diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index bfb6bb7..f0c0ed8 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -26716,7 +26716,7 @@ o )show DeRhamComplex \begin{chunk}{domain DERHAM DeRhamComplex} )abbrev domain DERHAM DeRhamComplex -++ Author: Larry A. Lambe +++ Author: Larry A. Lambe and Kurt Pagani ++ Date : 01/26/91. ++ Revised : 12/01/91. ++ Description: @@ -26783,6 +26783,14 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where ++ to the metric g. dot : (%,%,SMR) -> R ++ compute the inner product of two differential forms w.r.t. g + proj : (%,NNI) -> % + ++ projection to homogeneous terms of degree p + interiorProduct : (Vector(R),%,SMR) -> % + ++ calculates the interior product i_X(a) of the vector field X + ++ with the differential form a (w.r.t. metric g) + lieDerivative : (Vector(R),%,SMR) -> % + ++ calculates the Lie derivative L_X(a) of the differential + ++ form a with respect to the vector field X (w.r.t. metric g) Implement == ASY add Rep := ASY @@ -26824,6 +26832,7 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where err1:="CoefRing has no IntegralDomain" err2:="Not Implemented" err3:="Degenerate metric" + err4:="Index out of range" -- coord space dimension dim(f) == dim @@ -26838,36 +26847,12 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where y:= x pretend List(NNI) [j for j in 1..#y | y.j=a] - -- compute factors for hodgeStar - facs(cc:Record(k:EAB,c:R),g:SMR):R == - not CoefRing has IntegralDomain => error(err1) - not diagonal? g => error(err2) - G:=reduce("*",[g(j,j) for j in 1..dim]::List(R)) - G=0 => error(err3) - idx:=pos(cc.k,0) -- pos of 0 since already flipped - eps:=concat(pos(cc.k,1),pos(cc.k,0))::List(NNI) - dom:=[j for j in 1..dim]::List(NNI) - sgn:=sign(coercePreimagesImages([dom,eps])::Permutation(NNI))::R - if idx ^= [] then - fg:R:=sgn*reduce("*",[1/g(j,j) for j in idx]::List(R)) - else - fg:R:=sgn - fg*sqrt(abs(G))*cc.c - - -- export - hodgeStar(x,g) == - t:=terms(x) - s:=[copy(r) for r in t] -- we need a copy of x - for j in 1..#t repeat - s.j.k := flip(s.j.k) - s.j.c := facs(s.j,g) -- builtin g - s pretend % - -- compute dot of singletons dot1(r:Record(k:EAB,c:R),s:Record(k:EAB,c:R),g:SMR):R == not CoefRing has IntegralDomain => error(err1) test(r.k ^= s.k) => 0::R idx := pos(r.k,1) + idx = [] => r.c * s.c reduce("*",[1/g(j,j) for j in idx]::List(R))*r.c*s.c -- export @@ -26876,6 +26861,49 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where ty:=terms(y) reduce("+",[dot1(tx.j,ty.j,g) for j in 1..#tx]) + -- export + hodgeStar(x,g) == + not CoefRing has IntegralDomain => error(err1) + not diagonal? g => error(err2) + v := sqrt(abs(determinant(g))) -- volume factor + v = 0 => error(err3) + t:=terms(x) + s:=[copy(r) for r in t] -- we need a copy of x + for j in 1..#t repeat + s.j.k := flip(s.j.k) + fs:=[s.j] pretend % + ft:=[t.j] pretend % + s.j.c := s.j.c * v * dot1(t.j,t.j,g)/leadingCoefficient(ft*fs) + s pretend % + + -- export + proj(x,p) == + p < 0 or p > dim => error(err4) + t := terms(x) + idx := [j for j in 1..#t | #pos(t.j.k,1)=p] + s := [copy(t.j) for j in idx::List(NNI)] + s pretend % + + interiorProduct(v,x,g) == + not CoefRing has IntegralDomain => error(err1) + f := reduce("+",[generator(i)$% for i in 1..dim]::List(%)) + t := terms(f) + for j in 1..dim repeat + t.(dim-j+1).c := g(j,j)*v(j) -- reverse order + f -- term manipulations are destructive + dg:R := determinant(g) + sg:R := dg/abs(dg) + if odd?(dim) then + m:R := sg + else + m:R := (-1)**degree(x)*sg + m * hodgeStar(f*hodgeStar(x,g),g) + + lieDerivative(v,x,g) == + a:= exteriorDifferential(interiorProduct(v,x,g)) + b:= interiorProduct(v,exteriorDifferential(x),g) + a+b + coerce(a):O == a = 0$Rep => 0$I::O ta := terms a diff --git a/books/bookvol5.pamphlet b/books/bookvol5.pamphlet index 1b581e3..706a2b9 100644 --- a/books/bookvol5.pamphlet +++ b/books/bookvol5.pamphlet @@ -103,7 +103,7 @@ of effort. We would like to acknowledge and thank the following people: "Stephen Watt Jaap Weel Juergen Weiss" "M. Weller Mark Wegman James Wen" "Thorsten Werther Michael Wester R. Clint Whaley" -"James T. Wheeler" John M. Wiley Berhard Will" +"James T. Wheeler John M. Wiley Berhard Will" "Clifton J. Williamson Stephen Wilson Shmuel Winograd" "Robert Wisbauer Sandra Wityak Waldemar Wiwianka" "Knut Wolf Liu Xiaojun Clifford Yapp" diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 66d3083..2d07bec 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -6327,6 +6327,102 @@ Proc ISSAC 97 pp172-175 (1997) \end{chunk} +\index{Polyakov, S.P.} +\begin{chunk}{axiom.bib} +@article{Poly11, + author = "Polyadov, S.P.", + title = "Indefinite summation of rational functions with factorization + of denominators", + year = "2011", + month = "November", + journal = "Programming and Computer Software", + volume = "37", + number = "6", + pages = "322--325", + paper = "Poly11.pdf", + abstract = " + A computer algebra algorithm for indefinite summation of rational + functions based on complete factorization of denominators is + proposed. For a given$f$, the algorithm finds two rational functions +$g$,$r$such that$f=g(x+1)-g(x)+r$and the degree of the denominator + of$r$is minimal. A modification of the algorithm is also proposed + that additionally minimizes the degree of the denominator of +$g$. Computational complexity of the algorithms without regard to + denominator factorization is shown to be$O(m^2)$, where$m$is the + degree of the denominator of$f$." +} + +\end{chunk} + +\index{Gerhard, J.} +\index{Giesbrecht, M.} +\index{Storjohann, A.} +\index{Zima, E.V.} +\begin{chunk}{axiom.bib} +@inproceedings{Gerh03, + author = "Gerhard, J. and Giesbrecht, M. and Storjohann, A. and Zima, E.V.", + title = "Shiftless decomposition and polynomial-time rational summation", + booktitle = "Proceedings of ISSAC'03", + year = "2003", + pages = "119--126", + paper = "Gerh03.pdf", + abstract = " + New algorithms are presented for computing the dispersion set of two + polynomials over {\bf Q} and for {\sl shiftless} factorization. Together + with a summability criterion by Abramov, these are applied to get a + polynomial-time algorithm for indefinite rational summation, using a + sparse representation of the output." +} + +\end{chunk} + +\index{Abramov, S.A.} +\begin{chunk}{axiom.bib} +@article{Abra85, + author = "Abramov, S.A.", + title = "Separation of variables in rational functions", + year = "1985", + journal = "USSR Computational Mathematics and Mathematical Physics", + volume = "25", + number = "5", + pages = "99--102", + paper = "Abra85.pdf", + abstract = " +The problem of expanding a rational function of several variables into +terms with separable variables is formulated. An algorithm for solving +this problem is given. Programs which implement this algorithm can +occur in sets of algebraic alphabetical transformations on a computer +and can be used to reduce the multiplicity of sums and integrals of +rational functions for investigating differential equations with +rational right-hand sides etc." +} + +\end{chunk} + +\index{Abramov, S.A.} +\begin{chunk}{axiom.bib} +@article{Abra71, + author = "Abramov, S.A.", + title = "On the summation of rational functions", + year = "1971", + journal = "USSR Computational Mathematics and Mathematical Physics", + volume = "11", + number = "4", + pages = "324--330", + paper = "Abra71.pdf", + abstract = " + An algorithm is given for solving the following problem: let +$F(x_1,\ldots,x_n)$be a rational function of the variables +$x_i$with rational (read or complex) coefficients; to see if + there exists a rational function$G(v,w,x_2,\ldots,x_n)$with + coefficients from the same field, such that + $\sum_{x_1=v}^w{F(x_1,\ldots,x_n)} = G(v,w,x_2,\ldots,x_n)$ + for all integral values of$v \le w$. If$G$exists, to obtain it. + Realization of the algorithm in the LISP language is discussed." +} + +\end{chunk} + \index{Er\"ocal, Bur\c{c}in} \begin{chunk}{axiom.bib} @article{Eroc10, @@ -6436,6 +6532,8 @@ Proc ISSAC 97 pp172-175 (1997) implemented in Sigma to solve multi-summation problems efficiently. For instacne, the extended Sigma package has been applied successively to provide a computer-assisted proof of Stembridge's TSPP Theorem." +} + \end{chunk} \index{Kauers, Manuel} diff --git a/changelog b/changelog index 3af7c95..cd8b464 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20141010 kxp src/axiom-website/patches.html 20141010.01.kxp.patch +20141010 kxp books/bookvolbib add references +20141010 kxp src/input/derham3.input test Pagani's functions +20141010 kxp src/input/Makefile add derham3.input tests +20141010 kxp books/bookvol10.3 add Pagani's functions to DERHAM +20141010 tpd books/bookvol5 fix typo 20141008 tpd src/axiom-website/patches.html 20141008.03.tpd.patch 20141008 tpd books/bookvolbib add a section on Symbolic Summation 20141008 jtw src/axiom-website/patches.html 20141008.02.jtw.patch diff --git a/patch b/patch index 60bc62c..1583b4b 100644 --- a/patch +++ b/patch @@ -1,3 +1,3 @@ -books/bookvolbib add a section on Symbolic Summation +books/bookvol10.3 add Pagani's functions to DERHAM -Collect references to papers on symbolic summation using Karr's method +Additional functions in DERHAM diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index c79b4ca..7a66eeb 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4678,6 +4678,8 @@ books/bookvol10.3 DERHAM: add code for differential forms books/bookvol10.1 add chapter on differential forms 20141008.03.tpd.patch books/bookvolbib add a section on Symbolic Summation +20141010.01.kxp.patch +books/bookvol10.3 add Pagani's functions to DERHAM diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 6b9ee5e..72a999e 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -314,7 +314,8 @@ REGRESSTESTS= ackermann.regress \ danzwill.regress danzwill2.regress davenport.regress \ davis.regress \ decimal.regress defintef.regress defintrf.regress \ - derham.regress derham2.regress derivefail.regress \ + derham.regress derham2.regress derham3.regress \ + derivefail.regress \ dfloat.regress dftrig.regress \ dhmatrix.regress \ dhtri.regress directproduct.regress distexpr.regress \ @@ -730,7 +731,8 @@ FILES=${OUT}/ackermann.input \ ${OUT}/danzwill.input${OUT}/danzwill2.input ${OUT}/davenport.input \${OUT}/davis.input ${OUT}/decimal.input${OUT}/defs.input \ ${OUT}/defintef.input${OUT}/defintrf.input ${OUT}/derham.input \ -${OUT}/derham2.input ${OUT}/derivefail.input${OUT}/de2re.input \ + ${OUT}/derham2.input${OUT}/derham3.input \ + ${OUT}/derivefail.input${OUT}/de2re.input \ ${OUT}/dfloat.input${OUT}/dftrig.input ${OUT}/dhmatrix.input \${OUT}/dhtri.input ${OUT}/directproduct.input \${OUT}/distexpr.input ${OUT}/divisor.input${OUT}/donsimple.input \ @@ -1090,7 +1092,8 @@ DOCFILES= \ ${DOC}/de2re.input.dvi${DOC}/decimal.input.dvi \ ${DOC}/defintef.input.dvi${DOC}/defintrf.input.dvi \ ${DOC}/defs.input.dvi${DOC}/derham.input.dvi \ - ${DOC}/derham2.input.dvi${DOC}/derivefail.input.dvi \ + ${DOC}/derham2.input.dvi${DOC}/derham3.input.dvi \ + ${DOC}/derivefail.input.dvi \${DOC}/dfloat.input.dvi ${DOC}/dftrig.input.dvi \${DOC}/dhmatrix.input.dvi ${DOC}/dhtri.input.dvi \${DOC}/directproduct.input.dvi ${DOC}/distexpr.input.dvi \ diff --git a/src/input/derham3.input.pamphlet b/src/input/derham3.input.pamphlet new file mode 100644 index 0000000..ac1595f --- /dev/null +++ b/src/input/derham3.input.pamphlet @@ -0,0 +1,486 @@ +\documentclass{article} +\usepackage{axiom} +\setlength{\textwidth}{400pt} +\begin{document} +\title{\$SPAD/src/input derham3.input} +\author{Kurt Pagini} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\begin{chunk}{*} +)set break resume +)spool derham3.output +)set message test on +)set message auto off +)clear all + +--S 1 of 47 +coefRing:=Integer +--R +--R +--R (1) Integer +--R Type: Domain +--E 1 + +--S 2 of 47 +R3:List Symbol:=[x,y,z] +--R +--R +--R (2) [x,y,z] +--R Type: List(Symbol) +--E 2 + +--S 3 of 47 +X:=DERHAM(coefRing,R3) +--R +--R +--R (3) DeRhamComplex(Integer,[x,y,z]) +--R Type: Domain +--E 3 + +--S 4 of 47 +[dx,dy,dz]:=[generator(i)$X for i in 1..3] +--R +--R +--R (4) [dx,dy,dz] +--R Type: List(DeRhamComplex(Integer,[x,y,z])) +--E 4 + +--S 5 of 47 +f:BOP:=operator('f) +--R +--R +--R (5) f +--R Type: BasicOperator +--E 5 + +--S 6 of 47 +g:BOP:=operator('g) +--R +--R +--R (6) g +--R Type: BasicOperator +--E 6 + +--S 7 of 47 +h:BOP:=operator('h) +--R +--R +--R (7) h +--R Type: BasicOperator +--E 7 + +--S 8 of 47 +a:BOP:=operator('a) +--R +--R +--R (8) a +--R Type: BasicOperator +--E 8 + +--S 9 of 47 +b:BOP:=operator('b) +--R +--R +--R (9) b +--R Type: BasicOperator +--E 9 + +--S 10 of 47 +c:BOP:=operator('c) +--R +--R +--R (10) c +--R Type: BasicOperator +--E 10 + +--S 11 of 47 +U:BOP:=operator('U) +--R +--R +--R (11) U +--R Type: BasicOperator +--E 11 + +--S 12 of 47 +V:BOP:=operator('V) +--R +--R +--R (12) V +--R Type: BasicOperator +--E 12 + +--S 13 of 47 +W:BOP:=operator('W) +--R +--R +--R (13) W +--R Type: BasicOperator +--E 13 + +--S 14 of 47 +v:=vector[U(x,y,z),V(x,y,z),W(x,y,z)] +--R +--R +--R (14) [U(x,y,z),V(x,y,z),W(x,y,z)] +--R Type: Vector(Expression(Integer)) +--E 14 + +--S 15 of 47 +sigma:=f(x,y,z)*dx+g(x,y,z)*dy+h(x,y,z)*dz +--R +--R +--R (15) h(x,y,z)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 15 + +--S 16 of 47 +theta:=a(x,y,z)*dx*dy+b(x,y,z)*dx*dz+c(x,y,z)*dy*dz +--R +--R +--R (16) c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 16 + +--S 17 of 47 +d ==> exteriorDifferential +--R +--R Type: Void +--E 17 + +--S 18 of 47 +G:=diagonalMatrix([1,1,1]) +--R +--R +--R +1 0 0+ +--R | | +--R (18) |0 1 0| +--R | | +--R +0 0 1+ +--R Type: Matrix(Integer) +--E 18 + +--S 19 of 47 +interiorProduct(v,sigma,G) +--R +--R +--R (19) W(x,y,z)h(x,y,z) + V(x,y,z)g(x,y,z) + U(x,y,z)f(x,y,z) +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 19 + +--S 20 of 47 +interiorProduct(v,theta,G) +--R +--R +--R (20) +--R (W(x,y,z)b(x,y,z) + V(x,y,z)a(x,y,z))dx +--R + +--R (W(x,y,z)c(x,y,z) - U(x,y,z)a(x,y,z))dy +--R + +--R (- V(x,y,z)c(x,y,z) - U(x,y,z)b(x,y,z))dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 20 + +--S 21 of 47 +t1:=d interiorProduct(v,dx*dy*dz,G) +--R +--R +--R (21) (W (x,y,z) + V (x,y,z) + U (x,y,z))dx dy dz +--R ,3 ,2 ,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 21 + +--S 22 of 47 +hodgeStar(t1,G) -- should be div(v) +--R +--R +--R (22) W (x,y,z) + V (x,y,z) + U (x,y,z) +--R ,3 ,2 ,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 22 + +--S 23 of 47 +eta:=lieDerivative(v,theta,G) +--R +--R +--R (23) +--R - W(x,y,z)c (x,y,z) - V(x,y,z)c (x,y,z) - U(x,y,z)b (x,y,z) +--R ,3 ,2 ,2 +--R + +--R U(x,y,z)a (x,y,z) - c(x,y,z)W (x,y,z) - c(x,y,z)V (x,y,z) +--R ,3 ,3 ,2 +--R + +--R a(x,y,z)U (x,y,z) - b(x,y,z)U (x,y,z) +--R ,3 ,2 +--R * +--R dy dz +--R + +--R - V(x,y,z)c (x,y,z) - W(x,y,z)b (x,y,z) - U(x,y,z)b (x,y,z) +--R ,1 ,3 ,1 +--R + +--R - V(x,y,z)a (x,y,z) - b(x,y,z)W (x,y,z) - a(x,y,z)V (x,y,z) +--R ,3 ,3 ,3 +--R + +--R - c(x,y,z)V (x,y,z) - b(x,y,z)U (x,y,z) +--R ,1 ,1 +--R * +--R dx dz +--R + +--R 2W(x,y,z)c (x,y,z) - 2W(x,y,z)b (x,y,z) + W(x,y,z)a (x,y,z) +--R ,1 ,2 ,3 +--R + +--R - V(x,y,z)a (x,y,z) - U(x,y,z)a (x,y,z) - b(x,y,z)W (x,y,z) +--R ,2 ,1 ,2 +--R + +--R c(x,y,z)W (x,y,z) - a(x,y,z)V (x,y,z) - a(x,y,z)U (x,y,z) +--R ,1 ,2 ,1 +--R * +--R dx dy +--R + +--R (- V(x,y,z)c (x,y,z) + V(x,y,z)b (x,y,z) - V(x,y,z)a (x,y,z))dx dz +--R ,1 ,2 ,3 +--R + +--R (U(x,y,z)c (x,y,z) - U(x,y,z)b (x,y,z) + U(x,y,z)a (x,y,z))dy dz +--R ,1 ,2 ,3 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 23 + +--S 24 of 47 +proj(dx+dy*dz+dz*dy*dz,2) +--R +--R +--R (24) dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 24 + +--S 25 of 47 +proj(sigma+theta,1) +--R +--R +--R (25) h(x,y,z)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 25 + +--S 26 of 47 +dim(sigma) +--R +--R +--R (26) 3 +--R Type: PositiveInteger +--E 26 + +--S 27 of 47 +degree(sigma) +--R +--R +--R (27) 1 +--R Type: PositiveInteger +--E 27 + +)clear all + + +--S 28 of 47 +coefRing:=Integer +--R +--R +--R (1) Integer +--R Type: Domain +--E 28 + +--S 29 of 47 +S2:List Symbol := [r,theta] +--R +--R +--R (2) [r,theta] +--R Type: List(Symbol) +--E 29 + +--S 30 of 47 +Y:=DERHAM(coefRing,S2) +--R +--R +--R (3) DeRhamComplex(Integer,[r,theta]) +--R Type: Domain +--E 30 + +--S 31 of 47 +g:=diagonalMatrix([1,r^2]) +--R +--R +--R +1 0 + +--R (4) | | +--R | 2| +--R +0 r + +--R Type: Matrix(Polynomial(Integer)) +--E 31 + +--S 32 of 47 +F:=operator 'F +--R +--R +--R (5) F +--R Type: BasicOperator +--E 32 + +--S 33 of 47 +d ==> exteriorDifferential +--R +--R Type: Void +--E 33 + +--S 34 of 47 +[dr,dtheta]:=[generator(i)$Y for i in 1..2] +--R +--R +--R (7) [dr,dtheta] +--R Type: List(DeRhamComplex(Integer,[r,theta])) +--E 34 + +--S 35 of 47 +F0:=F(r,theta)*1::Y +--R +--R +--R (8) F(r,theta) +--R Type: DeRhamComplex(Integer,[r,theta]) +--E 35 + +--S 36 of 47 +F1:=d F0 +--R +--R +--R (9) F (r,theta)dtheta + F (r,theta)dr +--R ,2 ,1 +--R Type: DeRhamComplex(Integer,[r,theta]) +--E 36 + +--S 37 of 47 +F2:=hodgeStar(F1,g) +--R +--R +--R +-------+ +--R | 2 +--R \|abs(r ) F (r,theta) +-------+ +--R ,2 | 2 +--R (10) - ---------------------- dr + \|abs(r ) F (r,theta)dtheta +--R 2 ,1 +--R r +--R Type: DeRhamComplex(Integer,[r,theta]) +--E 37 + +--S 38 of 47 +F3:=d F2 +--R +--R +--R (11) +--R 2 2 2 2 +--R abs(r )F (r,theta) + r abs(r )F (r,theta) + r abs(r )F (r,theta) +--R ,2,2 ,1,1 ,1 +--R ----------------------------------------------------------------------- dr +--R +-------+ +--R 2 | 2 +--R r \|abs(r ) +--R * +--R dtheta +--R Type: DeRhamComplex(Integer,[r,theta]) +--E 38 + +--S 39 of 47 +LaplaceF:=hodgeStar(F3,g) +--R +--R +--R (12) +--R 2 2 2 2 +--R abs(r )F (r,theta) + r abs(r )F (r,theta) + r abs(r )F (r,theta) +--R ,2,2 ,1,1 ,1 +--R ----------------------------------------------------------------------- +--R 4 +--R r +--R Type: DeRhamComplex(Integer,[r,theta]) +--E 39 + +--S 40 of 47 +LapF:=LaplaceF::Expression(Integer) +--R +--R +--R (13) +--R 2 2 2 2 +--R abs(r )F (r,theta) + r abs(r )F (r,theta) + r abs(r )F (r,theta) +--R ,2,2 ,1,1 ,1 +--R ----------------------------------------------------------------------- +--R 4 +--R r +--R Type: Expression(Integer) +--E 40 + +--S 41 of 47 +subst(LapF,abs(r^2)=r^2) +--R +--R +--R 2 +--R F (r,theta) + r F (r,theta) + rF (r,theta) +--R ,2,2 ,1,1 ,1 +--R (14) ------------------------------------------------- +--R 2 +--R r +--R Type: Expression(Integer) +--E 41 + +--S 42 of 47 +R:=Expression(Integer) +--R +--R +--R (15) Expression(Integer) +--R Type: Domain +--E 42 + +--S 43 of 47 +terms:Y -> List Record(k:EAB,c:R) +--R +--R Type: Void +--E 43 + +--S 44 of 47 +terms(a)==a pretend List Record(k:EAB,c:R) +--R +--R Type: Void +--E 44 + +--S 45 of 47 +terms(dr) +--R +--R Compiling function terms with type DeRhamComplex(Integer,[r,theta]) +--R -> List(Record(k: ExtAlgBasis,c: Expression(Integer))) +--R +--R (18) [[k= [1,0],c= 1]] +--R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) +--E 45 + +--S 46 of 47 +terms(dtheta) +--R +--R +--R (19) [[k= [0,1],c= 1]] +--R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) +--E 46 + +--S 47 of 47 +terms(F1) +--R +--R +--R (20) [[k= [0,1],c= F (r,theta)],[k= [1,0],c= F (r,theta)]] +--R ,2 ,1 +--R Type: List(Record(k: ExtAlgBasis,c: Expression(Integer))) +--E 47 +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document}