# LaTeX2HTML 95.1 (Fri Jan 20 1995)
# Associate image original text (scrambled) with physical files.

$key = q/{_inline}$Avertv;SPM_gt;=lambdavertv;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img32.gif">'; 
$key = q/{_inline}$k=1ldotsn${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img6.gif">'; 
$key = q/{displaymath}underlinex=sum_kx_kunderlinee_k{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img10.gif">'; 
$key = q/{_inline}$f(a)=f(b)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img128.gif">'; 
$key = q/{_inline}$L=ifracddx${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img126.gif">'; 
$key = q/{_inline}$langlex|A|xrangle^*=langlex|A|xrangle${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img75.gif">'; 
$key = q/{displaymath}tildeA|v_k;SPM_gt;=lambda_k|v_k;SPM_gt;{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img101.gif">'; 
$key = q/{_inline}$left.v_kright)_i^*${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img66.gif">'; 
$key = q/{displaymath}mboxdetleftlbrackarraylllla_11-lambda;SPM_amp;a_12a_21;SPM_amp;a_22-lambda;SPM_amp;a_32;SPM_amp;;SPM_amp;;SPM_amp;a_nn-lambdaarrayrightrbrack{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img56.gif">'; 
$key = q/{_inline}$B=A-lambdaI${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img47.gif">'; 
$key = q/{_inline}$(A,B)^dagger=B^daggerA^dagger${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img131.gif">'; 
$key = q/{_inline}$fracddxf(x)=g(x)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img117.gif">'; 
$key = q/{_inline}$U^+U=I${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img97.gif">'; 
$key = q/{_inline}$A^+=A^T${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img108.gif">'; 
$key = q/{displaymath}A=leftlbrackarrayccccclambda_1;SPM_amp;lambda_2;SPM_amp;;SPM_amp;0;SPM_amp;;SPM_amp;lambda_3;SPM_amp;0;SPM_amp;;SPM_amp;ddots;SPM_amp;;SPM_amp;;SPM_amp;;SPM_amp;lambda_narrayrightrbrack{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img42.gif">'; 
$key = q/{_inline}$(-1)^n${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img60.gif">'; 
$key = q/{_inline}$;SPM_lt;v_1vertv_2^prime;SPM_gt;=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img92.gif">'; 
$key = q/{_inline}$fracd^2dx^2${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img133.gif">'; 
$key = q/{_inline}$U^+${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img95.gif">'; 
$key = q/{displaymath}I=int^b_ag(x)fracddxf(x)dx=langlegvertDvertfrangle{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img119.gif">'; 
$key = q/{_inline}$langlev_2|v_1rangle=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img83.gif">'; 
$key = q/{_inline}$(A-lambdaI)=${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img55.gif">'; 
$key = q/{_inline}$vertv_1;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img90.gif">'; 
$key = q/{_inline}$A^+=A${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img107.gif">'; 
$key = q/{displaymath}lambda=lambda_1,lambda_2,ldotslambda_n{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img61.gif">'; 
$key = q/{_inline}$Bunderlinev=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img52.gif">'; 
$key = q/{_inline}$(A-lambda_iI)|v_k;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img111.gif">'; 
$key = q/{_inline}$(beta)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img80.gif">'; 
$key = q/{_inline}$delta_lk=1${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img25.gif">'; 
$key = q/{_inline}$f(x)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img115.gif">'; 
$key = q/{_inline}$uparrow${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img100.gif">'; 
$key = q/{_inline}$lambda_1${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img36.gif">'; 
$key = q/{_inline}$tildeA${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img102.gif">'; 
$key = q/{displaymath}tildeA=U^*leftlbrackarraycccclambda_1;SPM_amp;lambda_2;SPM_amp;00;SPM_amp;;SPM_amp;;SPM_amp;lambda_narrayrightrbrackU{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img99.gif">'; 
$key = q/{_inline}$lambda_k=1${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img64.gif">'; 
$key = q/{_inline}$left(nabla^2right)^dagger=-L^daggerL^dagger=-L,L=nabla^2${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img130.gif">'; 
$key = q/{_inline}$ntimesn${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img2.gif">'; 
$key = q/{displaymath}A=U^+mboxDiag.leftlbrackarrayccclambda_1;SPM_amp;ddots;SPM_amp;;SPM_amp;lambda_narrayrightrbrackU{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img106.gif">'; 
$key = q/{_inline}$langleyvertAvertxrangle${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img17.gif">'; 
$key = q/{displaymath}|x;SPM_gt;=sum_k=1^nx_k|v_k;SPM_gt;{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img103.gif">'; 
$key = q/{_inline}$ifracddx${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img123.gif">'; 
$key = q/{_inline}$underlinex^TfifteenbxtiAunderlinex${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img16.gif">'; 
$key = q/{displaymath}left(lambda_1-lambda_2right)langlev_2|v_1rangle=0{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img81.gif">'; 
$key = q/{_inline}$underlinee_1=(1,0,0ldots0)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img35.gif">'; 
$key = q/{displaymath}lambda_k=fracsum_i,j=1^nleft.v_k^*right)_iA_ijleft.v_kright)_jsum_i=1^nleft.v_k^*right)_ileft.v_kright)_i{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img76.gif">'; 
$key = q/{_inline}$langlex|=langley|${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img74.gif">'; 
$key = q/{_inline}$A^T=mboxTransposehspace2emleft(A^Tright)_ij=A_ji${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img26.gif">'; 
$key = q/{displaymath}U^+=leftlbrackarrayccccccleft.v_1right)_1;SPM_amp;;SPM_amp;;SPM_amp;;SPM_amp;;SPM_amp;left.v_nright)_1left.v_1right)_2left.v_1right)_3left.v_1right)_n;SPM_amp;;SPM_amp;;SPM_amp;;SPM_amp;;SPM_amp;left.v_nright)_narrayrightrbrack{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img96.gif">'; 
$key = q/{_inline}$lbraceunderlinee_krbrace${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img21.gif">'; 
$key = q/{_inline}$langleyvert${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img18.gif">'; 
$key = q/{_inline}$lambda^n${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img59.gif">'; 
$key = q/{_inline}$A^T=A${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img109.gif">'; 
$key = q/{_inline}$nabla^2=fracd^2dx^2=-L,L${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img129.gif">'; 
$key = q/{_inline}$(A-lambdaI)=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img62.gif">'; 
$key = q/{_inline}$Aunderlinev=lambdaunderlinev${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img30.gif">'; 
$key = q/{_inline}$A^*=mboxComplexConjugatehspace1emleft(A^*right)_ij=A^*_ij${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img27.gif">'; 
$key = q/{_inline}$underlinee_2ldotsunderlinee_n${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img39.gif">'; 
$key = q/{_inline}$underlinee_underlinek${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img5.gif">'; 
$key = q/{_inline}$underlinev${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img33.gif">'; 
$key = q/{displaymath}langlev_k|v_krangle=sum_i=1^nleft.v_k^*right)_iv_kleft.right)_i=1{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img93.gif">'; 
$key = q/{_inline}$underlinex${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img13.gif">'; 
$key = q/{_inline}$v_i${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img49.gif">'; 
$key = q/{_inline}$langleyvertxrangle${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img19.gif">'; 
$key = q/{_inline}$A^+=mboxHermiteanConjugatehspace1emleft(A^+right)_ij=A^*_ji${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img28.gif">'; 
$key = q/{_inline}$(B)=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img53.gif">'; 
$key = q/{_inline}$underlinermboundaryconditions${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img127.gif">'; 
$key = q/{displaymath}langlev_k|v_lrangle=0hspace2emlneqk{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img94.gif">'; 
$key = q/{_inline}$A=A^+${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img29.gif">'; 
$key = q/{_inline}$vertv_2;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img91.gif">'; 
$key = q/{_inline}$bullet${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img63.gif">'; 
$key = q/{_inline}$lambda_i${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img43.gif">'; 
$key = q/{_inline}$underlinex=left(x_1,x_2,ldotsx_nright)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img9.gif">'; 
$key = q/{_inline}$lambda_k${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img112.gif">'; 
$key = q/{_inline}$delta_lk=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img24.gif">'; 
$key = q/{_inline}$nabla^2${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img132.gif">'; 
$key = q/{_inline}$(A-lambdaI)underlinev=0${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img45.gif">'; 
$key = q/{_inline}$A_ij${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img11.gif">'; 
$key = q/{_inline}$mboxLHS=lambda_k,mboxRHS${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img69.gif">'; 
$key = q/{_inline}$underlinee_k${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img7.gif">'; 
$key = q/{displaymath}v_1=v_2=ldots=v_k=ldots=v_n=0{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img50.gif">'; 
$key = q/{_inline}$|v_k;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img104.gif">'; 
$key = q/{_inline}$fracddx${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img120.gif">'; 
$key = q/{_inline}$n^2${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img3.gif">'; 
$key = q/{_inline}$(n)${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img114.gif">'; 
$key = q/{_inline}$lambda_1=lambda_2${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img86.gif">'; 
$key = q/{_inline}$lambda${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img34.gif">'; 
$key = q/{displaymath}int^b_ag(x)fracddxf(x),dx=-int^b_af(x)fracddxg(x),dx{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img125.gif">'; 
$key = q/{_inline}$vertv_2^prime;SPM_gt;${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img89.gif">'; 
$key = q/{displaymath}A=leftlbrackarrayccccccclambda_1;SPM_amp;0;SPM_amp;0;SPM_amp;0;SPM_amp;0;SPM_amp;ldots;SPM_amp;0000;SPM_amp;;SPM_amp;A;SPM_amp;N;SPM_amp;Y0;SPM_amp;;SPM_amp;V;SPM_amp;A;SPM_amp;L;SPM_amp;U;SPM_amp;E0arrayrightrbrack{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img38.gif">'; 
$key = q/{_inline}$lambda_2ldotslambda_n${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img40.gif">'; 
$key = q/{displaymath}underlinex=|x;SPM_gt;hboxcalledaket{displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img15.gif">'; 
$key = q/{_inline}$lbracee_krbrace${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img22.gif">'; 
$key = q/{displaymath}biglbrackf(x)g(x)bigrbrack^b_a=f(b),g(b)-f(a),g(a){displaymath}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img122.gif">'; 
$key = q/{_inline}$underlinermvectors${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=MIDDLE ALT="" SRC="img110.gif">'; 
$key = q/{_inline}$i=1ldotsn${_inline}/;
$cached_env_img{$key} ='<IMG  ALIGN=BOTTOM ALT="" SRC="img67.gif">'; 

1;

