{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 2 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 78 "Corrigendum to: ''A note on the \+ non-colorability threshold of a random graph''" }}{PARA 256 "" 0 "" {TEXT -1 58 "Ioannis Giotis, Alexis C. Kaporis and Lefteris M. Kirousi s" }}{PARA 256 "" 0 "" {TEXT -1 50 "Department of Computer Engineering and Informatics" }}{PARA 256 "" 0 "" {TEXT -1 66 "University of Patra s, University Campus, GR 256 04 Patras, Greece" }}{PARA 256 "" 0 "" {TEXT -1 51 "e-mail: \{giotis, kaporis, kirousis\}@ceid.upatras.gr" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 11 "Proposition" }}{PARA 0 "" 0 "" {TEXT -1 27 "The Hessian of the funcion " }{XPPEDIT 18 0 "-ln(alpha*beta*(1- alpha-beta)/((alpha*(1-x-y))^(1-x-y)*(beta*y)^y*((1-alpha-beta)*x)^x)) ;" "6#,$-%#lnG6#**%&alphaG\"\"\"%%betaGF),(F)F)F(!\"\"F*F,F)*()*&F(F), (F)F)%\"xGF,%\"yGF,F),(F)F)F1F,F2F,F))*&F*F)F2F)F2F))*&,(F)F)F(F,F*F,F )F1F)F1F)F,F," }{TEXT -1 21 " is positive definite" }}{PARA 0 "" 0 "" {TEXT -1 34 "for all points in the open region " }{TEXT 259 1 "\{" } {TEXT -1 1 "(" }{XPPEDIT 18 0 "alpha,beta,x,y" "6&%&alphaG%%betaG%\"xG %\"yG" }{TEXT -1 4 ") in" }{TEXT 258 1 " " }{XPPEDIT 18 0 "(0, 1)^4, a lpha+beta < 1, x+y < 1" "6%*$6$\"\"!\"\"\"\"\"%2,&%&alphaGF&%%betaGF&F &2,&%\"xGF&%\"yGF&F&" }{TEXT 257 1 "\}" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 5 "Proof" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f := -ln(alpha*beta*(1-alpha-beta)/(((alpha*(1-x-y))^(1-x-y))*((beta*y)^ y)*((1-alpha-beta)*x)^x)):" "6#>%\"fG,$-%#lnG6#**%&alphaG\"\"\"%%betaG F+,(F+F+F*!\"\"F,F.F+*()*&F*F+,(F+F+%\"xGF.%\"yGF.F+,(F+F+F3F.F4F.F+)* &F,F+F4F+F4F+)*&,(F+F+F*F.F,F.F+F3F+F3F+F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We now compute the Hessia n of f. The linalg package is needed." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace hav e been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Hf:=matrix(simplify(hessian(f,[x,y,beta,alpha])));" "6# >%#HfG-%'matrixG6#-%)simplifyG6#-%(hessianG6$%\"fG7&%\"xG%\"yG%%betaG% &alphaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#HfG-%'matrixG6#7&7&*(,&! \"\"\"\"\"%\"yGF-F-,(F,F-%\"xGF-F.F-F,F0F,,$*&F-F-F/F,F,*&F-F-,(F,F-%& alphaGF-%%betaGF-F,,$*(,&F,F-F6F-F-F5F,F4F,F,7&F1*(,&F,F-F0F-F-F/F,F.F ,*&F-F-F6F,,$*&F-F-F5F,F,7&F3F=,$*(,FS*(,:F0F-F.F-FDF-*(FFF-F0F-F6F-F,*(FFF-F6F- F.F-F,*(FFF-F0F-F5F-F,FKF-**FFF-F5F-F6F-F0F-F-**FFF-F5F-F.F-F6F-F-FOF- FPF-*(FFF-F5F-F.F-F,F-F5FRF4FR" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "We now verify that the leading principal \+ minor determinants are indeed " }{XPPEDIT 18 0 "A[1],A[2],A[3],A[4];" "6&&%\"AG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "A[1]:= (1-y)/( x*(1-x-y)):" "6#>&%\"AG6#\"\"\"*&,&F'F'%\"yG!\"\"F'*&%\"xGF',(F'F'F-F+ F*F+F'F+" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "A[2]:= 1/(x*y*(1-x-y)):" "6#>&%\"AG6#\"\"#*&\"\"\"F)*(%\"xGF)%\"yGF),(F)F)F+!\"\"F,F.F)F." }} {PARA 0 "> " 0 "" {XPPEDIT 19 1 "A[3]:= ( ((1-alpha-beta)*(1-y)-beta*( 1-x))^2+2*(1-alpha-beta)*beta*(1-x-y)) /((beta^2)*x*y*(1-x-y)*((1-alph a-beta)^2) ):" "6#>&%\"AG6#\"\"$*&,&*$,&*&,(\"\"\"F.%&alphaG!\"\"%%be taGF0F.,&F.F.%\"yGF0F.F.*&F1F.,&F.F.%\"xGF0F.F0\"\"#F.**F7F.,(F.F.F/F0 F1F0F.F1F.,(F.F.F6F0F3F0F.F.F.*,F1F7F6F.F3F.,(F.F.F6F0F3F0F.,(F.F.F/F0 F1F0F7F0" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "A[4]:= ((alpha*(1-y)^2-x *(1-beta-y)-beta*y*(1-x-y))^2 +2*x*y*(1-x-y)*( (1-beta-y)^2+ beta^2) ) /((alpha^2)*(beta^2)*x*y*(1-x-y)*((1-alpha-beta)^2)*(1-y)^2):" "6#>&% \"AG6#\"\"%*&,&*$,(*&%&alphaG\"\"\"*$,&F.F.%\"yG!\"\"\"\"#F.F.*&%\"xGF .,(F.F.%%betaGF2F1F2F.F2*(F7F.F1F.,(F.F.F5F2F1F2F.F2F3F.*,F3F.F5F.F1F. ,(F.F.F5F2F1F2F.,&*$,(F.F.F7F2F1F2F3F.*$F7F3F.F.F.F.*0F-F3F7F3F5F.F1F. ,(F.F.F5F2F1F2F.,(F.F.F-F2F7F2F3,&F.F.F1F2F3F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "is (A[1] = det(submatrix(Hf,1..1,1..1)));" "6#-%#isG 6#/&%\"AG6#\"\"\"-%$detG6#-%*submatrixG6%%#HfG;F*F*;F*F*" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "is (A[2] = det(submatrix(Hf,1..2,1..2)));" "6#- %#isG6#/&%\"AG6#\"\"#-%$detG6#-%*submatrixG6%%#HfG;\"\"\"F*;F3F*" }} {PARA 0 "> " 0 "" {XPPEDIT 19 1 "is (A[3] = det(submatrix(Hf,1..3,1..3 )));" "6#-%#isG6#/&%\"AG6#\"\"$-%$detG6#-%*submatrixG6%%#HfG;\"\"\"F*; F3F*" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "is (A[4] = det(submatrix(Hf,1 ..4,1..4)));" "6#-%#isG6#/&%\"AG6#\"\"%-%$detG6#-%*submatrixG6%%#HfG; \"\"\"F*;F3F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "7 5 13 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }