Content-Type: multipart/mixed; boundary="-------------1512090235259" This is a multi-part message in MIME format. ---------------1512090235259 Content-Type: text/plain; name="15-117.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="15-117.keywords" Schr\"odinger operator, $\delta$-interaction, conical surface, isoperimetric inequality, existence of bound states ---------------1512090235259 Content-Type: application/postscript; name="ExnerLotoreichik_final.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ExnerLotoreichik_final.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: ExnerLotoreichik_final.dvi %%CreationDate: Wed Dec 09 09:28:36 2015 %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: LMRoman10-Bold LMRoman9-Regular LMRomanCaps10-Regular %%+ LMMathItalic9-Regular LMRoman6-Regular LMRoman10-Regular %%+ LMMathItalic10-Regular LMMathSymbols10-Regular LMRoman10-Italic %%+ LMRoman8-Regular LMRoman9-Italic MSBM10 EUFM10 LMMathItalic8-Regular %%+ LMSans10-Regular EUSM10 LMMathExtension10-Regular %%+ LMMathItalic6-Regular LMMathSymbols8-Regular LMMathSymbols6-Regular %%+ MSAM10 LMRoman9-Bold LMMono9-Regular %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: "C:\Program Files\MiKTeX 2.9\miktex\bin\dvips.EXE" %+ ExnerLotoreichik_final.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2015.12.09:0928 %%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: lm-ec.enc 0 0 % This file belongs to the Latin Modern package. 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exch def /pgfdiffx pgfendx pgfstartx sub def /pgfdiffy pgfendy pgfstarty sub def dup /pgfdomb exch def }bind def /pgfR2{ newpath 0.5 add pgfcircx pgfcircy 3 2 roll 0 360 arc setrgbcolor fill pop}bind def /pgfR{ /pgfdiff 8 index round cvi 8 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 9 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 9 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 8 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 8 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 8 index 8 index 8 index 13 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll 4 index add 4 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 14 1 roll cleartomark exch pop }bind def /pgfsc{}bind def/pgffc{}bind def/pgfstr{stroke}bind def/pgffill{fill}bind def/pgfeofill{eofill}bind def/pgfe{a dup 0 rlineto exch 0 exch rlineto neg 0 rlineto closepath}bind def/pgfw{setlinewidth}bind def/pgfs{save pgfpd 72 Resolution div 72 VResolution div neg scale magscale{1 DVImag div dup scale}if pgfx neg pgfy neg translate pgffoa .setopacityalpha}bind def/pgfr{pgfsd restore}bind def userdict begin/pgfo{pgfsd /pgfx currentpoint /pgfy exch def def @beginspecial}bind def /pgfc{newpath @endspecial pgfpd}bind def /pgfsd{globaldict /pgfdelta /delta where {pop delta} {0} ifelse put}bind def/pgfpd{/delta globaldict /pgfdelta get def}bind def /.setopacityalpha where {pop} {/.setopacityalpha{pop}def} ifelse /.pgfsetfillopacityalpha{/pgffoa exch def /pgffill{gsave pgffoa .setopacityalpha fill 1 .setopacityalpha newpath fill grestore newpath}bind def /pgfeofill{gsave pgffoa .setopacityalpha eofill 1 .setopacityalpha newpath eofill grestore newpath}bind def}bind def /.pgfsetstrokeopacityalpha{/pgfsoa exch def /pgfstr{gsave pgfsoa .setopacityalpha stroke grestore newpath}bind def}bind def /pgffoa 1 def /pgfsoa 1 def end systemdict /pdfmark known{userdict /?pdfmark systemdict /exec get put}{userdict /?pdfmark systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put}ifelse /DvipsToPDF{72.27 mul Resolution div} def/PDFToDvips{72.27 div Resolution mul} def/BPToDvips{72 div Resolution mul}def/BorderArrayPatch{[exch{dup dup type/integertype eq exch type/realtype eq or{BPToDvips}if}forall]}def/HyperBorder {1 PDFToDvips} def/H.V {pdf@hoff pdf@voff null} def/H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def/H.S {currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF 72 add /pdf@hoff exch def HyperBorder sub /pdf@llx exch def} def/H.L {2 sub dup/HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def/pdf@urx exch def} def/H.A {H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def} def/H.R {currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def} def /pgf1{gsave exec 1.0 pgfw 2.00002 0.0 moveto -6.00006 4.00005 lineto -3.00003 0.0 lineto -6.00006 -4.00005 lineto pgffill grestore} bind def /pgf2{gsave exec 1.0 pgfw 0.8 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -3.00003 4.00005 moveto -2.75002 2.50002 0.0 0.24998 0.75 0.0 curveto 0.0 -0.24998 -2.75002 -2.50002 -3.00003 -4.00005 curveto pgfstr grestore} bind def /pgf3{gsave exec 1.0 pgfw [ ] 0.0 setdash 0.0 -5.00005 moveto 0.0 5.00005 lineto pgfstr grestore} bind def /pgf4{gsave exec 1.0 pgfw [ ] 0.0 setdash -3.00003 -5.00005 moveto 0.0 -5.00005 lineto 0.0 5.00005 lineto -3.00003 5.00005 lineto pgfstr grestore} bind def /pgf5{gsave exec 1.0 pgfw [ ] 0.0 setdash -2.00002 -5.00005 moveto 1.0 -3.00003 1.0 3.00003 -2.00002 5.00005 curveto pgfstr grestore} bind def /pgf6{gsave exec 1.0 pgfw [ ] 0.0 setdash -4.50003 -5.00005 moveto 0.49998 0.0 lineto -4.50003 5.00005 lineto pgfstr grestore} bind def /pgf7{gsave exec 1.0 pgfw -2.50002 0.0 translate [ ] 0.0 setdash 3.00003 0.0 moveto 3.00003 1.65689 1.65689 3.00003 0.0 3.00003 curveto -1.65689 3.00003 -3.00003 1.65689 -3.00003 0.0 curveto -3.00003 -1.65689 -1.65689 -3.00003 0.0 -3.00003 curveto 1.65689 -3.00003 3.00003 -1.65689 3.00003 0.0 curveto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def /pgf8{gsave exec 1.0 pgfw [ ] 0.0 setdash 1.0 0.0 moveto -5.00005 3.00003 lineto -11.00012 0.0 lineto -5.00005 -3.00003 lineto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def @fedspecial end %%BeginFont: LMMono9-Regular %!PS-AdobeFont-1.0: LMMono9-Regular 2.004 %%CreationDate: 7th October 2009 % Generated by MetaType1 (a MetaPost-based engine) % Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 778 222 0 %%EndComments FontDirectory/LMMono9-Regular known{/LMMono9-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman9-Bold known{/LMRoman9-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMSans10-Regular known{/LMSans10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 556 156 0 %%EndComments FontDirectory/LMMathItalic8-Regular known{/LMMathItalic8-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman8-Regular known{/LMRoman8-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman9-Italic known{/LMRoman9-Italic findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 694 194 0 %%EndComments FontDirectory/LMMathSymbols10-Regular known{/LMMathSymbols10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman10-Regular known{/LMRoman10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 694 194 0 %%EndComments FontDirectory/LMMathItalic10-Regular known{/LMMathItalic10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.003)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRomanCaps10-Regular known{/LMRomanCaps10-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman6-Regular known{/LMRoman6-Regular findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS). % Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG. % METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki % from GUST (http://www.gust.org.pl). % This work is released under the GUST Font License. % For the most recent version of this license see % This work has the LPPL maintenance status `maintained'. % The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki. % This work consists of the files listed in the MANIFEST-Latin-Modern.txt file. % ADL: 806 194 0 %%EndComments FontDirectory/LMRoman10-Bold known{/LMRoman10-Bold findfont dup/UniqueID known{dup /UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 17 dict begin /FontInfo 9 dict dup begin /version(2.004)readonly def /Notice(Copyright 2003--2009 by B. Jackowski and J.M. 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end 2855 2902 a 0 TeXcolorgray 41 w Fx(implies)41 b(that)g Fv(E)3489 2916 y Fr(1)3529 2902 y Fu(\()p Fk(H)3628 2916 y Fm(\013;)p Fr(\006)3749 2902 y Fu(\))i Fv(<)71 3014 y(E)138 3028 y Fr(1)177 3014 y Fu(\()p Fk(H)276 3030 y Fm(\013)321 3011 y Ff(0)345 3030 y Fm(;)p Fr(\006)420 3014 y Fu(\))25 b Ft(\024)g(\000)p Fv(\024)699 2981 y Fr(2)769 3014 y Fx(and)30 b(that)h Fu(#\()p Fv(\033)1306 3028 y Fr(d)1349 3014 y Fu(\()p Fk(H)1448 3028 y Fm(\013;)p Fr(\006)1569 3014 y Fu(\))20 b Ft(\\)g Fu(\()p Ft(\0001)p Fv(;)15 b Ft(\000)p Fv(\024)2065 2981 y Fr(2)2105 3014 y Fu(\)\))26 b Ft(\025)f Fu(1)p Fx(.)60 3160 y(\(ii\))32 b(F)-8 b(or)31 b(\020)p Ft(\))p Fx(\021,)g(w)m(e)g(supp)s(ose)e(that)j Fv(E)1349 3174 y Fr(1)1388 3160 y Fu(\()p Fk(H)1487 3174 y Fm(\013;)p Fr(\006)1608 3160 y Fu(\))26 b(=)g Ft(\000)p Fv(\024)1889 3127 y Fr(2)1928 3160 y Fx(.)42 b(Then)29 b(b)m(y)i(Theorem)p 0 0 1 TeXcolorrgb 2743 3160 a SDict begin H.S end 2743 3160 a 0 0 1 TeXcolorrgb Fx(2.1)p 0 0 1 TeXcolorrgb 2859 3103 a SDict begin H.R end 2859 3103 a 2859 3160 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.1) cvn H.B /ANN pdfmark end 2859 3160 a 0 TeXcolorgray 31 w Fx(w)m(e)g(ha)m(v)m(e)h Fu(1)p Fv(=\013)27 b Ft(2)e Fv(\033)3547 3174 y Fr(d)3591 3160 y Fu(\()p Fk(S)3677 3174 y Fr(\006)3732 3160 y Fu(\()p Fv(\024)p Fu(\)\))p Fx(.)68 3268 y(Therefore,)38 b(w)m(e)f(get)h Fv(\026)866 3282 y Fr(\006)921 3268 y Fu(\()p Fv(\024)p Fu(\))g Ft(\025)d Fu(1)p Fv(=\013)p Fx(,)40 b(but)d(the)g(inequalit)m(y)h Fv(\026)2220 3282 y Fr(\006)2275 3268 y Fu(\()p Fv(\024)p Fu(\))f Fv(>)f Fu(1)p Fv(=\013)i Fx(leads)f(to)h(a)f(con)m(tradiction)i (with)71 3376 y(item)32 b(\(i\))h(of)f(this)f(lemma,)i(b)s(ecause)f(w)m (e)g(w)m(ould)g(get)h Fv(E)1970 3390 y Fr(1)2009 3376 y Fu(\()p Fk(H)2108 3390 y Fm(\013;)p Fr(\006)2229 3376 y Fu(\))28 b Fv(<)f Ft(\000)p Fv(\024)2513 3343 y Fr(2)2552 3376 y Fx(.)45 b(Therefore,)32 b(the)g(equalit)m(y)h Fv(\026)3621 3390 y Fr(\006)3676 3376 y 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y Fr(1)1791 3736 y Fu(\()p Fk(H)1890 3750 y Fm(\013;)p Fr(\006)2011 3736 y Fu(\))d Fv(<)g Ft(\000)p Fv(\024)2306 3703 y Fr(2)2380 3736 y Fx(leads)i(to)h(a)f(con)m (tradiction)i(with)e(item)g(\(i\),)71 3848 y(b)s(ecause)30 b(w)m(e)h(w)m(ould)f(get)i Fv(\026)1013 3862 y Fr(\006)1068 3848 y Fu(\()p Fv(\024)p Fu(\))26 b Fv(>)f Fu(1)p Fv(=\013)p Fx(.)42 b(Therefore,)30 b(the)h(equalit)m(y)h Fv(E)2534 3862 y Fr(1)2573 3848 y Fu(\()p Fk(H)2672 3862 y Fm(\013;)p Fr(\006)2793 3848 y Fu(\))26 b(=)f Ft(\000)p Fv(\024)3073 3815 y Fr(2)3143 3848 y Fx(holds.)451 b Fe(\003)71 4070 y Fx(Next,)31 b(w)m(e)g(analyse)g(the)g(dep)s(endence)e(of)i Fv(\026)1576 4084 y Fr(\006)1631 4070 y Fu(\()p Fv(\024)p Fu(\))g Fx(on)f(the)h(parameter)g Fv(\024)p Fx(.)71 4245 y SDict begin H.S end 71 4245 a 71 4245 a SDict begin 13 H.A end 71 4245 a 71 4245 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.4) cvn /DEST pdfmark end 71 4245 a 0 TeXcolorgray FC(Lemma)46 b(2.4.)p 0 TeXcolorgray 47 w Fs(A)-5 b(ssume)40 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Ft(\024)25 b Fu(\()p Fk(S)2068 5006 y Fr(\006)2123 4992 y Fu(\()p Fv(\024)p Fu(\))p Ft(j)p Fv( )2329 5006 y Fr(\006)2385 4992 y Ft(j)p Fv(;)15 b Ft(j)p Fv( )2534 5006 y Fr(\006)2591 4992 y Ft(j)p Fu(\))2651 5006 y Fr(\006)2706 4992 y Fv(:)71 5158 y Fx(Let)31 b(us)e(write)1102 5285 y Fv( )1161 5299 y Fr(\006)1241 5285 y Fu(=)c Fv( )1399 5247 y Fr(+)1396 5313 y(\006)1479 5285 y Ft(\000)20 b Fv( )1632 5247 y Fg(\000)1629 5313 y Fr(\006)1873 5285 y Fx(and)181 b Ft(j)p Fv( )2285 5299 y Fr(\006)2341 5285 y Ft(j)26 b Fu(=)e Fv( )2549 5247 y Fr(+)2546 5313 y(\006)2629 5285 y Fu(+)c Fv( )2782 5247 y Fg(\000)2779 5313 y Fr(\006)2842 5285 y Fv(;)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 7 7 TeXDict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 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b(it)h(is)f(imp)s(ossible)g(to)h(ha)m(v)m(e)h(t)m(w)m(o)g (orthogonal)f(functions)f(in)g Fv(L)2818 2749 y Fr(2)2858 2782 y Fu(\(\006\))g Fx(that)h(are)g(b)s(oth)f(p)s(ositiv)m(e)71 2890 y(almost)g(ev)m(erywhere.)2996 b Fe(\003)71 3006 y SDict begin H.S end 71 3006 a 71 3006 a SDict begin 13 H.A end 71 3006 a 71 3006 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.2) cvn /DEST pdfmark end 71 3006 a 124 x Fx(2.2.)47 b FC(A)41 b(parametrization)g(of)h(cones)g(and)g(its)f (consequences.)47 b Fx(Let)37 b(the)f(cone)h Fu(\006)3206 3144 y Fm(R)3298 3130 y Fu(=)d(\006)3469 3144 y Fm(R)3527 3130 y Fu(\()p Fj(T)t Fu(\))h Ft(\032)f Fo(R)3860 3097 y Fr(3)71 3237 y Fx(b)s(e)d(as)g(in)h(\()p 0 0 1 TeXcolorrgb 451 3237 a SDict begin H.S end 451 3237 a 0 0 1 TeXcolorrgb Fx(1.2)p 0 0 1 TeXcolorrgb 568 3180 a SDict begin H.R end 568 3180 a 568 3237 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 568 3237 a 0 TeXcolorgray Fx(\).)44 b(In)31 b(this)g(subsection)h(w)m(e)g(pro)m (vide)f(an)h(e\036cien)m(t)h(parametrization)g(of)e Fu(\006)3164 3251 y Fm(R)3253 3237 y Fx(and)g(deriv)m(e)h(some)71 3345 y(consequences)23 b(of)g(it.)38 b(First,)25 b(note)e(that)g(the)g (cross-section)h Fj(T)i Fx(of)d Fu(\006)2364 3359 y Fm(R)2444 3345 y Fx(can)f(b)s(e)g(parametrized)h(b)m(y)g(its)f(arc-length)69 3453 y(via)37 b(the)g(unit-sp)s(eed)e Fv(C)907 3420 y Fr(2)946 3453 y Fx(-mapping)h Fv(\034)21 b Fu(:)32 b([0)p Fv(;)15 b(L)p Fu(])37 b Ft(!)f Fo(S)1886 3420 y Fr(2)1961 3453 y Fx(\()p Ft(j)15 b Fu(_)-40 b Fv(\034)10 b Ft(j)37 b(\021)e Fu(1)p Fx(\).)61 b(The)36 b(cone)h Fu(\006)2877 3467 y Fm(R)2971 3453 y Fx(can)g(b)s(e)f(corresp)s(ondingly)71 3561 y(parametrized)31 b(via)g(the)f(mapping)71 3729 y(\(2.6\))1113 3729 y SDict begin H.S end 1113 3729 a 1113 3729 a SDict begin 13 H.A end 1113 3729 a 1113 3729 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 1113 3729 a Fv(\033)13 b Fu(:)31 b([0)p Fv(;)15 b(R)q Fu(\))22 b Ft(\002)e Fu([0)p Fv(;)15 b(L)p Fu(])27 b Ft(!)e Fo(R)1968 3692 y Fr(3)2007 3729 y Fv(;)197 b(\033)s Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))27 b(:=)e Fv(r)s(\034)10 b Fu(\()p Fv(s)p Fu(\))p Fv(:)68 3897 y Fx(This)29 b(parametrization)j (de\034nes)e(natural)g(co-ordinates)i Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))32 b Fx(on)e Fu(\006)2480 3911 y Fm(R)2537 3897 y Fx(.)71 4038 y(Let)c(the)f(space)h Fv(L)680 4005 y Fr(1)720 4038 y Fu(\(\006)821 4052 y Fm(R)879 4038 y Fu(\))f Fx(b)s(e)g(in)m(tro)s(duced)g(as)h(usual,)g(b)m(y)f(means)h (of)f(the)h(surface)f(measure)h(on)f Fu(\006)3357 4052 y Fm(R)3414 4038 y Fx(.)39 b(A)26 b(function)71 4146 y Fv( )j Ft(2)d Fv(L)308 4113 y Fr(1)347 4146 y Fu(\(\006)448 4160 y Fm(R)506 4146 y Fu(\))31 b Fx(can)g(b)s(e)f(view)m(ed)h(as)g(a)g (function)f(of)h(the)g(argumen)m(ts)g Fv(r)i Fx(and)e Fv(s)f Fx(via)h(the)g(parametrization)i(\()p 0 0 1 TeXcolorrgb 3738 4146 a SDict begin H.S end 3738 4146 a 0 0 1 TeXcolorrgb Fx(2.6)p 0 0 1 TeXcolorrgb 3855 4089 a SDict begin H.R end 3855 4089 a 3855 4146 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 3855 4146 a 0 TeXcolorgray Fx(\).)68 4254 y(Th)m(us,)c(the)i(Leb)s(esgue)f (surface)g(in)m(tegral)i(of)f Fv( )j Fx(can)c(b)s(e)g(written)g(as)71 4607 y SDict begin H.S end 71 4607 a 71 4607 a SDict begin 13 H.A end 71 4607 a 71 4607 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 71 4607 a 528 4359 a Fi(Z)574 4548 y Fr(\006)625 4559 y Fh(R)695 4474 y Fv( )s Fu(\()p Fv(x)p Fu(\))p Fk(d)p Fv(\033)s Fu(\()p Fv(x)p Fu(\))d(=)1226 4359 y Fi(Z)1309 4386 y Fm(R)1272 4548 y Fr(0)1382 4359 y Fi(Z)1465 4386 y Fm(L)1428 4548 y Fr(0)1532 4474 y Fv( )s Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))p Ft(j)p Fv(\033)1862 4488 y Fm(r)1902 4474 y Fu(\()p Fv(r)m(;)g(s)p Fu(\))21 b Ft(\002)f Fv(\033)2257 4488 y Fm(s)2294 4474 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))p Ft(j)p Fk(d)p Fv(s)p Fk(d)p Fv(r)1130 4728 y Fu(=)1226 4614 y Fi(Z)1309 4640 y Fm(R)1272 4802 y Fr(0)1382 4614 y Fi(Z)1465 4640 y Fm(L)1428 4802 y Fr(0)1532 4728 y Fv( )s Fu(\()p Fv(r)m(;)g(s)p Fu(\))p Ft(j)p Fv(\033)1862 4742 y Fm(r)1902 4728 y Fu(\()p Fv(r)m(;)g(s)p Fu(\))p Ft(j)21 b(\001)g(j)p Fv(\033)2262 4742 y Fm(s)2299 4728 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))p Ft(j)p Fk(d)p Fv(s)p Fk(d)p Fv(r)29 b Fu(=)2818 4614 y Fi(Z)2901 4640 y Fm(R)2864 4802 y Fr(0)2974 4614 y Fi(Z)3057 4640 y Fm(L)3020 4802 y Fr(0)3124 4728 y Fv( )s Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))p Fv(r)s Fk(d)p Fv(s)p Fk(d)p Fv(r)s Fu(;)71 4594 y Fx(\(2.7\))68 4939 y(where,)30 b(\034rstly)-8 b(,)31 b(w)m(e)g(emplo)m(y)m(ed)h(that) g(the)f(v)m(ector)h Fv(\033)1851 4953 y Fm(s)1888 4939 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))27 b(=)e Fv(r)18 b Fu(_)-40 b Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))31 b Fx(is)g(of)g(length)g Fv(r)d(>)e Fu(0)31 b Fx(and)f(b)s(elongs)h(to)g(the)71 5047 y(tangen)m(t)c(plane)e Fv(T)687 5065 y Fm(\034)8 b Fr(\()p Fm(s)p Fr(\))818 5047 y Fu(\()p Fo(S)904 5014 y Fr(2)943 5047 y Fu(\))26 b Fx(of)f Fo(S)1153 5014 y Fr(2)1217 5047 y Fx(at)h(the)g(p)s(oin)m(t)f Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))p Fx(,)27 b(secondly)-8 b(,)27 b(w)m(e)f(used)e(that)i(the)g(v)m(ector)h Fv(\033)3297 5061 y Fm(r)3335 5047 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))26 b(=)f Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))26 b Fx(is)71 5166 y(of)31 b(unit)g(length)h(and)e(for)h(simple)h(geometric)h (reasons)e(is)g(orthogonal)i(to)f Fv(T)2697 5184 y Fm(\034)8 b Fr(\()p Fm(s)p Fr(\))2828 5166 y Fu(\()p Fo(S)2914 5133 y Fr(2)2953 5166 y Fu(\))p Fx(.)44 b(A)31 b(direct)h(consequence) 71 5285 y(of)38 b(\()p 0 0 1 TeXcolorrgb 217 5285 a SDict begin H.S end 217 5285 a 0 0 1 TeXcolorrgb Fx(2.7)p 0 0 1 TeXcolorrgb 333 5228 a SDict begin H.R end 333 5228 a 333 5285 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.7) cvn H.B /ANN pdfmark end 333 5285 a 0 TeXcolorgray Fx(\))31 b(is)f(the)h(follo)m(wing)h(isomorphism)d Fv(L)1624 5252 y Fr(2)1663 5285 y Fu(\(\006)1764 5299 y Fm(R)1822 5285 y Fu(\))d Ft(')f Fv(L)2041 5252 y Fr(2)2080 5285 y Fu(\(\(0)p Fv(;)15 b(R)q Fu(\);)g Fv(r)s Fk(d)p Fv(r)s Fu(\))22 b Ft(\012)e Fv(L)2725 5252 y Fr(2)2764 5285 y Fu(\()p Fj(T)t Fu(\))p Fx(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 8 8 TeXDict begin 8 7 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 63 -56 a SDict begin H.S end 63 -56 a 63 -56 a SDict begin H.R end 63 -56 a 63 -56 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 63 -56 a 0 TeXcolorgray 0 TeXcolorgray 71 19 a Fp(8)1066 b(P)-6 b(A)e(VEL)24 b(EXNER)f(AND)g(VLADIMIR)g(LOTOREICHIK)p 0 TeXcolorgray 71 218 a Fx(F)-8 b(urther,)25 b(with)f(the)h(aid)f(of)g (the)h(iden)m(tit)m(y)g Ft(j)p Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))e Ft(\000)g Fv(\034)i Fu(\()p Fv(s)1898 185 y Fg(0)1922 218 y Fu(\))p Ft(j)1982 185 y Fr(2)2047 218 y Fu(=)25 b(2)8 b Ft(\000)g Fu(2)p Ft(h)p Fv(\034)i Fu(\()p Fv(s)p Fu(\))p Fv(;)15 b(\034)10 b Fu(\()p Fv(s)2686 185 y Fg(0)2711 218 y Fu(\))p Ft(i)2781 238 y Fl(R)2829 219 y Fy(3)2892 218 y Fx(w)m(e)25 b(can)g(express)e(the)i(square)71 326 y(of)30 b(the)h(distance)g(b)s(et)m(w)m(een)g Fv(\033)s Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))32 b Fx(and)e Fv(\033)s Fu(\()p Fv(r)1625 293 y Fg(0)1648 326 y Fv(;)15 b(s)1731 293 y Fg(0)1754 326 y Fu(\))31 b Fx(through)f Fv(r)s Fx(,)g Fv(r)2307 293 y Fg(0)2329 326 y Fx(,)h(and)f Ft(j)p Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))21 b Ft(\000)f Fv(\034)10 b Fu(\()p Fv(s)2990 293 y Fg(0)3013 326 y Fu(\))p Ft(j)31 b Fx(as)71 643 y SDict begin H.S end 71 643 a 71 643 a SDict begin 13 H.A end 71 643 a 71 643 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 71 643 a 566 477 a Ft(j)p Fv(\033)s Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))22 b Ft(\000)d Fv(\033)s Fu(\()p Fv(r)1083 439 y Fg(0)1107 477 y Fv(;)c(s)1190 439 y Fg(0)1213 477 y Fu(\))p Ft(j)1273 439 y Fr(2)1338 477 y Fu(=)25 b Ft(j)p Fv(r)s(\034)10 b Fu(\()p Fv(s)p Fu(\))21 b Ft(\000)f Fv(r)1822 439 y Fg(0)1844 477 y Fv(\034)10 b Fu(\()p Fv(s)1972 439 y Fg(0)1996 477 y Fu(\))p Ft(j)2056 439 y Fr(2)2121 477 y Fu(=)25 b Fv(r)2261 439 y Fr(2)2320 477 y Fu(+)20 b(\()p Fv(r)2490 439 y Fg(0)2513 477 y Fu(\))2548 439 y Fr(2)2608 477 y Ft(\000)g Fu(2)p Fv(r)s(r)2832 439 y Fg(0)2855 477 y Ft(h)p Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))p Fv(;)15 b(\034)10 b Fu(\()p Fv(s)3221 439 y Fg(0)3245 477 y Fu(\))p Ft(i)3315 496 y Fl(R)3363 477 y Fy(3)1338 629 y Fu(=)25 b Fv(r)1478 591 y Fr(2)1538 629 y Fu(+)19 b(\()p Fv(r)1707 591 y Fg(0)1731 629 y Fu(\))1766 591 y Fr(2)1826 629 y Fu(+)h Fv(r)s(r)2005 591 y Fg(0)2027 560 y Fi(\000)2065 629 y Ft(j)p Fv(\034)10 b Fu(\()p Fv(s)p Fu(\))21 b Ft(\000)f Fv(\034)10 b Fu(\()p Fv(s)2493 591 y Fg(0)2516 629 y Fu(\))p Ft(j)2576 591 y Fr(2)2636 629 y Ft(\000)20 b Fu(2)2772 560 y 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b(orr)g(esp)g(onding)39 b(to)e(the)g(lar)-5 b(gest)63 1414 y(eigenvalue)32 b(of)h Fk(S)660 1428 y Fr(\000)704 1439 y Fh(R)758 1414 y Fu(\()p Fv(\024)p Fu(\))h Fs(is)e(r)-5 b(otational)7 b(ly)36 b(invariant;)e(i.e.)15 b(it)32 b(dep)-5 b(ends)34 b(on)f(the)g(distanc) -5 b(e)34 b(fr)-5 b(om)34 b(the)f(origin)g(\(tip)64 1522 y(of)g(the)g(c)-5 b(one\))33 b(only.)p 0 TeXcolorgray 71 1714 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 43 w Fx(Let)37 b Fv(L)f Ft(2)g Fu(\(0)p Fv(;)15 b Fu(2)p Fv(\031)s Fu(])39 b Fx(stand)d(for)h(the)g(length)h(of)f Fj(C)f Fx(and)h Fu(\()p Ft(\001)p Fv(;)15 b Ft(\001)p Fu(\))2393 1729 y Fd(C)2475 1714 y Fx(denote)38 b(the)f(scalar)h(pro)s(duct)e(in)g Fv(L)3725 1681 y Fr(2)3765 1714 y Fu(\()p Fj(C)p Fu(\))p Fx(.)68 1822 y(The)29 b(family)i(of)g(functions)965 2019 y Fv(\037)1022 2033 y Fm(m)1088 2019 y Fu(\()p Fv(s)p Fu(\))26 b(:=)1404 1957 y(1)p 1358 1998 138 4 v 1358 2015 a Ft(p)p 1434 2015 62 4 v 77 x Fv(L)1521 2019 y Fu(exp)1675 1900 y Fi(\022)1746 1957 y Fu(2)p 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b(w)m(e)f(form)m (ulate)g(a)g(sp)s(ecialised)g(v)m(ersion)f(of)h(this)f(result.)71 2514 y SDict begin H.S end 71 2514 a 71 2514 a SDict begin 13 H.A end 71 2514 a 71 2514 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.9) cvn /DEST pdfmark end 71 2514 a 0 TeXcolorgray FC(Prop)s(osition)42 b(2.9.)p 0 TeXcolorgray 45 w Fx([)p 0.2 0.2 0.6 TeXcolorrgb 874 2514 a SDict begin H.S end 874 2514 a 0.2 0.2 0.6 TeXcolorrgb Fx(18)p 0.2 0.2 0.6 TeXcolorrgb 965 2457 a SDict begin H.R end 965 2457 a 965 2514 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.EHL06) cvn H.B /ANN pdfmark end 965 2514 a 0 TeXcolorgray Fx(,)37 b(Thm.)55 b(2.2])p Fs(,)40 b Fx([)p 0.2 0.2 0.6 TeXcolorrgb 1533 2514 a SDict begin H.S end 1533 2514 a 0.2 0.2 0.6 TeXcolorrgb Fx(1)p 0.2 0.2 0.6 TeXcolorrgb 1578 2457 a SDict begin H.R end 1578 2457 a 1578 2514 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.ACGJ03) cvn H.B /ANN pdfmark end 1578 2514 a 0 TeXcolorgray Fx(,)d(Thm.)55 b(2])38 b Fs(L)-5 b(et)37 b Fv(f)43 b Ft(2)34 b Fv(C)7 b Fu(\([0)p Fv(;)15 b Ft(1)p Fu(\);)g Fo(R)p Fu(\))39 b Fs(b)-5 b(e)37 b(c)-5 b(onvex)38 b(and)g(de)-5 b(cr)g(e)g(asing.)66 2622 y(L)g(et)37 b(the)g(functional)h Fu(\010)880 2637 y Fm(f)962 2622 y Fs(b)-5 b(e)37 b(as)g(in)42 b Fx(\()p 0 0 1 TeXcolorrgb 1354 2622 a SDict begin H.S end 1354 2622 a 0 0 1 TeXcolorrgb Fx(2.10)p 0 0 1 TeXcolorrgb 1516 2565 a SDict begin H.R end 1516 2565 a 1516 2622 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 1516 2622 a 0 TeXcolorgray Fx(\))37 b Fs(with)h Fv(L)33 b Ft(2)f Fu(\(0)p Fv(;)15 b Fu(2)p Fv(\031)s Fu(])p Fs(.)57 b(L)-5 b(et)37 b Fj(C)c Ft(\032)g Fo(S)2712 2589 y Fr(2)2787 2622 y Fs(b)-5 b(e)37 b(a)g(cir)-5 b(cle)37 b(and)h Fj(T)f Ft(\032)c Fo(S)3659 2589 y Fr(2)3735 2622 y Fs(b)-5 b(e)37 b(a)71 2734 y Fv(C)143 2701 y Fr(2)182 2734 y Fs(-smo)-5 b(oth)33 b(non-cir)-5 b(cular)32 b(lo)-5 b(op)33 b(such)e(that)h Ft(j)p Fj(T)t Ft(j)26 b Fu(=)f Ft(j)p Fj(C)p Ft(j)g Fu(=)g Fv(L)p Fs(.)41 b(Then)32 b(the)f(fol)7 b(lowing)32 b(isop)-5 b(erimetric)33 b(ine)-5 b(quality)66 2842 y(holds)1693 2962 y Fu(\010)1759 2977 y Fm(f)1804 2962 y Fu([)p Fj(C)p Fu(])26 b Fv(<)f Fu(\010)2097 2977 y Fm(f)2142 2962 y Fu([)p Fj(T)t Fu(])p Fv(:)71 3210 y SDict begin H.S end 71 3210 a 71 3210 a SDict begin 13 H.A end 71 3210 a 71 3210 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 71 3210 a 1288 3351 a Fx(3.)46 b Fw(Pr)n(oofs)34 b(of)g(the)f(main)h(resul)-6 b(ts)71 3405 y SDict begin H.S end 71 3405 a 71 3405 a SDict begin 13 H.A end 71 3405 a 71 3405 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.1) cvn /DEST pdfmark end 71 3405 a 141 x Fx(3.1.)47 b FC(Pro)s(of)33 b(of)g(Theorem)p 0 0 1 TeXcolorrgb 1103 3546 a SDict begin H.S end 1103 3546 a 0 0 1 TeXcolorrgb FC(1.2)p 0 0 1 TeXcolorrgb 1237 3486 a SDict begin H.R end 1237 3486 a 1237 3546 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.1.2) cvn H.B /ANN pdfmark end 1237 3546 a 0 TeXcolorgray FC(.)45 b Fx(The)29 b(claim)g(of)g(\(i\))g(is)g(an)g(easy)g(consequence)g(of)g(Prop)s (osition)p 0 0 1 TeXcolorrgb 3477 3546 a SDict begin H.S end 3477 3546 a 0 0 1 TeXcolorrgb Fx(2.6)p 0 0 1 TeXcolorrgb 3593 3489 a SDict begin H.R end 3593 3489 a 3593 3546 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.6) cvn H.B /ANN pdfmark end 3593 3546 a 0 TeXcolorgray 28 w Fx(and,)g(in)71 3654 y(particular,)i Fv(\013)574 3668 y Fr(cr)662 3654 y Fu(=)25 b(1)p Fv(=\026)903 3668 y Fr(\000)947 3679 y Fh(R)1002 3654 y Fu(\(0\))p Fx(,)32 b(where)e Fv(\026)1492 3668 y Fr(\000)1536 3679 y Fh(R)1590 3654 y Fu(\(0\))d Fv(>)e Fu(0)30 b Fx(is)h(the)f(largest)i(eigen)m(v)-5 b(alue)32 b(of)f Fk(S)3034 3668 y Fr(\000)3078 3679 y Fh(R)3132 3654 y Fu(\(0\))p Fx(.)68 3795 y(T)-8 b(o)30 b(sho)m(w)h(\(ii\))g(consider)f(the)h(follo)m(wing)h(auxiliary)f (function)f Fv(f)35 b Ft(2)24 b Fv(C)7 b Fu(\([0)p Fv(;)15 b Fu(+)p Ft(1)p Fu(\);)g Fo(R)p Fu(\))32 b Fx(de\034ned)e(b)m(y)71 4030 y(\(3.1\))1623 4030 y SDict begin H.S end 1623 4030 a 1623 4030 a SDict begin 13 H.A end 1623 4030 a 1623 4030 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 1623 4030 a Fv(f)10 b Fu(\()p Fv(x)p Fu(\))26 b(:=)1957 3969 y Fv(e)1999 3936 y Fg(\000)p Fm(a)2091 3882 y Fg(p)p 2150 3882 156 3 v 54 x Fm(bx)p Fr(+)p Fm(c)p 1957 4009 354 4 v 1975 4026 a Ft(p)p 2051 4026 242 4 v 74 x Fv(bx)20 b Fu(+)g Fv(c)2320 4030 y(;)68 4235 y Fx(where)29 b Fv(a;)15 b(b;)g(c)27 b(>)e Fu(0)31 b Fx(are)f(some)h(parameters.)41 b(By)31 b(direct)g(computation)g(of)g (deriv)-5 b(ativ)m(es)31 b(w)m(e)g(get)710 4442 y Fv(f)765 4405 y Fg(0)787 4442 y Fu(\()p Fv(x)p Fu(\))26 b(=)f Ft(\000)p Fv(e)1144 4405 y Fg(\000)p Fm(a)1236 4351 y Fg(p)p 1295 4351 156 3 v 54 x Fm(bx)p Fr(+)p Fm(c)1455 4323 y Fi(\024)1644 4381 y Fv(ab)p 1509 4421 358 4 v 1509 4504 a Fu(2\()p Fv(bx)c Fu(+)f Fv(c)p Fu(\))1896 4442 y(+)2212 4381 y Fv(b)p 1997 4421 468 4 v 1997 4507 a Fu(2\()p Fv(bx)h Fu(+)f Fv(c)p Fu(\))2354 4481 y Fr(3)p Fm(=)p Fr(2)2475 4323 y Fi(\025)2544 4442 y Fv(<)25 b Fu(0)p Fv(;)691 4703 y(f)746 4665 y Fg(00)787 4703 y Fu(\()p Fv(x)p Fu(\))h(=)f Fv(e)1073 4665 y Fg(\000)p Fm(a)1165 4612 y Fg(p)p 1225 4612 156 3 v 1225 4665 a Fm(bx)p Fr(+)p Fm(c)1384 4584 y Fi(\024)1589 4641 y Fv(a)1637 4608 y Fr(2)1676 4641 y Fv(b)1715 4608 y Fr(2)p 1438 4681 468 4 v 1438 4768 a Fu(4\()p Fv(bx)c Fu(+)f Fv(c)p Fu(\))1795 4741 y Fr(3)p Fm(=)p Fr(2)1936 4703 y Fu(+)2149 4641 y(3)p Fv(ab)2281 4608 y Fr(2)p 2037 4681 398 4 v 2037 4765 a Fu(4\()p Fv(bx)h Fu(+)e Fv(c)p Fu(\))2393 4739 y Fr(2)2464 4703 y Fu(+)2737 4641 y(3)p Fv(b)2821 4608 y Fr(2)p 2565 4681 468 4 v 2565 4768 a Fu(4\()p Fv(bx)i Fu(+)f Fv(c)p Fu(\))2922 4741 y Fr(5)p Fm(=)p Fr(2)3042 4584 y Fi(\025)3111 4703 y Fv(>)25 b Fu(0)p Fv(:)71 4912 y Fx(Hence,)37 b(the)f(function)f Fv(f)44 b Fx(satis\034es)36 b(the)f(conditions)h(of)f(Prop)s(osition)p 0 0 1 TeXcolorrgb 2528 4912 a SDict begin H.S end 2528 4912 a 0 0 1 TeXcolorrgb Fx(2.9)p 0 0 1 TeXcolorrgb 2644 4854 a SDict begin H.R end 2644 4854 a 2644 4912 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.9) cvn H.B /ANN pdfmark end 2644 4912 a 0 TeXcolorgray Fx(.)55 b(F)-8 b(urther,)36 b(let)g Fv(\034)3261 4927 y Fd(C)3340 4912 y Fx(and)f Fv(\034)3562 4927 y Fd(T)3643 4912 y Fx(b)s(e)f(the)71 5019 y(unit-sp)s(eed)22 b(mappings)h(whic)m(h)h(parametrise)g Fj(C)f Fx(and)g Fj(T)t Fx(,)i(resp)s(ectiv)m(ely)-8 b(.)40 b(Th)m(us,)24 b(for)g(the)f(functional)h Fu(\010)3546 5034 y Fm(f)3615 5019 y Fx(de\034ned)71 5127 y(in)29 b(\()p 0 0 1 TeXcolorrgb 211 5127 a SDict begin H.S end 211 5127 a 0 0 1 TeXcolorrgb Fx(2.10)p 0 0 1 TeXcolorrgb 373 5070 a SDict begin H.R end 373 5070 a 373 5127 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 373 5127 a 0 TeXcolorgray Fx(\))g(with)f Fv(L)d Fu(=)g Ft(j)p Fj(C)p Ft(j)h Fu(=)f Ft(j)p Fj(T)t Ft(j)k Fx(and)f Fv(f)38 b Fx(as)28 b(in)h(\()p 0 0 1 TeXcolorrgb 1695 5127 a SDict begin H.S end 1695 5127 a 0 0 1 TeXcolorrgb Fx(3.1)p 0 0 1 TeXcolorrgb 1812 5070 a SDict begin H.R end 1812 5070 a 1812 5127 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1812 5127 a 0 TeXcolorgray Fx(\))g(w)m(e)g(ha)m(v)m(e)h(b)m(y)e (Prop)s(osition)p 0 0 1 TeXcolorrgb 2828 5127 a SDict begin H.S end 2828 5127 a 0 0 1 TeXcolorrgb Fx(2.9)p 0 0 1 TeXcolorrgb 2944 5070 a SDict begin H.R end 2944 5070 a 2944 5127 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.9) cvn H.B /ANN pdfmark end 2944 5127 a 0 TeXcolorgray 29 w Fx(the)g(follo)m(wing)j(inequalit)m(y)71 5285 y(\(3.2\))1693 5285 y SDict begin H.S end 1693 5285 a 1693 5285 a SDict begin 13 H.A end 1693 5285 a 1693 5285 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 1693 5285 a Fu(\010)1759 5300 y Fm(f)1804 5285 y Fu([)p Fj(C)p Fu(])26 b Fv(<)f Fu(\010)2097 5300 y Fm(f)2142 5285 y Fu([)p Fj(T)t Fu(])p Fv(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 63 -56 a SDict begin H.S end 63 -56 a 63 -56 a SDict begin H.R end 63 -56 a 63 -56 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end 63 -56 a 0 TeXcolorgray 0 TeXcolorgray 71 19 a Fp(10)1031 b(P)-6 b(A)e(VEL)24 b(EXNER)f(AND)g(VLADIMIR)g(LOTOREICHIK)p 0 TeXcolorgray 71 218 a Fx(Next,)39 b(let)e Fv(\024)f Ft(2)f Fu([0)p Fv(;)15 b Fu(+)p Ft(1)p Fu(\))38 b Fx(b)s(e)d(\034xed,)j (let)f Fv(\026)1566 232 y Fr(\000)1610 243 y Fh(R)1665 218 y Fu(\()p Fv(\024)p Fu(\))f Fv(>)f Fu(0)i Fx(b)s(e)f(the)g(largest) i(eigen)m(v)-5 b(alue)38 b(of)f Fk(S)3204 232 y Fr(\000)3248 243 y Fh(R)3302 218 y Fu(\()p Fv(\024)p Fu(\))g Fx(and)f Fv( )3703 232 y Fr(\000)3747 243 y Fh(R)3837 218 y Ft(2)71 328 y Fv(L)133 295 y Fr(2)172 328 y Fu(\(\000)264 342 y Fm(R)322 328 y Fu(\))h Fx(b)s(e)f(the)h(corresp)s(onding)e (normalized)i(eigenfunction)h(of)e Fk(S)2473 342 y Fr(\000)2517 353 y Fh(R)2571 328 y Fu(\()p Fv(\024)p Fu(\))p Fx(.)61 b(Let)37 b(also)g Fv(\026)3191 344 y Fr(\003)3240 355 y Fh(R)3294 328 y Fu(\()p Fv(\024)p Fu(\))g Fv(>)e Fu(0)i Fx(b)s(e)f(the)71 436 y(largest)d(eigen)m(v)-5 b(alue)34 b(of)e Fk(S)958 452 y Fr(\003)1007 463 y Fh(R)1061 436 y Fu(\()p Fv(\024)p Fu(\))p Fx(.)47 b(By)32 b(Prop)s(ositions)p 0 0 1 TeXcolorrgb 1926 436 a SDict begin H.S end 1926 436 a 0 0 1 TeXcolorrgb Fx(2.7)p 0 0 1 TeXcolorrgb 2042 379 a SDict begin H.R end 2042 379 a 2042 436 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.7) cvn H.B /ANN pdfmark end 2042 436 a 0 TeXcolorgray 32 w Fx(and)p 0 0 1 TeXcolorrgb 2252 436 a SDict begin H.S end 2252 436 a 0 0 1 TeXcolorrgb Fx(2.8)p 0 0 1 TeXcolorrgb 2368 379 a SDict begin H.R end 2368 379 a 2368 436 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.2.8) cvn H.B /ANN pdfmark end 2368 436 a 0 TeXcolorgray 32 w Fx(the)g(function)g Fv( )2975 450 y Fr(\000)3019 461 y Fh(R)3105 436 y Fx(can)h(b)s(e)e(c)m(hosen)i(to)f (b)s(e)71 544 y(p)s(ositiv)m(e)j(and)f(dep)s(ending)f(on)h(the)h (distance)g(from)f(the)h(tip)f(of)h(the)g(cone)g Fu(\000)2744 558 y Fm(R)2836 544 y Fx(only)-8 b(.)53 b(Let)35 b(us)f(in)m(tro)s (duce)g(the)71 652 y(follo)m(wing)e(test)f(function)f Fv( )1047 668 y Fr(\003)1096 679 y Fh(R)1161 652 y Fu(:)g(\003)1279 666 y Fm(R)1362 652 y Ft(!)c Fo(R)1545 666 y Fr(+)1634 652 y Fx(b)m(y)1283 812 y Fv( )1342 828 y Fr(\003)1391 839 y Fh(R)1446 812 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))26 b(:=)g Fv( )1844 826 y Fr(\000)1888 837 y Fh(R)1942 812 y Fu(\()p Fv(r)s Fu(\))p Fv(;)198 b(r)28 b Ft(2)d Fu(\(0)p Fv(;)15 b(R)q Fu(\))p Fv(:)71 972 y Fx(Using)30 b(the)h(form)m(ula)g (in)f(\()p 0 0 1 TeXcolorrgb 958 972 a SDict begin H.S end 958 972 a 0 0 1 TeXcolorrgb Fx(2.7)p 0 0 1 TeXcolorrgb 1075 915 a SDict begin H.R end 1075 915 a 1075 972 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.7) cvn H.B /ANN pdfmark end 1075 972 a 0 TeXcolorgray Fx(\),)h(w)m(e)g(v)m(erify)f (that)421 1189 y Ft(k)p Fv( )525 1205 y Fr(\003)574 1216 y Fh(R)629 1189 y Ft(k)674 1151 y Fr(2)674 1212 y(\003)723 1223 y Fh(R)803 1189 y Fu(=)899 1074 y Fi(Z)982 1100 y Fm(L)946 1263 y Fr(0)1050 1074 y Fi(Z)1133 1100 y Fm(R)1096 1263 y Fr(0)1206 1189 y Ft(j)p Fv( )1290 1205 y Fr(\003)1339 1216 y Fh(R)1393 1189 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))p Ft(j)1609 1151 y Fr(2)1650 1189 y Fv(r)s Fk(d)p Fv(r)s Fk(d)p Fv(s)24 b Fu(=)1995 1074 y Fi(Z)2078 1100 y Fm(L)2041 1263 y Fr(0)2145 1074 y Fi(Z)2229 1100 y Fm(R)2192 1263 y Fr(0)2301 1189 y Ft(j)p Fv( )2385 1203 y Fr(\000)2429 1214 y Fh(R)2484 1189 y Fu(\()p Fv(r)s Fu(\))p Ft(j)2623 1151 y Fr(2)2663 1189 y Fv(r)s Fk(d)p Fv(r)s Fk(d)p Fv(s)g Fu(=)h Ft(k)p Fv( )3112 1203 y Fr(\000)3156 1214 y Fh(R)3211 1189 y Ft(k)3256 1151 y Fr(2)3256 1211 y(\000)3300 1222 y Fh(R)3380 1189 y Fu(=)g(1)p Fv(:)71 1391 y Fx(Let)h Fv(\033)281 1405 y Fr(\000)325 1416 y Fh(R)405 1391 y Fx(and)f Fv(\033)629 1407 y Fr(\003)678 1418 y Fh(R)758 1391 y Fx(based)h(on)f Fv(\034)1168 1406 y Fd(C)1238 1391 y Fx(and)g Fv(\034)1450 1406 y Fd(T)1496 1391 y Fx(,)i(resp)s(ectiv)m(ely)-8 b(,)28 b(parametrize)f Fu(\000)2610 1405 y Fm(R)2693 1391 y Fx(and)f Fu(\003)2929 1405 y Fm(R)3012 1391 y Fx(as)g(in)g(\()p 0 0 1 TeXcolorrgb 3256 1391 a SDict begin H.S end 3256 1391 a 0 0 1 TeXcolorrgb Fx(2.6)p 0 0 1 TeXcolorrgb 3373 1334 a SDict begin H.R end 3373 1334 a 3373 1391 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 3373 1391 a 0 TeXcolorgray Fx(\).)39 b(Emplo)m(ying)71 1499 y(the)30 b(iden)m(tit)m(y)j(\()p 0 0 1 TeXcolorrgb 599 1499 a SDict begin H.S end 599 1499 a 0 0 1 TeXcolorrgb Fx(2.8)p 0 0 1 TeXcolorrgb 716 1442 a SDict begin H.R end 716 1442 a 716 1499 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.8) cvn H.B /ANN pdfmark end 716 1499 a 0 TeXcolorgray Fx(\),)d(w)m(e)h(\034nd)71 1753 y SDict begin H.S end 71 1753 a 71 1753 a SDict begin 13 H.A end 71 1753 a 71 1753 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 71 1753 a 859 1662 a Ft(j)p Fv(\033)936 1676 y Fr(\000)980 1687 y Fh(R)1035 1662 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))21 b Ft(\000)f Fv(\033)1390 1676 y Fr(\000)1434 1687 y Fh(R)1489 1662 y Fu(\()p Fv(r)1568 1625 y Fg(0)1591 1662 y Fv(;)15 b(s)1674 1625 y Fg(0)1697 1662 y Fu(\))p Ft(j)1757 1625 y Fr(2)1823 1662 y Fu(=)24 b(\()p Fv(r)g Ft(\000)19 b Fv(r)2152 1625 y Fg(0)2175 1662 y Fu(\))2210 1625 y Fr(2)2270 1662 y Fu(+)h Fv(r)s(r)2449 1625 y Fg(0)2472 1662 y Ft(j)p Fv(\034)2537 1678 y Fd(C)2581 1662 y Fu(\()p Fv(s)p Fu(\))g Ft(\000)g Fv(\034)2845 1678 y Fd(C)2889 1662 y Fu(\()p Fv(s)2967 1625 y Fg(0)2990 1662 y Fu(\))p Ft(j)3050 1625 y Fr(2)3090 1662 y Fv(;)850 1815 y Ft(j)p Fv(\033)927 1831 y Fr(\003)976 1842 y Fh(R)1030 1815 y Fu(\()p Fv(r)m(;)15 b(s)p Fu(\))22 b Ft(\000)d Fv(\033)1385 1831 y Fr(\003)1434 1842 y Fh(R)1489 1815 y Fu(\()p Fv(r)1568 1777 y Fg(0)1591 1815 y Fv(;)c(s)1674 1777 y Fg(0)1697 1815 y Fu(\))p Ft(j)1757 1777 y Fr(2)1823 1815 y Fu(=)24 b(\()p Fv(r)g Ft(\000)19 b Fv(r)2152 1777 y Fg(0)2175 1815 y Fu(\))2210 1777 y Fr(2)2270 1815 y Fu(+)h Fv(r)s(r)2449 1777 y Fg(0)2472 1815 y Ft(j)p Fv(\034)2537 1830 y Fd(T)2582 1815 y Fu(\()p Fv(s)p Fu(\))h Ft(\000)f Fv(\034)2847 1830 y Fd(T)2892 1815 y Fu(\()p Fv(s)2970 1777 y Fg(0)2994 1815 y Fu(\))p Ft(j)3054 1777 y Fr(2)3094 1815 y Fv(:)71 1740 y Fx(\(3.3\))71 1974 y(F)-8 b(urther,)30 b(de\034ne)g(for)g Fv(\024)c Ft(\025)f Fu(0)30 b Fx(the)h(function)71 2184 y(\(3.4\))1010 2184 y SDict begin H.S end 1010 2184 a 1010 2184 a SDict begin 13 H.A end 1010 2184 a 1010 2184 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.4) cvn /DEST pdfmark end 1010 2184 a Fv(K)1087 2198 y Fm(\024)1132 2184 y Fu(\()p Fv(r)m(;)15 b(r)1289 2146 y Fg(0)1313 2184 y Fu(\))25 b(:=)1505 2122 y Fv(r)s(r)1593 2089 y Fg(0)p 1505 2162 111 4 v 1510 2246 a Fu(4)p Fv(\031)1625 2114 y Fi(\000)1663 2184 y Fu(\010)1729 2199 y Fm(f)1774 2184 y Fu([)p Fj(T)t Fu(])c Ft(\000)f Fu(\010)2059 2199 y Fm(f)2104 2184 y Fu([)p Fj(C)p Fu(])2209 2114 y Fi(\001)2248 2184 y Fv(;)197 b(r)m(;)15 b(r)2592 2146 y Fg(0)2641 2184 y Ft(2)25 b Fu([0)p Fv(;)15 b(R)q Fu(])p Fv(;)68 2382 y Fx(where)26 b Fu(\010)393 2397 y Fm(f)466 2382 y Fx(is)h(as)h(in)g(\()p 0 0 1 TeXcolorrgb 802 2382 a SDict begin H.S end 802 2382 a 0 0 1 TeXcolorrgb Fx(2.10)p 0 0 1 TeXcolorrgb 964 2325 a SDict begin H.R end 964 2325 a 964 2382 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 964 2382 a 0 TeXcolorgray Fx(\))f(and)g Fv(f)37 b Fx(as)27 b(in)h(\()p 0 0 1 TeXcolorrgb 1529 2382 a SDict begin H.S end 1529 2382 a 0 0 1 TeXcolorrgb Fx(3.1)p 0 0 1 TeXcolorrgb 1646 2325 a SDict begin H.R end 1646 2325 a 1646 2382 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1646 2382 a 0 TeXcolorgray Fx(\))f(with)g Fv(a)p Fu(\()p Fv(r)m(;)15 b(r)2117 2349 y Fg(0)2141 2382 y Fu(\))26 b(:=)f Fv(\024)p Fx(,)k Fv(b)p Fu(\()p Fv(r)m(;)15 b(r)2625 2349 y Fg(0)2649 2382 y Fu(\))26 b(:=)f Fv(r)s(r)2919 2349 y Fg(0)2941 2382 y Fx(,)j(and)f Fv(c)p Fu(\()p Fv(r)m(;)15 b(r)3364 2349 y Fg(0)3389 2382 y Fu(\))25 b(:=)g(\()p Fv(r)17 b Ft(\000)d Fv(r)3792 2349 y Fg(0)3815 2382 y Fu(\))3850 2349 y Fr(2)3890 2382 y Fx(.)71 2490 y(Note)36 b(that)g(the)g(function) f Fv(K)1097 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Fk(S)1575 2937 y Fr(\000)1619 2948 y Fh(R)1674 2923 y Fu(\()p Fv(\024)p Fu(\))p Fv( )1855 2937 y Fr(\000)1899 2948 y Fh(R)1955 2923 y Fv(;)15 b( )2054 2937 y Fr(\000)2098 2948 y Fh(R)2153 2923 y Fu(\))2188 2937 y Fr(\000)2232 2948 y Fh(R)2307 2923 y Fu(+)2398 2808 y Fi(Z)2481 2834 y Fm(R)2444 2997 y Fr(0)2553 2808 y Fi(Z)2636 2834 y Fm(R)2600 2997 y Fr(0)2709 2923 y Fv(K)2786 2937 y Fm(\024)2831 2923 y Fu(\()p Fv(r)m(;)g(r)2988 2885 y Fg(0)3012 2923 y Fu(\))p Fv( )3106 2937 y Fr(\000)3150 2948 y Fh(R)3206 2923 y Fu(\()p Fv(r)s Fu(\))p Fv( )3379 2937 y Fr(\000)3423 2948 y Fh(R)3477 2923 y Fu(\()p Fv(r)3556 2885 y Fg(0)3580 2923 y Fu(\))p Fk(d)p Fv(r)s Fk(d)p Fv(r)3797 2885 y Fg(0)456 3124 y Fv(>)25 b Fu(\()p Fk(S)638 3138 y Fr(\000)682 3149 y Fh(R)736 3124 y Fu(\()p Fv(\024)p Fu(\))p Fv( )917 3138 y Fr(\000)961 3149 y Fh(R)1017 3124 y Fv(;)15 b( )1116 3138 y Fr(\000)1160 3149 y Fh(R)1215 3124 y Fu(\))1250 3138 y Fr(\000)1294 3149 y Fh(R)1374 3124 y Fu(=)25 b Fv(\026)1525 3138 y Fr(\000)1569 3149 y Fh(R)1623 3124 y Fu(\()p Fv(\024)p Fu(\))p Ft(k)p Fv( )1849 3138 y Fr(\000)1893 3149 y Fh(R)1949 3124 y Ft(k)1994 3086 y Fr(2)1994 3146 y(\000)2038 3157 y Fh(R)2119 3124 y Fu(=)g Fv(\026)2270 3138 y Fr(\000)2314 3149 y Fh(R)2368 3124 y Fu(\()p Fv(\024)p Fu(\))p Fv(:)71 3330 y Fx(No)m(w,)36 b(let)f Fv(\013)d Ft(\025)f Fv(\013)693 3344 y Fr(cr)788 3330 y Fu(=)g(\()p Fv(\026)980 3344 y Fr(\000)1024 3355 y Fh(R)1078 3330 y Fu(\(0\)\))1228 3297 y Fg(\000)p Fr(1)1324 3330 y Fx(.)52 b(Then)33 b(w)m(e)i(get)g(b)m(y)f(the)g(ab)s(o)m(v)m(e)i(inequalit)m(y) f(for)f Fv(\024)e Fu(=)f(0)j Fx(that)h Fv(\026)3576 3346 y Fr(\003)3625 3357 y Fh(R)3679 3330 y Fu(\(0\))e Fv(>)71 3438 y(\026)126 3452 y Fr(\000)170 3463 y Fh(R)224 3438 y Fu(\(0\))27 b(=)e(1)p Fv(=\013)610 3452 y Fr(cr)699 3438 y Ft(\025)g Fu(1)p Fv(=\013)32 b Fx(and)d(using)h(Prop)s(osition)p 0 0 1 TeXcolorrgb 1879 3438 a SDict begin H.S end 1879 3438 a 0 0 1 TeXcolorrgb Fx(2.6)p 0 0 1 TeXcolorrgb 1995 3381 a SDict begin 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3989 y Fr(1)1468 3975 y Fu(\()p Fk(H)1567 3991 y Fm(\013;)p Fr(\003)1681 4002 y Fh(R)1736 3975 y Fu(\))26 b Fv(<)f Ft(\000)p Fv(\024)2016 3937 y Fr(2)2081 3975 y Fu(=)g Fv(E)2244 3989 y Fr(1)2283 3975 y Fu(\()p Fk(H)2382 3989 y Fm(\013;)p Fr(\000)2491 4000 y Fh(R)2546 3975 y Fu(\))p Fv(:)71 4086 y SDict begin H.S end 71 4086 a 71 4086 a SDict begin 13 H.A end 71 4086 a 71 4086 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.2) cvn /DEST pdfmark end 71 4086 a 113 x Fx(3.2.)47 b FC(Pro)s(of)33 b(of)g(Theorem)p 0 0 1 TeXcolorrgb 1103 4199 a SDict begin H.S end 1103 4199 a 0 0 1 TeXcolorrgb FC(1.3)p 0 0 1 TeXcolorrgb 1236 4138 a SDict begin H.R end 1236 4138 a 1236 4199 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.1.3) cvn H.B /ANN pdfmark end 1236 4199 a 0 TeXcolorgray FC(.)46 b Fx(Before)29 b(pro)m(ving)g(Theorem)p 0 0 1 TeXcolorrgb 2302 4199 a SDict begin H.S end 2302 4199 a 0 0 1 TeXcolorrgb Fx(1.3)p 0 0 1 TeXcolorrgb 2418 4141 a SDict 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TeXcolorgray Fx(,)g(Satz)h(9.26)17 b(\(b\)])33 b(w)m(e)g(obtain)g(that)h Fu(#)p Fv(\033)2094 2497 y Fr(d)2137 2483 y Fu(\()p Fk(H)2236 2497 y Fm(\013;)p Fr(\000)2345 2508 y Fh(R)2400 2483 y Fu(\))p Fv(;)15 b Fu(#)p Fv(\033)2603 2497 y Fr(d)2647 2483 y Fu(\()p Fk(H)2746 2500 y Fm(\013;)p Fr(\003)2860 2511 y Fh(R)2915 2483 y Fu(\))30 b Ft(\025)f Fu(1)k Fx(for)f(all)i Fv(R)c(>)f Fu(0)k Fx(large)71 2591 y(enough)d(and)g(that)71 2849 y SDict begin H.S end 71 2849 a 71 2849 a SDict begin 13 H.A end 71 2849 a 71 2849 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 71 2849 a 1166 2759 a Fv(E)1233 2773 y Fr(1)1273 2759 y Fu(\()p Fk(H)1372 2773 y Fm(\013;)p Fr(\000)1481 2784 y Fh(R)1536 2759 y Fu(\))25 b Ft(!)h Fu(inf)c Fv(\033)s Fu(\()p Fk(H)1993 2773 y Fm(\013;)p Fr(\000)2102 2781 y Ff(1)2172 2759 y Fu(\))p Fv(;)182 b(R)26 b Ft(!)f Fu(+)p Ft(1)p Fv(;)1156 2899 y(E)1223 2913 y Fr(1)1263 2899 y Fu(\()p Fk(H)1362 2915 y Fm(\013;)p Fr(\003)1476 2926 y Fh(R)1531 2899 y 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Fr(\003)1816 3761 y Ff(1)1886 3737 y Fu(\))j Ft(\024)g Fu(inf)d Fv(\033)s Fu(\()p Fk(H)2322 3751 y Fm(\013;)p Fr(\000)2431 3759 y Ff(1)2501 3737 y Fu(\))p Fv(:)71 3918 y Fx(Moreo)m(v)m(er,)45 b(b)m(y)40 b([)p 0.2 0.2 0.6 TeXcolorrgb 669 3918 a SDict begin H.S end 669 3918 a 0.2 0.2 0.6 TeXcolorrgb Fx(4)p 0.2 0.2 0.6 TeXcolorrgb 715 3861 a SDict begin H.R end 715 3861 a 715 3918 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.BEL14_JPA) cvn H.B /ANN pdfmark end 715 3918 a 0 TeXcolorgray Fx(,)j(Thm.)69 b(3.2])42 b(holds)e Fv(\033)1552 3932 y Fr(d)1595 3918 y Fu(\()p Fk(H)1694 3932 y Fm(\013;)p Fr(\000)1803 3940 y Ff(1)1873 3918 y Fu(\))i Ft(6)p Fu(=)g Fo(?)e Fx(and)g(the)g(p)s(oin)m(t)h Fu(inf)21 b Fv(\033)s Fu(\()p Fk(H)3054 3932 y Fm(\013;)p Fr(\000)3163 3940 y Ff(1)3234 3918 y Fu(\))40 b Fx(is,)j(in)d(fact,)k(the)71 4029 y(lo)m(w)m(est)32 b(eigen)m(v)-5 b(alue)32 b Fv(E)845 4043 y Fr(1)885 4029 y Fu(\()p Fk(H)984 4043 y Fm(\013;)p 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