p Fm(k)24561 72381 y Fo(\()p Fn(t)p Fo(\))p Fn(;)202 b(\033)27173 72563 y Fm(j)27660 72381 y Fl(g)274 b Fp(is)g(singular)f(system)h(of)g Fn(D)34 b(F)168 b Fo([)p Fn(q)42467 72563 y Fu(0)42992 72381 y Fo(])p Fn(;)274 b Fp(the)f(F)-55 b(r)47123 72359 y(\264)46988 72381 y(echet)-189 74203 y(der)18 b(iv)-30 b(ativ)g(e)336 b(of)h Fn(F)506 b Fp(at)337 b(positiv)-30 b(e)336 b(constant)h Fn(q)19148 74385 y Fu(0)19674 74203 y Fn(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop -189 715 a Fd(Theorem)335 b(3.6.)553 b Fj(Assume)337 b(that)g Fo(0)g Fn(<)f(q)17679 897 y Fu(0)18205 715 y Fo(\()p Fn(t)p Fo(\))h(=)g Fn(q)21743 897 y Fu(0)22605 715 y Fn(<)g Fo(1)p Fn(;)f Fj(a)h(positiv)-30 b(e)337 b(constant.)417 b(Then)336 b(the)h(oper)-12 b(ator)14463 3397 y Fn(f)15056 3579 y Fm(p)15586 3397 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)18920 3579 y Fu(0)19446 3397 y Fo(])19783 2896 y Ff(\003)20308 3397 y Fn(D)34 b(F)168 b Fo([)p Fn(q)23171 3579 y Fu(0)23697 3397 y Fo(]\))337 b(:)f Fn(H)26621 2896 y Fm(s)27111 3397 y Fo(\()p Fg(T)p Fo(\))i Fl(!)f Fn(H)31854 2896 y Fm(s)p Fu(+)p Fm(p)33550 3397 y Fo(\()p Fg(T)p Fo(\))-189 6079 y Fj(is)g(bounded)g(and)f (boundedly)h(in)-24 b(v)-30 b(er)48 b(tib)-24 b(le)-18 b(.)-189 8599 y(Proof.)606 b Fp(By)337 b(the)g(help)g(of)g(the)g (singular)f(system)h(of)g Fn(D)34 b(F)168 b Fo([)p Fn(q)26389 8781 y Fu(0)26914 8599 y Fo(])p Fn(;)337 b Fp(w)-12 b(e)337 b(ha)-24 b(v)-30 b(e)16239 11281 y Fn(f)16832 11463 y Fm(p)17361 11281 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)20695 11463 y Fu(0)21221 11281 y Fo(])21558 10780 y Ff(\003)22084 11281 y Fn(D)34 b(F)168 b Fo([)p Fn(q)24947 11463 y Fu(0)25472 11281 y Fo(]\))p Fn(\017)26772 10780 y Fm(s)26772 11580 y(j)27599 11281 y Fo(=)337 b Fn(f)29472 11463 y Fm(p)30001 11281 y Fo(\()p Fn(\033)31208 10780 y Fu(2)31165 11580 y Fm(j)31735 11281 y Fo(\))p Fn(\017)32698 10780 y Fm(s)32698 11580 y(j)33188 11281 y Fn(:)-189 13963 y Fp(Choosing)g Fn(\025)5975 14145 y Fu(0)6838 13963 y Fo(=)f Fl(k)p Fn(D)34 b(F)168 b Fo([)p Fn(q)11586 14145 y Fu(0)12112 13963 y Fo(])p Fl(k)336 b Fp(in)h(the)g(de\002nition)g(of)g Fn(f)23822 14145 y Fm(p)24351 13963 y Fn(;)g Fp(w)-12 b(e)337 b(obtain)7455 16645 y Fn(f)8048 16827 y Fm(p)8577 16645 y Fo(\()p Fn(\033)9784 16144 y Fu(2)9741 16944 y Fm(j)10310 16645 y Fo(\))g(=)g(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)21757 16833 y Fm(R)22795 16645 y Fo(+)g Fn(s)202 b Fo(ln)q(\(1)270 b(+)f Fn(j)28915 16144 y Fu(2)29441 16645 y Fo(\))h Fl(\000)f Fo(ln)202 b Fl(j)p Fn(j)69 b Fl(j)p Fn(;)2626 b(R)346 b Fo(:=)336 b(1)p Fn(=q)41446 16827 y Fu(0)41973 16645 y Fn(:)-189 19327 y Fp(Since)410 22009 y Fo(\()p Fl(j)p Fn(j)69 b Fl(j)269 b Fo(+)h Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)9769 22197 y Fm(R)10807 22009 y Fo(+)g Fn(s)202 b Fo(ln)q(\(1)270 b(+)f Fn(j)16927 21508 y Fu(2)17453 22009 y Fo(\))h Fl(\000)19539 21189 y Fo(1)p 19539 21730 607 49 v 19539 22840 a(2)20480 22009 y(ln\(1)f(+)h Fn(j)24617 21508 y Fu(2)25143 22009 y Fo(\))p Fl(j)337 b Fn(<)f Fo(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)36926 22197 y Fm(R)37965 22009 y Fo(+)g Fn(s)202 b Fo(ln\(1)270 b(+)f Fn(j)44084 21508 y Fu(2)44610 22009 y Fo(\))h Fl(\000)f Fo(ln)202 b Fl(j)p Fn(j)69 b Fl(j)p Fn(;)-189 24690 y Fp(then)337 b(there)f(e)-36 b(xists)337 b Fn(c)f(>)h Fo(0)g Fp(a)f(constant,)h(and)g(f)-36 b(or)336 b(all)h Fn(j)406 b Fl(2)337 b Fg(Z)10250 27372 y Fn(c)p Fo(\()11246 26237 y Fi(p)p 12458 26237 3182 49 v 1135 x Fo(1)269 b(+)g Fn(j)15113 27022 y Fu(2)15640 27372 y Fo(\))337 b Fl(\024)f Fo(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)27086 27560 y Fm(R)28125 27372 y Fo(+)g Fn(s)202 b Fo(ln\(1)270 b(+)f Fn(j)34244 26872 y Fu(2)34770 27372 y Fo(\))h Fl(\000)f Fo(ln)202 b Fl(j)p Fn(j)69 b Fl(j)p Fn(;)-189 30054 y Fp(theref)-36 b(ore)18186 31876 y Fn(f)18779 32058 y Fm(p)19308 31876 y Fo(\()p Fn(\033)20515 31375 y Fu(2)20472 32176 y Fm(j)21042 31876 y Fo(\))337 b Fl(\024)f Fn(c)23654 31375 y Ff(\000)p Fm(p)24915 31876 y Fo(\(1)269 b(+)h Fn(j)28042 31375 y Fu(2)28568 31876 y Fo(\))29039 31375 y Ff(\000)p Fm(p=)p Fu(2)31241 31876 y Fn(:)-189 34238 y Fp(This)337 b(leads)g(to)13757 36060 y Fl(k)p Fn(f)14956 36242 y Fm(p)15485 36060 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)18819 36242 y Fu(0)19345 36060 y Fo(])19682 35560 y Ff(\003)20208 36060 y Fn(D)34 b(F)168 b Fo([)p Fn(q)23071 36242 y Fu(0)23597 36060 y Fo(]\))p Fl(k)25011 36319 y Fm(H)25852 36067 y Fk(s)26295 36319 y Fu(\()p Fe(T)p Fu(\))p Ff(!)p Fm(H)29399 36067 y Fk(s)p Fh(+)p Fk(p)30891 36319 y Fu(\()p Fe(T)p Fu(\))32605 36060 y Fl(\024)337 b Fn(c)34410 35560 y Ff(\000)p Fm(p)35670 36060 y Fn(:)-189 38423 y Fp(Using)g(the)g(f)-36 b(act)336 b Fl(\000)202 b Fo(ln)g Fl(j)p Fn(j)69 b Fl(j)337 b Fn(<)f Fo(0)h Fp(f)-36 b(or)337 b(all)g Fn(j)406 b Fl(2)337 b Fg(Z)19803 38605 y Fu(0)20329 38423 y Fp(,)f(and)h(use)g(of)g(the)g (f)-36 b(ollo)-18 b(wing)336 b(inequality:)4423 41104 y Fo(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)13782 41292 y Fm(R)14821 41104 y Fo(+)g Fn(s)202 b Fo(ln\(1)270 b(+)f Fn(j)20940 40604 y Fu(2)21466 41104 y Fo(\))h Fl(\000)f Fo(ln)202 b Fl(j)p Fn(j)69 b Fl(j)337 b Fn(<)f Fo(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)36848 41292 y Fm(R)37887 41104 y Fo(+)g Fn(s)202 b Fo(ln\(1)270 b(+)f Fn(j)44006 40604 y Fu(2)44533 41104 y Fo(\))p Fn(;)-189 43786 y Fp(then)337 b(there)f(e)-36 b(xists)337 b Fn(C)424 b(>)336 b Fo(0)h Fp(a)g(constant,)g(and)f(f)-36 b(or)336 b(all)i Fn(j)406 b Fl(2)337 b Fg(Z)27265 43968 y Fu(0)12003 46468 y Fo(\()p Fl(j)p Fn(j)69 b Fl(j)270 b Fo(+)f Fl(j)p Fn(k)38 b Fl(j)p Fo(\))202 b(ln)g Fn(R)279 b Fo(+)269 b Fn(c)21362 46656 y Fm(R)22401 46468 y Fo(+)g Fn(s)202 b Fo(ln\(1)270 b(+)f Fn(j)28520 45968 y Fu(2)29046 46468 y Fo(\))338 b Fl(\024)e Fn(C)87 b Fo(\()32558 45333 y Fi(p)p 33771 45333 V 33771 46468 a Fo(1)269 b(+)g Fn(j)36426 46118 y Fu(2)36952 46468 y Fo(\))p Fn(;)-189 49150 y Fp(theref)-36 b(ore)17972 50972 y Fn(C)18925 50471 y Ff(\000)p Fm(p)20186 50972 y Fo(\(1)269 b(+)h Fn(j)23313 50471 y Fu(2)23839 50972 y Fo(\))24310 50471 y Ff(\000)p Fm(p=)p Fu(2)26849 50972 y Fl(\024)336 b Fn(f)28721 51154 y Fm(p)29251 50972 y Fo(\()p Fn(\033)30458 50471 y Fu(2)30415 51272 y Fm(j)30984 50972 y Fo(\))p Fn(:)-189 53334 y Fp(So)h(w)-12 b(e)337 b(conclude)12914 55156 y Fl(k)p Fn(f)14113 55338 y Fm(p)14642 55156 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)17976 55338 y Fu(0)18502 55156 y Fo(])18839 54655 y Ff(\003)19365 55156 y Fn(D)34 b(F)168 b Fo([)p Fn(q)22228 55338 y Fu(0)22754 55156 y Fo(]\))23562 54655 y Ff(\000)p Fu(1)24819 55156 y Fl(k)25425 55415 y Fm(H)26266 55163 y Fk(s)p Fh(+)p Fk(p)27758 55415 y Fu(\()p Fe(T)p Fu(\))p Ff(!)p Fm(H)30862 55163 y Fk(s)31306 55415 y Fu(\()p Fe(T)p Fu(\))33020 55156 y Fl(\024)336 b Fn(C)35252 54655 y Ff(\000)p Fm(p)36513 55156 y Fn(:)-189 57518 y Fp(Hence)h Fn(f)4244 57700 y Fm(p)4773 57518 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)8107 57700 y Fu(0)8633 57518 y Fo(])8970 57079 y Ff(\003)9496 57518 y Fn(D)34 b(F)168 b Fo([)p Fn(q)12359 57700 y Fu(0)12885 57518 y Fo(]\))337 b Fp(is)g(boundedly)f(in)-24 b(v)-30 b(er)48 b(tib)-24 b(le)-18 b(.)p 49073 57518 45 819 v 49118 56745 728 45 v 49118 57518 V 49845 57518 45 819 v -189 60277 a Fj(Remar)18 b(k)613 b Fp(3.1)p Fj(.)638 b Fp(The)492 b(abo)-18 b(v)-30 b(e)491 b(result,)530 b(highlights)492 b(the)g(deg)-12 b(ree)491 b(of)h(ill-posedness)g(of)g (the)f(prob)-24 b(lem.)881 b(The)-189 62098 y(condition)488 b Fn(q)5691 62280 y Fu(0)6588 62098 y Fl(\000)371 b Fn(q)8486 61659 y Fm(])9498 62098 y Fo(=)590 b Fn(f)11624 62280 y Fm(p)12153 62098 y Fo(\()p Fn(D)34 b(F)168 b Fo([)p Fn(q)15487 62280 y Fu(0)16013 62098 y Fo(])16350 61659 y Ff(\003)16876 62098 y Fn(D)34 b(F)168 b Fo([)p Fn(q)19739 62280 y Fu(0)20265 62098 y Fo(]\))p Fn(w)521 b Fp(f)-36 b(or)487 b(some)h Fn(w)623 b Fl(2)591 b Fn(H)31776 61659 y Fm(s)32266 62098 y Fo(\()p Fg(T)p Fo(\))489 b Fp(is)g(equiv)-30 b(alent)488 b(to)h(the)f(f)-36 b(act)488 b(that)-189 63920 y Fn(q)352 64102 y Fu(0)1021 63920 y Fl(\000)143 b Fn(q)2691 63480 y Fm(])3450 63920 y Fl(2)336 b Fn(H)5700 63480 y Fm(s)p Fu(+)p Fm(p)7396 63920 y Fo(\()p Fg(T)p Fo(\))p Fn(:)300 b Fp(Moreo)-18 b(v)-30 b(er)-61 b(,)306 b(there)299 b(are)g(constants)g Fn(c;)202 b(C)386 b Fp(such)299 b(that)h Fn(c)p Fl(k)p Fn(w)33 b Fl(k)36172 64167 y Fm(H)37013 63915 y Fk(s)37455 64167 y Fu(\()p Fe(T)p Fu(\))39170 63920 y Fl(\024)336 b(k)p Fn(q)41596 64102 y Fu(0)42265 63920 y Fl(\000)143 b Fn(q)43935 63480 y Fm(])44357 63920 y Fl(k)44963 64179 y Fm(H)45804 63927 y Fk(s)p Fh(+)p Fk(p)47296 64179 y Fu(\()p Fe(T)p Fu(\))49010 63920 y Fl(\024)-189 65742 y Fn(C)87 b Fl(k)p Fn(w)33 b Fl(k)2877 65988 y Fm(H)3718 65736 y Fk(s)4161 65988 y Fu(\()p Fe(T)p Fu(\))-189 67912 y Fj(Remar)18 b(k)474 b Fp(3.2)p Fj(.)562 b Fp(F)-55 b(rom)352 b(the)h(kno)-18 b(wledge)353 b(of)f(the)h (singular)g(system)f(of)h Fn(D)34 b(F)168 b Fo([)p Fn(q)34839 68094 y Fu(0)35365 67912 y Fo(])p Fp(,)357 b(w)-12 b(e)353 b(could)f(obtain)h(the)g(inf)-36 b(or-)-189 69734 y(mation)273 b(of)g Fj(modi\002ed)g(source)g(condition)g Fp(needed)g(f)-36 b(or)273 b(the)g(class)g(of)h(iter)-12 b(ativ)-30 b(e)272 b(method)h(using)g(\002x)-36 b(ed)273 b(F)-55 b(r)47123 69712 y(\264)46988 69734 y(echet)-189 71556 y(der)18 b(iv)-30 b(ativ)g(e)306 b(such)g(as)h(simpli\002ed)f(IRGNM)h(as)g (studied)f(b)-24 b(y)306 b(Mahale)h(&)g(Nair)f(\(Mahale)g(and)g(Nair) -61 b(,)307 b(2009\))f(and)-189 73377 y(Jin)327 b(\(Jin,)f(2010\))g(b) -24 b(y)327 b(f)-36 b(ollo)-18 b(wing)326 b(the)h(argument)e(in)i (\(Hohage)-18 b(,)326 b(2001\))g(in)h(the)f(discussion)h(on)g(source)f (condi-)-189 75199 y(tion)357 b(f)-36 b(or)356 b(the)h(in)-24 b(v)-30 b(erse)357 b(\(constant\))f(source)h(prob)-24 b(lem)356 b(case)-18 b(.)477 b(This)357 b(issue)g(will)h(be)f (addressed)f(in)h(the)g(future)-189 77021 y(w)-12 b(or)18 b(ks)-18 b(.)p eop end %%Page: 13 13 TeXDict begin 13 12 bop -189 715 a Fj(Remar)18 b(k)455 b Fp(3.3)p Fj(.)549 b Fp(In)334 b(the)f(case)h(inclusion)g(of)g(the)g (f)-36 b(or)30 b(m)333 b Fn(a)p Fo(\()p Fn(r)34 b Fo(\))p Fn(;)333 b Fp(that)h(is)g(a)g(kno)-18 b(wn)334 b(r)-12 b(adial)333 b(function,)h(the)g(f)-36 b(orw)-18 b(ard)-189 2537 y(map)337 b(\(2.8\))f(will)h(be)g(of)g(the)f(f)-36 b(or)30 b(m)7173 5976 y Fn(F)168 b Fo(\()p Fn(q)43 b Fo(\)\()p Fn(t)p Fo(\))340 b(=)13673 4825 y Fi(X)12645 7460 y Fm(n)p Ff(6)p Fu(=0)p Fm(;n)p Ff(2)p Fu(Z)16788 5156 y Fl(j)p Fn(n)p Fl(j)p 16788 5697 1401 49 v 16818 6807 a Fo(2)p Fn(\031)18523 4326 y Fi(Z)19736 4677 y Fu(2)p Fm(\031)19197 7075 y Fu(0)21036 5976 y Fn(e)21600 5476 y Fm(i)157 b(n)p Fu(\()p Fm(t)p Ff(\000)p Fm(s)p Fu(\))24942 5976 y Fn(e)25506 5476 y Fm(i)g(k)24 b(s)27189 3903 y Fi( )28149 4326 y(Z)29361 4677 y Fm(q)32 b Fu(\()p Fm(s)p Fu(\))28822 7075 y(0)31237 5976 y Fn(r)31818 5476 y Ff(j)p Fm(n)p Ff(j)p Fu(+)p Ff(j)p Fm(k)24 b Ff(j\000)p Fu(2)35937 5976 y Fn(a)p Fo(\()p Fn(r)34 b Fo(\))p Fn(r)g(dr)39894 3903 y Fi(!)41055 5976 y Fn(ds:)-189 9728 y Fp(Obser)i(v)-30 b(e)499 b(that)f(in)i(our)e(case)h(f)-36 b(or)498 b(starshape)h (inclusion,)539 b(it)499 b(is)h(necessar)36 b(y)498 b(that)h(r)-12 b(adially)499 b(supp)p Fo(\()p Fn(a)p Fo(\()p Fn(r)34 b Fo(\)\))500 b Fp(is)-189 11549 y(an)k(inter)36 b(v)-30 b(al)505 b Fo([0)p Fn(;)202 b(b)p Fo(])616 b Fl(\022)h Fo([0)p Fn(;)202 b Fo(1])p Fn(;)g Fo(0)617 b Fn(<)g(b)g Fl(\024)g Fo(1)p Fn(:)504 b Fp(This)h(condition)f(r)18 b(ules)504 b(out)h(the)f(f)-36 b(ollo)-18 b(wing)504 b(studies)h(on)f(non-)-189 13371 y(uniqueness)337 b(result)f(:)1224 15970 y(1.)606 b(Counter)337 b(e)-36 b(xample)336 b(of)h(constant)f(r) -12 b(adial)337 b(object)g(b)-24 b(y)336 b(Kang)i(&)f(Seo)g(\(Kang)g (and)g(Seo)-48 b(,)337 b(2001\).)1224 18700 y(2.)606 b(Identi\002cation)337 b(of)g(r)-12 b(adial)336 b(function)h(b)-24 b(y)337 b(El)g(Badia)h(&)f(Ha-Duong)g(\(Badia)g(and)f(Ha-Duong,)h (1998\).)-189 21299 y(Assuming)423 b(that)f Fn(a)p Fo(\()p Fn(r)34 b Fo(\))423 b Fp(is)g(bounded)f(o)-18 b(v)-30 b(er)422 b(its)h(suppor)48 b(t)422 b(and)h(with)g(\002nite)f(jump)h (discontin)-12 b(uity)-121 b(,)444 b(the)422 b(results)-189 23121 y(f)-36 b(or)345 b(starshape)h(constant)g(conductivity)g(suppor) 48 b(t)346 b(still)g(v)-30 b(alid)346 b(without)g(essential)g(change)g (in)g(the)g(proof)-36 b(.W)g(e)-189 24942 y(don't)316 b(pose)g(the)g(gener)-12 b(al)315 b(r)-12 b(adial)316 b(case)g(result)g(in)g(this)g(w)-12 b(or)18 b(k,)320 b(as)c(to)g(maintain)g(the)g(simplicity)g(of)g(e)-36 b(xposition)-189 26764 y(in)337 b(mind.)-189 30808 y Fv(4)1328 b(Numerical)368 b(Implementation.)-189 34087 y Fp(W)-36 b(e)534 b(ma)-36 b(y)533 b(f)-36 b(ollo)-18 b(w)534 b(the)f(implementation)g(of)h(either)g(\(Ring,)f(1995\))g(or)h (\(Hohage)-18 b(,)533 b(2001\))g(f)-36 b(or)533 b(starshape)-189 35909 y(suppor)48 b(t)364 b(of)g(in)-24 b(v)-30 b(erse)363 b(source)h(prob)-24 b(lem,)370 b(f)-36 b(or)363 b(n)-12 b(umer)18 b(ical)363 b(implementation)h(to)g(reconstr)18 b(ucts)363 b(the)h(shape)g(of)-189 37731 y(conductivity)449 b(inclusion.)753 b(Rather)448 b(than)h(w)-12 b(or)18 b(king)449 b(in)g(comple)-36 b(x)448 b(ar)18 b(ithmetic)448 b(as)h(in)g(\(Ring,)g(1995\),)476 b(in)449 b(this)-189 39552 y(w)-12 b(or)18 b(k)502 b(w)-12 b(e)503 b(f)-36 b(ollo)-18 b(w)502 b(the)g(n)-12 b(umer)18 b(ical)502 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Fh(0)20817 54081 y Fn(n)p 19270 54622 3821 49 v 19270 55732 a Fo(\()p Fn(n)269 b Fo(+)g Fn(k)38 b Fo(\))23426 53192 y Fi(\022)24318 54901 y Fo(\()24789 53251 y Fi(Z)26002 53602 y Fu(2)p Fm(\031)25463 56000 y Fu(0)27302 54901 y Fn(q)43 b Fo(\()p Fn(s)p Fo(\))29396 54400 y Fm(n)p Fu(+)p Fm(k)31471 54901 y Fo(cos)202 b(\()p Fn(k)308 b Fl(\000)269 b Fn(n)p Fo(\))p Fn(s)337 b(ds)p Fo(\))539 b(cos)202 b Fn(nt)336 b Fl(\000)15567 58937 y Fo(\()16038 57287 y Fi(Z)17250 57637 y Fu(2)p Fm(\031)16712 60036 y Fu(0)18550 58937 y Fn(q)43 b Fo(\()p Fn(s)p Fo(\))20644 58436 y Fm(n)p Fu(+)p Fm(k)22720 58937 y Fo(sin)202 b(\()p Fn(k)308 b Fl(\000)269 b Fn(n)p Fo(\))p Fn(s)337 b(ds)p Fo(\))539 b(sin)202 b Fn(nt)34200 57228 y Fi(\023)35295 58937 y Fn(:)11829 b Fp(\(4.1\))1629 62484 y Fl(\017)606 b Fp(Der)18 b(iv)-30 b(ativ)g(e)337 b(:)6728 65439 y Fo(\()p Fn(D)34 b(F)9184 64939 y Fm(c)9016 65748 y(k)9647 65439 y Fo([)p Fn(q)43 b Fo(])p Fn(h)p Fo(\)\()p Fn(t)p Fo(\))1109 b(=)16809 64619 y(1)p 16745 65160 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Fm(k)2426 34490 y Fn(:)-189 36311 y(P)589 36499 y Fm(N)1486 36311 y Fo(\()p Fn(F)2904 35871 y Ff(0)3215 36311 y Fo([)p Fn(q)4093 36493 y Fu(0)4619 36311 y Fo(])p Fn(h)5654 36508 y Fm(k)6544 36311 y Fo(+)320 b Fn(F)168 b Fo(\()p Fn(q)9766 36508 y Fm(k)10337 36311 y Fo(\))321 b Fl(\000)f Fn(g)13013 35871 y Fm(\016)13519 36311 y Fo(\))414 b Fp(is)g(easily)g(e)-36 b(v)-30 b(aluated)413 b(b)-24 b(y)414 b(tr)18 b(uncating)413 b(the)h(ser)18 b(ies)414 b(in)g(\(4.1\))f(and)g(\(4.2\).)647 b(W)-36 b(e)-189 38133 y(appro)g(ximate)282 b(the)g(integ)-12 b(r)g(als)283 b(in)g(these)f(f)-36 b(or)30 b(m)-12 b(ulas)282 b(using)h(the)g(tr)-12 b(apez)-18 b(oidal)282 b(r)18 b(ule)283 b(with)g Fn(N)39982 38315 y Fm(t)40660 38133 y Fp(g)-12 b(r)18 b(id)283 b(points)-18 b(.)398 b(Since)-189 39954 y(the)337 b Fn(L)2658 40136 y Fu(2)3184 39954 y Fl(\000)p Fp(nor)30 b(m)336 b(of)h(a)f(function)h Fn(f)130 b Fo(\()p Fn(t)p Fo(\))338 b(=)f Fn(a)18557 40136 y Fu(0)19352 39954 y Fo(+)20564 39045 y Fi(P)21843 39396 y Fm(N)21843 40313 y(j)51 b Fu(=1)23734 39954 y Fn(a)24375 40136 y Fm(j)25064 39954 y Fo(cos\()p Fn(j)69 b(t)p Fo(\))270 b(+)f Fn(b)30637 40136 y Fm(j)31327 39954 y Fo(sin\()p Fn(j)69 b(t)p Fo(\))338 b Fp(pro)-18 b(vided)336 b(b)-24 b(y)17507 44055 y Fl(k)p Fn(f)130 b Fl(k)19442 43554 y Fu(2)19442 44355 y Fm(L)20082 44478 y Fh(2)20936 44055 y Fo(:=)336 b Fn(a)23193 43554 y Fu(2)23193 44355 y(0)23988 44055 y Fo(+)25333 43235 y(1)p 25333 43776 607 49 v 25333 44886 a(2)26729 42540 y Fm(N)26274 42903 y Fi(X)26332 45509 y Fm(j)51 b Fu(=1)28227 44055 y Fn(a)28868 43554 y Fu(2)28868 44355 y Fm(j)29662 44055 y Fo(+)270 b Fn(b)31395 43554 y Fu(2)31395 44355 y Fm(j)31921 44055 y Fn(;)-189 48040 y Fp(w)-12 b(e)447 b(obtain)g Fo(\(2)p Fn(N)475 b Fo(+)343 b(1\))447 b Fp(linear)g(equations)f(f)-36 b(or)446 b(the)h Fo(2)p Fn(M)475 b Fo(+)343 b(1)447 b Fp(F)-36 b(our)18 b(ier)445 b(coef\002cients)i(of)g Fn(h)41479 48237 y Fm(k)42495 48040 y Fp(from)f(the)h(ter)30 b(m)-189 49862 y Fl(k)p Fn(P)1195 50050 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currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc 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Operating System: Microsoft Windows XP. %%Title: C:\Documents and Settings\MathIPB\My Documents\ADG_Workfiles\SupportIdentification\Data\ConstantConductivityInclusionPaper\DentCircFig1a.eps %%CreationDate: 11/04/2010 10:33:25 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 96 238 515 553 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 96 238 515 553 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 1152 6636 csm 0 0 5039 3780 rc 85 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef /c3 { 0.000000 0.820000 0.000000 sr} bdef /c4 { 0.000000 0.000000 0.800000 sr} bdef /c5 { 0.910000 0.820000 0.320000 sr} bdef /c6 { 1.000000 0.260000 0.820000 sr} bdef /c7 { 0.000000 0.820000 0.820000 sr} bdef c0 1 j 1 sg 0 0 5040 3781 rf 6 w gs 1267 283 2681 3082 rc 8 -84 13 -83 19 -82 23 -81 29 -79 33 -77 38 -75 43 -73 47 -69 52 -67 55 -63 60 -60 63 -55 67 -52 69 -47 73 -43 75 -38 77 -33 79 -29 81 -23 82 -19 83 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43 -75 38 -77 33 -79 29 -81 23 -82 19 -83 13 -84 8 -84 3 -84 -3 -84 -8 -83 -13 -82 -19 -81 -23 -79 -29 -77 -33 -75 -38 -73 -43 -69 -47 -67 -52 -63 -55 -60 -60 -55 -63 -52 -67 -47 -69 -43 -73 -38 -75 -33 -77 -29 -79 -23 -81 -19 -82 -13 -83 -8 -84 -3 -84 3947 1823 102 MP stroke DO 12 w 1 -17 1 -17 3 -16 4 -17 4 -16 6 -16 7 -15 7 -15 9 -15 9 -13 11 -14 11 -12 12 -12 12 -11 14 -11 13 -9 15 -9 15 -7 15 -7 16 -6 16 -4 17 -4 16 -3 17 -1 17 -1 17 1 17 1 16 3 17 4 16 4 16 6 15 7 15 7 15 9 13 9 14 11 12 11 12 12 11 12 11 14 9 13 9 15 7 15 7 15 6 16 4 16 4 17 3 16 1 17 1 17 -1 17 -1 17 -3 16 -4 17 -4 16 -6 16 -7 15 -7 15 -9 15 -9 13 -11 14 -11 12 -12 12 -12 11 -14 11 -13 9 -15 9 -15 7 -15 7 -16 6 -16 4 -17 4 -16 3 -17 1 -17 1 -17 -1 -17 -1 -16 -3 -17 -4 -16 -4 -16 -6 -15 -7 -15 -7 -15 -9 -13 -9 -14 -11 -12 -11 -12 -12 -11 -12 -11 -14 -9 -13 -9 -15 -7 -15 -7 -15 -6 -16 -4 -16 -4 -17 -3 -16 -1 -17 -1 -17 2875 1823 101 MP stroke gr 12 w 0 sg DO %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 2646 1517 mt ( 0.2) s gs 1267 283 2681 3082 rc 1 -34 3 -33 6 -33 7 -33 9 -33 12 -31 13 -31 15 -30 18 -29 19 -28 20 -27 22 -25 24 -24 25 -22 27 -20 28 -19 29 -18 30 -15 31 -13 31 -12 33 -9 33 -7 33 -6 33 -3 34 -1 34 1 33 3 33 6 33 7 33 9 31 12 31 13 30 15 29 18 28 19 27 20 25 22 24 24 22 25 20 27 19 28 18 29 15 30 13 31 12 31 9 33 7 33 6 33 3 33 1 34 -1 34 -3 33 -6 33 -7 33 -9 33 -12 31 -13 31 -15 30 -18 29 -19 28 -20 27 -22 25 -24 24 -25 22 -27 20 -28 19 -29 18 -30 15 -31 13 -31 12 -33 9 -33 7 -33 6 -33 3 -34 1 -34 -1 -33 -3 -33 -6 -33 -7 -33 -9 -31 -12 -31 -13 -30 -15 -29 -18 -28 -19 -27 -20 -25 -22 -24 -24 -22 -25 -20 -27 -19 -28 -18 -29 -15 -30 -13 -31 -12 -31 -9 -33 -7 -33 -6 -33 -3 -33 -1 -34 3143 1823 101 MP stroke gr 2683 1252 mt ( 0.4) s gs 1267 283 2681 3082 rc 2 -50 5 -51 7 -50 12 -49 14 -48 17 -48 20 -46 23 -45 25 -44 29 -41 31 -40 33 -38 36 -36 38 -33 40 -31 41 -29 44 -25 45 -23 46 -20 48 -17 48 -14 49 -12 50 -7 51 -5 50 -2 50 2 51 5 50 7 49 12 48 14 48 17 46 20 45 23 44 25 41 29 40 31 38 33 36 36 33 38 31 40 29 41 25 44 23 45 20 46 17 48 14 48 12 49 7 50 5 51 2 50 -2 50 -5 51 -7 50 -12 49 -14 48 -17 48 -20 46 -23 45 -25 44 -29 41 -31 40 -33 38 -36 36 -38 33 -40 31 -41 29 -44 25 -45 23 -46 20 -48 17 -48 14 -49 12 -50 7 -51 5 -50 2 -50 -2 -51 -5 -50 -7 -49 -12 -48 -14 -48 -17 -46 -20 -45 -23 -44 -25 -41 -29 -40 -31 -38 -33 -36 -36 -33 -38 -31 -40 -29 -41 -25 -44 -23 -45 -20 -46 -17 -48 -14 -48 -12 -49 -7 -50 -5 -51 -2 -50 3411 1823 101 MP stroke gr 2721 987 mt ( 0.6) s gs 1267 283 2681 3082 rc 2 -67 7 -67 10 -67 15 -66 19 -64 23 -64 26 -61 31 -60 34 -58 38 -56 41 -53 45 -51 47 -47 51 -45 53 -41 56 -38 58 -34 60 -31 61 -26 64 -23 64 -19 66 -15 67 -10 67 -7 67 -2 67 2 67 7 67 10 66 15 64 19 64 23 61 26 60 31 58 34 56 38 53 41 51 45 47 47 45 51 41 53 38 56 34 58 31 60 26 61 23 64 19 64 15 66 10 67 7 67 2 67 -2 67 -7 67 -10 67 -15 66 -19 64 -23 64 -26 61 -31 60 -34 58 -38 56 -41 53 -45 51 -47 47 -51 45 -53 41 -56 38 -58 34 -60 31 -61 26 -64 23 -64 19 -66 15 -67 10 -67 7 -67 2 -67 -2 -67 -7 -67 -10 -66 -15 -64 -19 -64 -23 -61 -26 -60 -31 -58 -34 -56 -38 -53 -41 -51 -45 -47 -47 -45 -51 -41 -53 -38 -56 -34 -58 -31 -60 -26 -61 -23 -64 -19 -64 -15 -66 -10 -67 -7 -67 -2 -67 3679 1823 101 MP stroke gr 2758 722 mt ( 0.8) s gs 1267 283 2681 3082 rc SO 3 -84 8 -84 13 -83 19 -82 23 -81 29 -79 33 -77 38 -75 43 -73 47 -69 52 -67 55 -63 60 -60 63 -55 67 -52 69 -47 73 -43 75 -38 77 -33 79 -29 81 -23 82 -19 83 -13 84 -8 84 -3 84 3 84 8 83 13 82 19 81 23 79 29 77 33 75 38 73 43 69 47 67 52 63 55 60 60 55 63 52 67 47 69 43 73 38 75 33 77 29 79 23 81 19 82 13 83 8 84 3 84 -3 84 -8 84 -13 83 -19 82 -23 81 -29 79 -33 77 -38 75 -43 73 -47 69 -52 67 -55 63 -60 60 -63 55 -67 52 -69 47 -73 43 -75 38 -77 33 -79 29 -81 23 -82 19 -83 13 -84 8 -84 3 -84 -3 -84 -8 -83 -13 -82 -19 -81 -23 -79 -29 -77 -33 -75 -38 -73 -43 -69 -47 -67 -52 -63 -55 -60 -60 -55 -63 -52 -67 -47 -69 -43 -73 -38 -75 -33 -77 -29 -79 -23 -81 -19 -82 -13 -83 -8 -84 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65 2 66 -2 66 -5 65 -8 66 -11 64 -15 64 -17 64 -21 62 -23 62 -27 60 -30 58 -32 58 -35 55 -38 54 -41 52 -42 50 -46 47 -47 46 -50 42 -52 41 -54 38 -55 35 -58 32 -58 30 -60 27 -62 23 -62 21 -64 17 -64 15 -64 11 -66 8 -65 5 -66 2 -66 -2 -65 -5 -66 -8 -64 -11 -64 -15 -64 -17 -62 -21 -62 -23 -60 -27 -58 -30 -58 -32 -55 -35 -54 -38 -52 -41 -50 -42 -47 -46 -46 -47 -42 -50 -41 -52 -38 -54 -35 -55 -32 -58 -30 -58 -27 -60 -23 -62 -21 -62 -17 -64 -15 -64 -11 -64 -8 -66 -5 -65 -2 -66 3947 1823 129 MP stroke 1 -39 3 -40 5 -39 7 -39 8 -38 11 -38 12 -38 14 -37 16 -36 18 -35 20 -34 21 -34 22 -32 25 -31 25 -30 28 -28 28 -28 30 -25 31 -25 32 -22 34 -21 34 -20 35 -18 36 -16 37 -14 38 -12 38 -11 38 -8 39 -7 39 -5 40 -3 39 -1 39 1 40 3 39 5 39 7 38 8 38 11 38 12 37 14 36 16 35 18 34 20 34 21 32 22 31 25 30 25 28 28 28 28 25 30 25 31 22 32 21 34 20 34 14 37 -5 45 -12 45 -12 41 -8 39 -6 37 -5 35 -2 34 -2 33 0 66 2 33 2 34 5 35 6 37 8 39 12 41 12 45 5 45 -14 37 -20 34 -21 34 -22 32 -25 31 -25 30 -28 28 -28 28 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Operating System: Microsoft Windows XP. %%Title: C:\Documents and Settings\MathIPB\My Documents\ADG_Workfiles\SupportIdentification\Data\ConstantConductivityInclusionPaper\CircPertFig1b.eps %%CreationDate: 11/05/2010 17:41:40 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 96 238 515 553 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 96 238 515 553 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 1152 6636 csm 0 0 5039 3780 rc 85 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef /c3 { 0.000000 0.820000 0.000000 sr} bdef /c4 { 0.000000 0.000000 0.800000 sr} bdef /c5 { 0.910000 0.820000 0.320000 sr} bdef /c6 { 1.000000 0.260000 0.820000 sr} bdef /c7 { 0.000000 0.820000 0.820000 sr} bdef c0 1 j 1 sg 0 0 5040 3781 rf 6 w gs 1267 283 2681 3082 rc 8 -84 13 -83 19 -82 23 -81 29 -79 33 -77 38 -75 43 -73 47 -69 52 -67 55 -63 60 -60 63 -55 67 -52 69 -47 73 -43 75 -38 77 -33 79 -29 81 -23 82 -19 83 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Operating System: Microsoft Windows XP. %%Title: C:\Documents and Settings\MathIPB\My Documents\ADG_Workfiles\SupportIdentification\Data\ConstantConductivityInclusionPaper\DentCircFig1b.eps %%CreationDate: 11/04/2010 10:30:12 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 96 238 515 553 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri 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Operating System: Microsoft Windows XP. %%Title: C:\Documents and Settings\MathIPB\My Documents\ADG_Workfiles\SupportIdentification\Data\ConstantConductivityInclusionPaper\CircPertFig1c.eps %%CreationDate: 11/05/2010 17:42:44 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 96 238 515 553 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 96 238 515 553 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 1152 6636 csm 0 0 5039 3780 rc 85 dict begin %Colortable dictionary /c0 { 0.000000 0.000000 0.000000 sr} bdef /c1 { 1.000000 1.000000 1.000000 sr} bdef /c2 { 0.900000 0.000000 0.000000 sr} bdef /c3 { 0.000000 0.820000 0.000000 sr} bdef /c4 { 0.000000 0.000000 0.800000 sr} bdef /c5 { 0.910000 0.820000 0.320000 sr} bdef /c6 { 1.000000 0.260000 0.820000 sr} bdef /c7 { 0.000000 0.820000 0.820000 sr} bdef c0 1 j 1 sg 0 0 5040 3781 rf 6 w gs 1267 283 2681 3082 rc 8 -84 13 -83 19 -82 23 -81 29 -79 33 -77 38 -75 43 -73 47 -69 52 -67 55 -63 60 -60 63 -55 67 -52 69 -47 73 -43 75 -38 77 -33 79 -29 81 -23 82 -19 83 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b(of)h(a)g(core)g(from)f(boundar)36 b(y)388 b(data,)402 b Fj(SIAM)390 b(J)-36 b(.)388 b(on)h(Applied)h (Mathematics)2303 68973 y Fd(55)p Fp(\(3\):)336 b(677\226706.)-189 71791 y(Wloka,)h(J)-36 b(.)337 b(1987.)470 b Fj(P)-48 b(ar)48 b(tial)337 b(Diff)-36 b(erential)336 b(Equations)p Fp(,)h(Cambr)18 b(idge)337 b(Univ)-30 b(ersity)336 b(Press)-18 b(.)p eop end %%Page: 18 18 TeXDict begin 18 17 bop -189 715 a Fv(Appendix)367 b(A:)j(Nomenc)-27 b(lature)367 b(and)h(facts)-189 3995 y Fp(Let)338 b Fn(H)2940 3555 y Fu(1)3466 3995 y Fo(\(\012)4812 4177 y Fu(1)5339 3995 y Fo(\))h Fp(is)g(the)f(sobole)-36 b(v)338 b(space)h(of)f(all)h (functions)f Fn(u)h Fl(2)h Fn(L)28533 3555 y Fu(2)29059 3995 y Fo(\(\012)30405 4177 y Fu(1)30931 3995 y Fo(\))f Fp(f)-36 b(or)338 b(which)37196 3517 y Fm(@)52 b(u)p 37027 3715 1429 49 v 37027 4413 a(@)g(x)38105 4548 y Fk(i)38929 3995 y Fl(2)339 b Fn(L)40901 3555 y Fu(2)41427 3995 y Fo(\(\012)42773 4177 y Fu(1)43299 3995 y Fo(\))g Fp(f)-36 b(or)338 b Fn(i)h Fo(=)g(1)p 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Fo(=)22807 36693 y Fi(X)22788 39323 y Fm(n)p Ff(2)p Fe(Z)24576 37845 y Fo(\(1)270 b(+)f Fn(n)27863 37344 y Fu(2)28389 37845 y Fo(\))28860 37344 y Fm(l)29468 37525 y Fo(~)29207 37845 y Fn(f)130 b Fo(\()p Fn(n)p Fo(\))31871 37457 y Fi(d)31641 37845 y Fo(~)-647 b Fn(g)44 b Fo(\()p Fn(n)p Fo(\))13569 b Fp(\(5.5\))-189 41617 y(By)364 b Fn(H)2695 41177 y Ff(\000)p Fm(l)3774 41617 y Fo(\()p Fg(T)p Fo(\)\()p Fn(l)406 b Fl(\025)381 b Fo(0\))365 b Fp(w)-12 b(e)363 b(denote)h(the)f(dual)h(of)f Fn(H)22683 41177 y Fm(l)23030 41617 y Fo(\()p Fg(T)p Fo(\))p Fn(:)h Fp(With)g(the)g(F)-36 b(our)18 b(ier)362 b(tr)-12 b(ansf)-36 b(or)30 b(m)362 b(de\002ned)i(on)f Fn(H)47123 41177 y Ff(\000)p Fm(l)48202 41617 y Fo(\()p Fg(T)p Fo(\))-189 43439 y Fp(b)-24 b(y)1665 43119 y Fo(^)1404 43439 y Fn(f)131 b Fo(\()p Fn(n)p Fo(\))337 b(:=)f Fl(h)p Fn(f)63 b(;)202 b(e)7981 42999 y Ff(\000)p Fm(int)10000 43439 y Fl(i)10471 43734 y Fm(H)11312 43482 y Fc(\000)p Fk(l)12274 43734 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)14698 43482 y Fk(l)15021 43734 y Fu(\()p Fe(T)p Fu(\))16736 43439 y Fp(w)-12 b(e)337 b(\002nd)15412 46211 y Fl(h)p Fn(f)63 b(;)202 b(e)17642 45711 y Ff(\000)p Fm(int)19661 46211 y Fl(i)20132 46507 y Fm(H)20973 46255 y Fc(\000)p Fk(l)21935 46507 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)24359 46255 y Fk(l)24682 46507 y Fu(\()p Fe(T)p Fu(\))26396 46211 y Fo(=)27694 45060 y Fi(X)27676 47690 y Fm(n)p Ff(2)p Fe(Z)29927 45891 y Fo(~)29666 46211 y Fn(f)131 b Fo(\()p Fn(n)p Fo(\))32331 45824 y Fi(d)32101 46211 y Fo(~)-647 b Fn(g)44 b Fo(\()p Fn(n)p Fo(\))13109 b Fp(\(5.6\))-189 49983 y(f)-36 b(or)428 b(the)g(duality)h(pair)18 b(ing)428 b Fl(h)p Fn(:;)202 b(:)p Fl(i)13902 50279 y Fm(H)14743 50027 y Fc(\000)p Fk(l)15705 50279 y Fu(\()p Fe(T)p Fu(\))p Fm(;H)18129 50027 y Fk(l)18451 50279 y Fu(\()p Fe(T)p Fu(\))20257 49983 y Fp(and)429 b(f)-36 b(or)428 b Fn(f)620 b Fl(2)490 b Fn(H)28132 49544 y Ff(\000)p Fm(l)29211 49983 y Fo(\()p Fg(T)p Fo(\))429 b Fp(and)g Fn(g)533 b Fl(2)490 b Fn(H)37356 49544 y Fm(l)37703 49983 y Fo(\()p Fg(T)p Fo(\))p Fn(:)429 b Fp(Moreo)-18 b(v)-30 b(er)-61 b(,)450 b Fn(H)47123 49544 y Ff(\000)p Fm(l)48202 49983 y Fo(\()p Fg(T)p Fo(\))-189 51805 y Fp(is)405 b(char)-12 b(acter)18 b(iz)-18 b(ed)404 b(b)-24 b(y)405 b(5.3)g(and)g(the)g(inner) f(product)h(on)g Fn(H)27433 51365 y Ff(\000)p Fm(l)28511 51805 y Fo(\()p Fg(T)p Fo(\))h Fp(is)g(giv)-30 b(en)404 b(b)-24 b(y)405 b(5.6,)422 b(in)405 b(both)g(cases)g(with)g Fn(l)-189 53627 y Fp(replaced)336 b(b)-24 b(y)337 b Fl(\000)p Fn(l)24 b(:)336 b Fp(W)-36 b(e)337 b(ha)-24 b(v)-30 b(e)20594 55448 y Fn(f)468 b Fo(=)22953 54297 y Fi(X)22934 56927 y Fm(n)p Ff(2)p Fe(Z)25186 55128 y Fo(~)24925 55448 y Fn(f)131 b Fo(\()p Fn(n)p Fo(\))p Fn(e)27883 54948 y Fm(int)47461 55448 y Fp(\(5.7\))-189 58856 y(f)-36 b(or)520 b Fn(f)774 b Fl(2)644 b Fn(H)5635 58416 y Fm(l)5982 58856 y Fo(\()p Fg(T)p Fo(\))p Fn(;)202 b(l)667 b Fl(2)644 b Fg(R)p Fn(;)520 b Fp(where)g(the)g(ser)18 b(ies)521 b(5.7)f(con)-24 b(v)-30 b(erges)519 b(in)i Fn(H)33114 58416 y Fm(l)33460 58856 y Fo(\()p Fg(T)p Fo(\))p Fn(:)g Fp(W)-36 b(e)520 b(de\002ne)h(the)f(diff)-36 b(erential)-189 60678 y(oper)-12 b(ator)336 b Fn(D)370 b Fo(:)336 b Fn(H)7803 60238 y Fm(l)8150 60678 y Fo(\()p Fg(T)p Fo(\))h Fl(!)g Fn(H)12892 60238 y Fm(l)11 b Ff(\000)p Fu(1)14441 60678 y Fo(\()p Fg(T)p Fo(\))338 b Fp(b)-24 b(y)19335 64321 y Fn(D)34 b(f)466 b Fo(=)22731 63169 y Fi(X)22712 65800 y Fm(n)p Ff(2)p Fe(Z)24702 64321 y Fn(in)26108 64001 y Fo(~)25848 64321 y Fn(f)130 b Fo(\()p Fn(n)p Fo(\))p Fn(e)28805 63820 y Fm(int)30093 64321 y Fn(:)17031 b Fp(\(5.8\))-189 67583 y(It)433 b(is)g(a)g(bounded)f(linear)h(oper)-12 b(ator)431 b(with)i Fo(k)-34 b(er\()p Fn(D)34 b Fo(\))433 b Fp(giv)-30 b(en)432 b(b)-24 b(y)433 b(the)g(constant)f(functions)h (on)g Fg(T)p Fn(:)f Fp(The)h(space)-189 69404 y Fl(C)520 68964 y Ff(1)1516 69404 y Fo(\()p Fg(T)p Fo(\))338 b Fp(of)f(all)g(in\002nitely)g(diff)-36 b(erentiab)-24 b(le)336 b(functions)g(on)h Fg(T)g Fp(is)g(char)-12 b(acter)18 b(iz)-18 b(ed)336 b(b)-24 b(y)9458 72177 y Fl(C)10167 71676 y Ff(1)11163 72177 y Fo(\()p Fg(T)p Fo(\))337 b(=)g Fl(f)p Fn(f)468 b Fo(:)336 b Fg(T)h Fl(!)f Fg(C)i Fl(j)p Fn(n)p Fl(j)22178 71676 y Fm(k)22745 72177 y Fl(j)23343 71857 y Fo(~)23082 72177 y Fn(f)131 b Fo(\()p Fn(n)p Fo(\))337 b Fl(!)g Fo(0)p Fp(as)p Fn(n)f Fl(!)h(1)p Fp(f)-36 b(or)336 b(all)h Fn(k)375 b Fl(2)337 b Fg(N)p Fl(g)p Fn(:)7155 b Fp(\(5.9\))-189 74949 y(Those)486 b(f)-36 b(acts)487 b(f)-36 b(ollo)-18 b(ws)487 b(as)g(a)f(special)h(case)g(of)g (theorems)f(on)h(Sobole)-36 b(v)487 b(spaces)g(on)g(smooth)f(compact) -189 76771 y(manif)-36 b(olds)336 b(as)h(giv)-30 b(en)337 b(f)-36 b(or)336 b(e)-36 b(xample)336 b(in)h(Wloka)g(\(Wloka,)g (1987\).)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------1106030305746--