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255 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 255 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 "Normal" -1 258 1 {CSTYLE "" -1 -1 "Helve tica" 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 } } {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 26 "\nThis is clifford6_13.m ws\n" }}{PARA 258 "" 0 "" {TEXT -1 61 "(Created: October 9, 2002)\n(La st revised: December 27, 2002)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1092 "################################################################ #############\n# \+ #\n#DISCLAIMER: \+ #\n# \+ #\n#THERE IS NO WARRANTY FOR TH E CLIFFORD, BIGEBRA, Cliplus, Octonion, GTP #\n#PACKAGES TO THE EX TENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE #\n#STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE #\n# PROGRAM \"AS IS\" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IM PLIED, #\n#INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF ME RCHANTABILITY #\n#AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE R ISK AS TO THE QUALITY #\n#AND PERFORMANCE OF THE PROGRAM IS WITH YO U. SHOULD THE PROGRAM PROVE #\n#DEFECTIVE, YOU ASSUME THE COST O F ALL NECESSARY SERVICING, REPAIR OR #\n#CORRECTION. \+ #\n############### ##############################################################\n" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 322 "This is a listing (without examples) of all procedures in a Maple package \+ called 'CLIFFORD' (Version 6, Copyright 1995-2003 by Rafal Ablamowic z, Tennessee Technological University), and Bertfried Fauser, Univers it\"at Konstanz, for Maple 6. User will know which version he/she is u sing by using the 'version()' function." }}{PARA 0 "" 0 "" {TEXT -1 258 "\nThe following new procedures have been added on October 20, 200 2:\n\nrd_basmonom - generates a random Grassmann basis monomial \nrd_monom - generates a random Grassmann monomial \nrd_c lipolynom - generates a random Grassmann polynomial \n" }}{PARA 0 "" 0 "" {TEXT 277 55 "The following procedures can use index such as K or -K:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "cmul[K](p1,p2,...,pn); ##Clifford product of p1,p2,...,pn in Cl(K) " }}{PARA 0 "" 0 "" {TEXT -1 81 "&c[K](p1,p2,...,pn); ##Clifford produ ct of p1,p2,...,pn in Cl(K) (ampersand form)" }}{PARA 0 "" 0 "" {TEXT -1 112 "cmulQ[K](p1,p2,...,pn); ##Clifford product of p1,p2,...,pn in \+ Cl(K) (here K is expected to be a diagonal matrix)" }}{PARA 0 "" 0 "" {TEXT -1 126 "&cQ[K](p1,p2,...,pn); ##Clifford product of p1,p2,...,pn in Cl(K) (here K is expected to be a diagonal matrix), ampersand form " }}{PARA 0 "" 0 "" {TEXT -1 56 "cexp[K](p,N); ## exponential of p in \+ Cl(K) up to order N" }}{PARA 0 "" 0 "" {TEXT -1 102 "cexpQ[K](p,N); ## exponential of p in Cl(K) up to order N (here K is expected to be a d iagonal matrix)" }}{PARA 0 "" 0 "" {TEXT -1 53 "climinpoly[K](p); ## m inimal polynomial of p in Cl(K)" }}{PARA 0 "" 0 "" {TEXT -1 91 "sexp[K ](p,N); ## exponential of p in Cl(K) up to order N modulo the minimal \+ polynomial of p" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 96 "The following procedures can use name K or a numeric mul tiple of a name as an optional argument:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "LC(p1,p2,K); ##left contraction o f p2 by p1 w.r.t. K\nRC(p1,p2,K); ##right contraction of p1 by p2 w.r. t. K" }}{PARA 0 "" 0 "" {TEXT -1 68 "cmulNUM(m1,m2,K); ##Clifford (num eric) product of m1 and m2 in Cl(K)" }}{PARA 0 "" 0 "" {TEXT -1 41 "re version(p,K); ##reversion of p in Cl(K)" }}{PARA 0 "" 0 "" {TEXT -1 43 "cinv(p,K); ##Clifford inverse of p in Cl(K)" }}{PARA 0 "" 0 "" {TEXT -1 73 "LCQ(p1,p2,K); ##left contraction of p2 by p1 w.r.t. diago nal entries in K" }}{PARA 0 "" 0 "" {TEXT -1 74 "RCQ(p1,p2,K); ##right contraction of p1 by p2 w.r.t. diagonal entries in K" }}{PARA 0 "" 0 "" {TEXT -1 46 "conjugation(p,K); ## conjugation of p in Cl(K)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 86 "The foll lowing procedures can pass on name or a numeric multiple of a name via a list:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "type([p,K],nilpotent); ## checks if p is nilpotent in Cl(K)\ntype ([p,K],idempotent); ## checks if p is idempotent in Cl(K)" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 580 "\nProcedur es that define types: `type/climon`, `type/clipolynom`, `type/climatri x` as well as other procedures such as 'reorder', 'wedge', etc., have \+ been substantially revised to improve efficiency and speed of the pack age. This work has been done together with Bertfried Fauser, Universit \"at Konstanz, in Cookeville on October 5, 2001. \n\nThis version incl udes \"Bigebra\" package that has been created together with Bertfried Fauser, Universit\"at Konstanz, Konstanz, Germany. Additional help pa ges have been written and added to the database that explain the usage of this package." }{TEXT 276 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" } }{PARA 258 "" 0 "" {TEXT -1 150 "An additional feature in this version is an ability to display and change environmental variables. They can be displayed with procedure CLIFFORD_ENV.\n" }}{PARA 258 "" 0 "" {TEXT -1 387 "NOTE: Big change from version 4 is that now types clibas mon, climon, and clipolynom are all exclusive whereas in version 4 the y were inclusive, that is, `type/clipolynom` included `type/climon` wh ich in turn included `type/clibasmon`. \n\nThis package is made to run under Maple 6. It is available on a server of the Department of Mat hematics, Tennessee Technological University, at: \n" }}{PARA 258 " " 0 "" {TEXT -1 69 " http://math.tntech.ed u/rafal/clifford/ " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 130 "In order to create a Maple file 'Clifford.m' contai ning the 'CLIFFORD' package, execute this worksheet.\n\nTo load the pa ckage type:" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 17 ">with(Clifford); " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 189 "You will know if the package has been \+ loaded because a list with Clifford procedures will be displayed on th e screen. To check the current version of the package, at the Maple p rompt type " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 12 ">version( );" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 35 "Rafal Ablamowicz, Ph.D. and Chair " }}{PARA 258 "" 0 "" {TEXT -1 35 "Department of Mathematics, Box 5054" }}{PARA 258 "" 0 "" {TEXT -1 36 "Tennessee Technological University " }} {PARA 258 "" 0 "" {TEXT -1 21 "Cookeville, TN 38505 " }}{PARA 258 "" 0 "" {TEXT -1 24 "rablamowicz@tntech.edu " }}{PARA 258 "" 0 "" {TEXT -1 25 "phone: USA (931) 372-3569" }}{PARA 258 "" 0 "" {TEXT -1 23 "fax : USA (931) 372-6353" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "restart:\nunprotect('Clifford','e','qi','qj',' qk','Id','w');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 989 "Clifford:=module ()\n###################################\nexport `&m`, Bsignature, CLIF FORD_ENV, Kfield, LC, LCQ, RC, RCQ, RHnumber, adfmatrix, all_sigs, bet a_minus, beta_plus, buildm, bygrade, c_conjug, cbasis, cdfmatrix, cexp , cexpQ, cinv, clibilinear, clicollect, clidata, clilinear, climinpoly , cliparse, cliremove, clisolve, clisort, cliterms, cmul, cmulNUM, cmu lQ, cmulRS, cmulgen, cocycle, commutingelements, conjugation,ddfmatrix , diagonalize, displayid, extract, factoridempotent, find1str, findbas is, gradeinv, init, isVahlenmatrix, isproduct, makealiases, makeclibas mon, matKrepr, maxgrade, maxindex, mdfmatrix, minimalideal, ord, perms ign, pseudodet, q_conjug, qdisplay, qinv, qmul, qnorm, reorder, revers ion, rmulm, rot3d, scalarpart, sexp, specify_constants, spinorKbasis, \+ spinorKrepr, squaremodf, subs_clipolynom, useproduct, vectorpart, vers ion, wedge, wexp, rd_clibasmon, rd_climon, rd_clipolynom;\n########### ########################\nlocal setup;\noption package, load=setup;\n " }}{PARA 258 "" 0 "" {TEXT -1 84 "No. 1. Name 'version' stores inform ation about the current version of the package. " }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 25 "Typical use: version( ); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1521 "version:= proc()\noptions `Copyright (c) 1995-2003 by Rafal Ab lamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `La st revised: November 1, 2002`;\nprint(`+++++++++++++++++++++++++++++++ ++++++++++++`);\nprint(`CLIFFORD - A Maple 6 Package for Clifford Alge bras`); \nprint(`(Version 6 with global variable _prolevel and \"Bigeb ra\" package)`);\nprint(`\"Bigebra\" package written with Bertfried Fa user, Universit\"at Konstanz`);\nprint(`Last revised: November 1, 2002 (Source file: clifford6_12.mws)`);\nprint(`Copyright 1995-2003 by Raf al Ablamowicz (*) and Bertfried Fauser ($)`);\nprint(``);\nprint(`(*) \+ Department of Mathematics, Box 5054`);\nprint(` Tennessee Technolog ical University, Cookeville, TN 38505`);\nprint(` tel: USA (931) 37 2-3569, fax: USA (931) 372-6353`);\nprint(` rablamowicz@tntech.edu` );\nprint(` http://math.tntech.edu/rafal/Clifford/`);\nprint(`($) U niversit\"at Konstanz, Fachbereich Physik, Fach M678`);\nprint(` 78 457 Konstanz, Germany`);\nprint(` Bertfried.Fauser@uni-konstanz.de` );\nprint(` http://kaluza.physik.uni-konstanz.de/~fauser/`); \+ \nprint(``);\nprint(`If you are a Clifford algebra pro, assign 'true' \+ to '_prolevel' and see`);\nprint(`how much faster your computations wi ll be! But watch your syntax!`);\nprint(`Use 'useproduct' to change va lue of _default_Clifford_product in Cl(B) from`);\nprint(`cmulRS when \+ B is symbolic to cmulNUM when B is numeric. Type ?cmul for help.`); \n print(`++++++++This is CLIFFORD version 6, library file : Clifford.m++ ++++++`);\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 17 "No. 2. Proced ure " }{TEXT 282 17 "specify_constants" }{TEXT -1 503 " allows user to specify any new symbolic constants, e.g., a, b, c, B, e.t.c, which a re to be known to Maple. The originally known constants are stored in a global, non-protected variable 'constants' and must be saved separa tely, if needed. This procedure is needed when sorting or collecting \+ multivariate Clifford polynomials containing expressions like 'aa*eiwe j' in which 'aa' is intended to be a constant and 'eiwej' is intended \+ to be a Clifford basis monomial with indices i and j. Before using " }{TEXT 281 7 "clisort" }{TEXT -1 4 " or " }{TEXT 280 10 "clicollect" } {TEXT -1 350 " user should make any additional constants of length 2 o r more known to Maple as shown below. If these constants of length 2 \+ or more are not defined as Maple constants, then some procedures might yield error messages (although an attempt has been made to avoid this problem). Constants of length one are automatically assumed to be Map le constants. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 46 "Typical use: specify_constants(a, b, B, aa); " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 186 "NOTE: from now on, extra spaces have been added for the Reader's convenience in \+ the sequence of input variables as in the above example. These spaces \+ are not needed or required by Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "specify_constants:=proc(a1::anyt hing) global constants;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\nconstants:=op(\{constants,args\});\nprintf(\"Maple now knows t he following constant(s): %q\\n\",constants);\nreturn NULL;\nend proc: \n" }}{PARA 258 "" 0 "" {TEXT -1 21 "No. 3. The procedure " }{TEXT 283 6 "cbasis" }{TEXT -1 793 " writes a canonical basis for a Clifford algebra Cl(B) over a vector space V endowed with a bilinear form B. \+ The dimension of V is specified by a Maple global variable 'dim' where 1 <= dim <= 9. This procedure can be used with one or two arguments \+ as, for example, in cbasis(4) or cbasis(4, 2). In the first case, it \+ returns a list of all basis elements in the Clifford algebra Cl(4). In the second case, it returns a list of basis elements in the 2-vector \+ subspace of Cl(4). Below, 'Id' stands for the algebra unit element and 'w' denotes wedge/exterior product in the Clifford algebra. An option 'even' allows one to create a basis in the even subalgebra of the giv en Clifford algebra as in cbasis(3, 'even'). In fact, 'even' can be r eplaced with any name which evaluates to a string. \n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1875 "cbasis:=proc(a1::nonnegint,a2::\{string,symb ol,nonnegint\})\nlocal i,k,X,XX,YY,L,Leven,Lodd,bas,nxt,ind,start; glo bal choose,e;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz an d Bertfried Fauser. All rights reserved.`,remember;\ndescription `Last revised: November 1, 2002`;\n######################################## #####\nif a1>9 then \n error \"first argument must be between 0 and \+ 9 inclusive but received %1 instead\",a1 \nend if;\nif a1=0 and nargs= 1 then return [Id] end if;\nif nargs=2 and type(a2,\{string,symbol\}) \+ then do\n L:=procname(a1):\n Leven:=[Id]:Lodd:=[]:\n if nops(L) \+ > 1 then\n for i from 2 to nops(L) do\n if type(length(L[i]),o dd) then Leven:=[op(Leven),L[i]] else\n \+ Lodd:=[op(Lodd),L[i]]\n end if \n end do \n end if; \+ \nif args[2]='even' then return Leven \n elif args[2]='odd' then ret urn Lodd\n else error \"second argument must be an integer or a stri ng 'even' or 'odd' but received %1 instead\",args[2]\nend if\nend do \+ \nend if;\nfor k from 0 to a1 do \n X[k]:=combinat[choose]([seq(i,i =1..a1)],k) \nend do;\nif not nargs = 1 and not nargs = 2 then \n er ror \"one or two arguments are needed as input but received %0 instead \",args\nelif nargs = 1 then XX:=[seq(op(X[k]),k=0..a1)] \nelse if not a2 >= 0 or not a2 <= a1 then \n error \"second argument must satisfy: 0 <= 'a2' <= %1 but received %2 instead\",a1,a2 \nelse XX:=X[a2] \nen d if \nend if;\nYY:=array(1..nops(XX),[]);start:=1:\nif XX[1] = [] th en \n YY[1]:=Id; \n start:=2 \nend if;\nfor k from start to nops(X X) do\n ind:=XX[k][1];\n if ind=10 then \n bas:=e||0 else \+ bas:=e||ind \n end if;\nfor i from 2 to nops(XX[k]) do \n ind:=X X[k][i]:\n if ind=10 then nxt:=e||0 else nxt:=e||ind end if:\n \+ bas:=cat(bas,\"w\",nxt): \n end do;\nYY[k]:=bas;\nend do:\nYY:=co nvert(YY,list);\nprotect(op(YY)); #protect basis monomials\nreturn YY \nend proc:\n " }}{PARA 258 "" 0 "" {TEXT -1 17 "No. 4. Procedure " } {TEXT 284 8 "find1str" }{TEXT -1 327 " finds all locations of the firs t string of length one in the second string of length at least one. It returns a set of these positions. If the first string is not found t hen it returns \{0\}. This procedure is primarily for internal use in \+ 'type/clibasmon' and 'cliparse'. \nTypical use: find1str(e,e1we2we3); \+ find1str(w,e1we2);" }{MPLTEXT 0 21 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 662 "find1str:=proc(a1::symbol,a2::symbol) local ns,p,p1, ap,le2;\nglobal _prolevel;\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`,remember;\ndesc ription `Last revised: November 1, 2002`;\n########################### ##################\nle2:=length(a2):\nif _prolevel=false then\nif leng th(a1) <> 1 or le2<1 then \n error \"first string must be of length \+ 1 but received %1 instead\",a1 \nend if;\nend if;\np:=SearchText(a1,a 2):\nap:=\{p\}:p1:=p:\nwhile p<>0 and p10 then p1:=p1+p;\n \+ ap:=ap union \{p1\} \n end if;\nend do;\nreturn ap\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 16 "No. 5. Function " }{TEXT 285 8 "cliparse" }{TEXT -1 349 " checks user's input for correct spell ing of basis monomials. When unable to decide if the given input is c orrect, it tells the user to check spelling or define the given string as a Maple constant. If the spelling is correct, it returns true; if \+ it is not correct, it returns a set of suspect words.\n \nTypical use: cliparse(e1+e2we3+2*Pi*B[1,2]);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1179 "cliparse:=proc(a1::anything) local x,S1,S2,p,S;\nglobal _proleve l,_scalartypes;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz \+ and Bertfried Fauser. All rights reserved.`;\ndescription `Last revise d: November 1, 2002`;\n#############################################\n if _prolevel then return true end if;\nif type(a1,_scalartypes) then r eturn true end if;\np:=remove(type,a1,_scalartypes):S1:=\{op(p)\}:\nfo r x in S1 do \n if type(x,_scalartypes) or type(x,clibasmon) then S 1:=S1 minus \{x\} end if;\nend do; \nS2:=map(op,S1); \nfor x in S2 do \+ \n if type(x,_scalartypes) or type(x,clibasmon) then S2:=S2 minus \+ \{x\} end if;\nend do;\nS:=remove(hastype,map(op,\{op(expand(p))\}),\{ op(_scalartypes),clibasmon\});\nfor x in S do \n if find1str(e,x)= \{0\} and x<>'Id' then S:=S minus \{x\} end if;\nend do;\nif S=\{\} th en return true end if;\nS1:=select(type,S,procedure):\nif S1 <> \{\} t hen\n error \"procedure name %1 that has been found in input is not \+ allowed as a symbolic coefficient\",op(S1)\nend if;\nif nops(S)=1 then \n error \"check spelling of %1 or define it as a constant or an al ias\",op(S)\nelse \n error \"check spelling of %1 or define them as \+ constants or aliases\",op(S) \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 16 "No. 6. Function " }{TEXT 286 9 "displayid" }{TEXT -1 186 " replaces a user-entered Clifford scalar with the scalar times th e unit element 'Id'. It may also be applied to matrices with Clifford \+ algebra entries.\n\nTypical use: displayid(e1+2*Pi);\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 621 "displayid:=proc(a1::\{array,matrix,algebraic \}) local KK,p;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz \+ and Bertfried Fauser. All rights reserved.`;\ndescription `Last revise d: November 1, 2002`;\n#############################################\n KK:=proc() if type(args[1],cliscalar) then return args[1]*Id \n \+ elif hastype(args[1],clibasmon) then return args[1] \n e nd if \nend proc:\nif type(a1,\{array,matrix\}) then return map(procna me,a1) end if;\np:=expand(a1):\nif type(p,\{`*`,cliscalar,clibasmon,cl imon\}) then return KK(p) \nelif type(p,\{`+`\}) then return map(KK,p) \nelse return a1 \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 17 "No. 7. Procedure " }{TEXT 287 8 "cliterms" }{TEXT -1 222 " iden tifies Clifford basis elements in the given Clifford polynomial.\n\nNO TE: 'cliterms' also works with terms of type cliprod and it finds corr ectly terms involving such expressions. \n\nTypical use: cliterms(2*Pi +2*e1we2);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1019 "cliterms:= proc(a 1::anything) local S1,S2,S3,x,p,Cliplusflag;\noptions `Copyright (c) 1 995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved .`;\ndescription `Last revised: November 1, 2002`;\n################## ###########################\nCliplusflag:=assigned(Cliplus):\nif hasty pe(a1,cliprod) and not Cliplusflag and _warnings_flag then \n WARNIN G(`argument to 'cliterms' contains type cliprod. Load 'Cliplus' to ext end functionality of CLIFFORD. Type ?cliprod for help.`)\nend if;\nif type(a1,\{clibasmon,cliprod\}) then return \{a1\} end if;\np:=display id(simplify(a1)):\nif hastype(p,cliprod) then \n S1:=remove(type,\{o p(p)\},cliscalar);\n S2:=select(hastype,S1,\{clibasmon,climon,clipro d\});\n S3:=\{\}:\n while not S2=\{\} do\n S3:=S3 union se lect(type,S2,\{clibasmon,cliprod\});\n S2:=select(hastype,map( op,remove(type,S2,\{clibasmon,cliprod\})),\{clibasmon,cliprod\});\n \+ end do;\nreturn S3\nend if;\nx:='x':\nS1:=remove(type,\{op(p)\},clisca lar);\nreturn \{seq(select(hastype,x,clibasmon),x=S1)\}\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 17 "No. 8. Procedure " }{TEXT 288 11 "cli bilinear" }{TEXT -1 360 " makes any procedure K specified as the third argument bilinear with respect to Clifford scalars in the first two a rguments. The first two arguments are of the type clipolynom, i.e., Cl ifford polynomials. The third argument is a string or a procedure.\nIt can handle terms involving elements of type cliprod.\n\nTypical use: \+ clibilinear(e1+2*e2we3,Id+2*e2+e3,K);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 922 "clibilinear:=proc(a1,a2,a3::\{procedure,name,symbol,matrix,ar ray\}) \n local tail,p1,p2,S1,S2,S12,res,x,y,cli1,cli2,co1 ,co2;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfr ied Fauser. All rights reserved.`;\ndescription `Last revised: Novembe r 1, 2002`;\n#############################################\nif simplif y(a1)=0 or simplify(a2)=0 then return 0 end if; \np1:=clicollect(a1): \np2:=clicollect(a2):\n tail:=args[4..-1];\n if type(p1,\{climon,cli prod\}) then S1:=[p1] else S1:=[op(p1)] end if:\n if type(p2,\{climon ,cliprod\}) then S2:=[p2] else S2:=[op(p2)] end if:\n S12:=[seq(seq([ x,y],x=S1),y=S2)];#this list will be huge for long polynomials\n res: =0:\n for x in S12 do \n cli1:=select(type,x[1],\{cliprod,clibasmo n\}):\n cli2:=select(type,x[2],\{cliprod,clibasmon\}):\n co1:=co eff(x[1],cli1):\n co2:=coeff(x[2],cli2):\n res:=res+co1*co2*a3(c li1,cli2,tail):\n end do:\n return res;\nend proc:\n" }}{PARA 258 " " 0 "" {TEXT -1 17 "No. 9. Procedure " }{TEXT 289 9 "clilinear" } {TEXT -1 336 " makes any procedure K specified as the second argument \+ linear with respect to Clifford scalars (elements of type cliscalar). \+ It can now distribute over Clifford polynomials with elements of `type /cliprod`. Any additional parameters are passed on to the procedure en tered as the second argument.\nTypical use: clilinear(a*e1+2*e2we3,K); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "clilinear:=proc(a1::\{symbol ,cliscalar,clibasmon,climon,clipolynom\},a2::\{name,procedure\}) \nloc al tail,p1,S1,res,x,cli1,co1;\noptions `Copyright (c) 1995-2003 by Raf al Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescriptio n `Last revised: November 1, 2002`;\n################################# ############\ntail:=args[3..-1];\nif type(a1,cliscalar) then return a1 *a2(Id,tail) end if;\np1:=displayid(a1):\nif type(p1,climon) then S1:= [p1] else S1:=[op(p1)] end if:\nres:=0:\nfor x in S1 do\n cli1:=sel ect(hastype,x,\{clibasmon,cliprod\}):\n co1:=coeff(x,cli1); \nres:= res+co1*a2(cli1,tail):\nend do:\nreturn res\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 10. Procedure " }{TEXT 290 7 "clisort" } {TEXT -1 312 " sorts the given multivariate Clifford polynomial with r espect to the Clifford indetereminates found in the expression via the procedure 'cliterms'. It puts scalar coefficients of the type cliscal ar in front of the Clifford basis monomials. It may also be applied to matrices with entries in a Clifford algebra. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 40 "Typical use: clisort(2* e1we2 - e1*b); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 427 "clisort:=pro c(p::algebraic) local L,N;\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription ` Last revised: November 1, 2002`;\n#################################### #########\nif type(p,matrix) then return map(procname,p) end if;\nif t ype(p,\{climon,clipolynom\}) or hastype(p,cliprod) then\n L:=cliterm s(expand(displayid(p)));\n return sort(p,L);\nend if:\nreturn p\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 11. Procedure " } {TEXT 291 10 "clicollect" }{TEXT -1 382 " reorders monomial terms in s tandard order and then collects them in a multivariate Clifford polyno mial. It may also be applied to matrices with entries in a Clifford al gebra. It will simplify 6 + 7*Id to 13*Id. It collects now terms of t ype cliprod, if present.\n\nNOTE: 'clicollect' also works with terms o f type cliprod and it collects correctly terms involving such expressi ons. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 46 "Typical use: clicollect(e1 + a*e1 - e1we2); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "clicollect:=proc(a1::algebraic) local p,L; \nopti ons `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`; \n#############################################\nif type(a1,matrix) th en return map(procname,a1) end if;\np:=expand(a1):\nif type(p,cliscala r) then return p*Id\nelif type(p,clipolynom) then \n L:=cliterms(p );\n return map(simplify,collect(displayid(p),L,'distributed'))\ne lse return args[1] \nend if\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 23 "No. 12. The procedure " }{TEXT 292 3 "ord" }{TEXT -1 319 " ret urns an ordered list of positions in a monomial, e.g., e1we2, where v ector indices are found. Then, nops(ord(e1we2)) can be used to find \+ the order of the monomial. Note that for consistency we have ord(Id) \+ = ord(numeric) = ord(numeric*Id) = ord(cliscalar)=[] where cliscalar i s any object of the type cliscalar." }}{PARA 258 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 35 "This procedure is for internal use. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 386 "ord:=proc(a1) local v,k;\noptions `Copyright (c) 1995-2003 by Raf al Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescriptio n `Last revised: November 1, 2002`;\n################################# ############\nif type(a1,cliscalar) then return [] end if;\nv:=select( type,a1,clibasmon);\nif v = Id then return [] end if;\nk:='k':\nreturn [seq(2+3*k,k=0..((length(v)+1)/3-1))]\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 13. Procedure " }{TEXT 293 9 "cliremove" }{TEXT -1 193 " removes one symbol 'ei' from the location specified by the pr ocedure 'ord'. \n(NOTE: procedure 'ord' specifies location of the inde x 'i' in 'ei'.) This procedure is primarily for internal use." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 578 "cl iremove:=proc(p::posint,s::symbol) local S1,S2;global _prolevel;\nopti ons `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`,remember;\ndescription `Last revised: November \+ 1, 2002`;\n#############################################\nif not _prol evel then\n if s=Id then error \"second argument must be Grassmann b asis monomial of rank >= 1\" end if;\nend if;\nS2:=substring(s,(p+2).. length(s));\nS1:=substring(s,1..(p-3));\nif length(S2)=0 and S1 <> s t hen return S1 \n elif S1 = s then return S2 \n else return cat(S1, \"w\",S2); \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "N o. 14. Procedure " }{TEXT 294 7 "extract" }{TEXT -1 445 " extracts ind ices of a monomial (or a constant times a monomial) and it returns the m as a list of strings. If necessary, they can be returned as a list \+ of integers if option 'integers' is selected (in fact, any name which \+ evaluates to a string may be used as the option). Indices could be no w integers, letters, or they could be mixed. Note that extract(Id) = [ ] and extract(numeric) = extract(numeric*Id) = [] results in no vecto r indices. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 63 "Typical use: extract(2*e1we2); or extract(e2we3, \"intege rs\"); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 780 "extract:=proc(a1::\{symbol,cliscalar,clibasmon,climo n\},a2::symbol) \nlocal v,k,inds;global _prolevel,str_to_int;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. Al l rights reserved.`,remember;\ndescription `Last revised: November 1, \+ 2002`;\n#############################################\nif type(a1,clis calar) or (type(a1,symbol) and length(a1)=1) then return [] \nelif\n \+ type(a1,\{climon,clibasmon\}) then v:=select(type,a1,clibasmon):\nels e \n error \"wrong argument: %1\",a1 \nend if;\nif v = Id then retur n [] end if;\nk:='k':\ninds:=[seq(substring(v,(2+3*k)..(2+3*k)),k=0..( (length(v)+1)/3-1))];\nif nargs=1 then return inds \n elif type(a2,s ymbol) then return map(convert,inds,str_to_int)\n else error \"wrong option or number of arguments\" \nend if;\nend proc:\n" }}{PARA 258 " " 0 "" {TEXT -1 19 "No. 15. Procedure " }{TEXT 295 7 "reorder" } {TEXT -1 330 " reorders Clifford monomials in the given Clifford polyn omial using standard ordering and calculates sign of each permutation, e.g., reorder(e1we3we2) = -e1we2we3, reorder(e2we1 + 2*e1we5we2) = -e 1we2 - 2*e1we2we5. If any one of the indices of the monomial is a lett er, e.g., reorder(eiwe3) = eiwe3, reorder returns its argument. " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 139 "Reor der now can order monomials and polynomials with symbolic coefficients , e.g. reorder(ejwei) = -eiwej, using the lexicographic order. " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 48 "Typic al use: reorder(e2we1 + 2*Id + e4we3we1); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1075 "reorder:=proc(a1::algeb raic) \n local L1,L2,N,newbas,f,a,x,K,dummy_set,n12,s12,ss;\n \+ global B,dim_V;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\nif type(a1,\{matrix,`+`,`*`\}) then return map(procname,a1) en d if; \nL1:=Clifford:-extract(a1);\nN:=nops(L1);\nif N>9 then error \" detected basis monomial of grade higher than 9 in the input\" end if; \nif N=0 or N=1 then return a1 end if;\nn12,s12:=selectremove(member,L 1,\{`1`,`2`,`3`,`4`,`5`,`6`,`7`,`8`,`9`\}):\n#s12:=remove(member,L1,\{ `1`,`2`,`3`,`4`,`5`,`6`,`7`,`8`,`9`\}):\nL2:=[op(sort(n12)),op(sort(s1 2))];\nf:=proc() end proc:\nfor ss from 1 to N do\n f(L2[ss]):=L1[ss] ;\nend do:\ndummy_set:=convert(L1,set):\nK:=0:\nwhile dummy_set <> \{ \} do\n a:=dummy_set[1]:\n dummy_set:=dummy_set[2..-1];\n x:=a:\n \+ while f(x)<>a do\n x:=f(x);\n dummy_set:=dummy_set minus \{x \};\n K:=K+1;\n end do:\nend do:\nnewbas:=cat(e||(op(L2[1..-2])) ||w,e,L2[-1]):\nreturn (-1)^K*newbas\nend proc:\n" }}{PARA 258 "" 0 " " {TEXT -1 35 "No. 16. Defining a useful function " }{TEXT 296 8 "maxi ndex" }{TEXT -1 226 " which finds the greatest index in the given Clif ford polynomial or in the given list or set of Clifford monomials. It \+ returns 0 for a Clifford scalar (an element of type cliscalar).\n\nTyp ical use: maxindex(a*Id+6+2*Pi*e1we2);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "maxindex:=proc(a1::\{cliscalar,clibasmon,climon,clip olynom,list,set\}) \nlocal inds,mons,symbinds;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserv ed.`;\ndescription `Last revised: November 1, 2002`;\n################ #############################\nif type(a1,cliscalar) or a1=Id then ret urn 0 elif\n type(a1,list) then return max(op(convert(map(procname,a 1),set))) elif\n type(a1,set) then return max(op(map(procname,a1))) \+ else \n mons:=cliterms(a1);\n inds:=map(op,map(Clifford:-extract,m ons,'integers'));\n symbinds:=remove(type,inds,integer);\n if symb inds = \{\} then\n if inds=\{\} then return 0 else return max(op( inds)) end if;\n else\n error \"cannot determine maximum index \+ because input contains symbolic index or indices\"\n end if;\n end \+ if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 35 "No. 17. Defining a \+ useful function " }{TEXT 297 8 "maxgrade" }{TEXT -1 176 " which finds \+ the maximum grade in the given Clifford polynomial. It returns 0 for \+ a Clifford scalar (an element of type cliscalar).\n\nTypical use: maxg rade(a*Id+6+2*Pi*e1we2);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "maxg rade:=proc(a1::\{cliscalar,clibasmon,climon,clipolynom\}) local S;\nop tions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fause r. All rights reserved.`;\ndescription `Last revised: November 1, 2002 `;\n#############################################\nif type(a1,cliscala r) then return 0 end if;\nS:=\{op(cliterms(a1))\}:\nreturn max(op(map( nops,map(Clifford:-extract,S))))\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 18. Procedure " }{TEXT 298 2 "LC" }{TEXT -1 233 " de fines a left contraction between any multivector u and a multivector v , i.e., multivector u acts on the multivector v from the left. This p rocedure is now bilinear in both arguments. It can accept third argum ent such as K or -K." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 46 "Typical use: LC(e1 + 2*e2, e1we3 + b*e2we3); \n" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 2276 "LC:=proc(x::\{cliscalar,clibasmo n,climon,clipolynom\},\n y::\{cliscalar,clibasmon,climon,clipo lynom\})\n local N1,N2,lst1,lst2,i,j,cf,term,lname,res,coB,nameB;\n global _CLIENV,B;\noptions `Copyright (c) 1995-2003 by Rafal Ablam owicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last \+ revised: November 1, 2002`;\n######################################### ####\nif nargs=2 then\n coB:=1:\n nameB:=`B`: \n lname:=`B`: \+ \nelif nargs=3 then\n if type(args[3],\{name,symbol,matrix,array\}) then\n coB:=1:\n nameB:=args[3];\n lname:=args[3]; \n elif type(args[3],`&*`(numeric,\{name,symbol,matrix,array\})) th en\n coB:=op(select(type,\{op(args[3])\},numeric));\n name B:=op(remove(type,\{op(args[3])\},numeric));\n lname:=args[3]:\n else \n error \"wrong type of third argument in LC. See ?LC \+ for more help.\" \n end if;\nelse\n error \"two or three argument s expected in LC. See ?LC for more help.\"\n end if;\n################ ################\n if type(x,clibasmon) then\n if type(y,clibasmon ) then\n lst1:=Clifford:-extract(x,'integers');\n lst2:=Clif ford:-extract(y,'integers');\n N1:=nops(lst1);N2:=nops(lst2);\n \+ if N1>N2 then return 0 end if;\n if N1=0 then return y end if ;\n if N1=1 then \n res:=`+`(seq(coB*nameB[lst1[1],lst2[j ]]*_CLIENV[_QDEF_PREFACTOR]^(j-1)*\n makeclib asmon([op(subs(lst2[j]=NULL,lst2))]),j=1..N2));\n return reord er(res) \n else\n res:=\nprocname(makeclibasmon(lst1[1..-2]) ,procname(makeclibasmon([lst1[-1]]),y,lname),lname);\n return \+ reorder(res)\n end if;\n elif type(y,climon) then\n ter m,cf:=selectremove(type,y,clibasmon);\n return expand(cf*procn ame(x,term,lname))\n elif type(y,clipolynom) then\n retur n add(procname(x,i,lname),i=[op(y)])\n elif type(y,cliscalar) the n \n return displayid(scalarpart(x)*y)\n end if; \n elif type(x,climon) then\n term,cf:=selectremove(type,x,clibasmon);\n \+ return expand(cf*procname(term,y,lname))\n elif type(x,clipolynom) \+ then\n return add(procname(i,y,lname),i=[op(x)])\n elif type(x,cli scalar) then \n return x*reorder(y)\n end if;\nerror \"Got input \+ %1 and %2 but LC can only process constants and Clifford numbers\",x,y ;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 19. Procedure " } {TEXT 299 3 "LCQ" }{TEXT -1 270 " is a special version of 'LC' and giv es left contraction in the orthogonal Clifford algebra Cl(Q) of the qu adratic form Q defined via the symmetric part g of B as Q(x) = g(x, x) = B(x, x). It can accept name as a third optional argument or a nume ric multiple of a name." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 76 "Proposed by Yvon Siret, Universite Joseph Fou rier, Grenoble, France. Thanks!" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 82 "Typical use: LCQ(e1 + 2*e2, e1we3 + b*e 2we3);\nLCQ(e1 + 2*e2, e1we3 + b*e2we3,K); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1794 "LCQ:=proc(x::\{cliscalar,clibasmon,climon,clipolyno m\},\n y::\{cliscalar,clibasmon,climon,clipolynom\}) \n l ocal ii,N,L,m,Sxy,symbxy,lname,coB,nameB;global B:\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights re served.`;\ndescription `Last revised: November 1, 2002`;\n############ #################################\nif nargs=2 then\n coB:=1:\n n ameB:=`B`: \n lname:=`B`: \nelif nargs=3 then\n if type(args[3], \{name,symbol,matrix,array\}) then\n coB:=1:\n nameB:=args [3];\n lname:=args[3];\n elif type(args[3],`&*`(numeric,\{nam e,symbol,matrix,array\})) then\n coB:=op(select(type,\{op(args[3 ])\},numeric));\n nameB:=op(remove(type,\{op(args[3])\},numeric) );\n lname:=args[3]:\n else \n error \"wrong type of th ird argument in LCQ. See ?LCQ for more help.\" \n end if;\nelse\n \+ error \"two or three arguments expected in LCQ. See ?LCQ for more hel p.\"\nend if;\n################################\nSxy:=remove(type,map( op,\{op(x),op(y)\}),cliscalar);\nSxy:=map(op,map(Clifford:-extract,Sxy ,'integers'));\nsymbxy:=remove(type,Sxy,posint);\nif symbxy <> \{\} th en \n return LC(x,y,lname) \nend if;\nm:=max(op(Sxy),1);# 1 is neede d when both x and y have maxindex=0\nif type(evalm(lname),matrix) then \n N:=linalg[coldim](evalm(lname)):\n if m>N then \n error \+ \"input contains index larger than size of bilinear form %1\",lname \n end if;\nend if:\nif type(lname,\{name,symbol,array,matrix\}) then \n L:=seq(lname[ii,ii],ii=1..m);\n return LC(x,y,linalg[diag](L)) \nelif \n type(lname,`&*`(numeric,\{name,symbol,array,matrix\})) the n\n coB:=op(select(type,\{op(lname)\},numeric));\n nameB:=op(selec t(type,\{op(lname)\},\{name,symbol,array,matrix\}));\n L:=seq(coB*na meB[ii,ii],ii=1..m);\n return LC(x,y,linalg[diag](L))\n end if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 20. Procedure " } {TEXT 300 2 "RC" }{TEXT -1 241 " defines a right contraction between a ny multivector u and a multivector v, i.e., multivector u acts on the \+ multivector v from the right. This procedure is now bilinear in both \+ arguments. It can accept third optional argument like B or -B." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 46 "Typical \+ use: RC(e1 + 2*e2, e1we3 + b*e2we3); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2279 "RC:=proc(x::\{cliscalar,clibasmon,climon,clipolynom\},\n \+ y::\{cliscalar,clibasmon,climon,clipolynom\})\n local N1,N2,lst1 ,lst2,i,j,cf,term,lname,res,coB,nameB;\n global _CLIENV,B;\noptions ` Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All \+ rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n### ##########################################\nif nargs=2 then\n coB:= 1:\n nameB:=`B`: \n lname:=`B`: \nelif nargs=3 then\n if type (args[3],\{name,symbol,matrix,array\}) then\n coB:=1:\n na meB:=args[3];\n lname:=args[3];\n elif type(args[3],`&*`(nume ric,\{name,symbol,matrix,array\})) then\n coB:=op(select(type,\{ op(args[3])\},numeric));\n nameB:=op(remove(type,\{op(args[3])\} ,numeric));\n lname:=args[3]:\n else \n error \"wrong t ype of third argument in RC. See ?RC for more help.\" \n end if;\ne lse\n error \"two or three arguments expected in RC. See ?RC for mor e help.\"\nend if;\n################################\n if type(x,clib asmon) then\n if type(y,clibasmon) then\n lst1:=Clifford:-extr act(x,'integers');\n lst2:=Clifford:-extract(y,'integers');\n \+ N1:=nops(lst1);N2:=nops(lst2);\n if N2>N1 then return 0 end if; \n if N2=0 then return x end if;\n if N2=1 then \n r es:=`+`(seq(coB*nameB[lst1[-i],lst2[1]]*_CLIENV[_QDEF_PREFACTOR]^(i-1) *\n makeclibasmon([op(subs(lst1[-i]=NULL,lst1))]),i =1..N1));\n return reorder(res) \n else\n res: =procname(procname(x,makeclibasmon([lst2[1]]),lname),\n \+ makeclibasmon(lst2[2..-1]),lname);\n return reorder(res)\n end if;\n elif type(y,climon) then\n term ,cf:=selectremove(type,y,clibasmon);\n return expand(cf*procname( x,term,lname))\n elif type(y,clipolynom) then\n return add(pro cname(x,i,lname),i=[op(y)])\n elif type(y,cliscalar) then return re order(x)*y \n end if;\n elif type(x,climon) then\n term,cf:=se lectremove(type,x,clibasmon);\n return expand(cf*procname(term,y,ln ame))\n elif type(x,clipolynom) then\n return add(procname(i,y,lna me),i=[op(x)])\n elif type(x,cliscalar) then \n return displayid(x *scalarpart(y))\n end if;\nerror \"Got input %1 and %2 but can only \+ process constants and Clifford numbers\",x,y\nend proc:\n" }}{PARA 0 " " 0 "" {TEXT 259 18 "No. 21. Procedure " }{TEXT 301 3 "RCQ" }{TEXT 302 85 ": Right contraction in Cl(Q). It can accept third optional arg ument such as K or -K.\n" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1799 "RCQ:=proc(x::\{cliscalar,clibasmon,climon,clipolynom\},\n \+ y::\{cliscalar,clibasmon,climon,clipolynom\}) \n local ii,N ,L,m,Sxy,symbxy,lname,coB,nameB;global B:\noptions `Copyright (c) 1995 -2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`; \ndescription `Last revised: November 1, 2002`;\n##################### ######################## \nif nargs=2 then\n coB:=1:\n nameB:=` B`: \n lname:=`B`: \nelif nargs=3 then\n if type(args[3],\{name, symbol,matrix,array\}) then\n coB:=1:\n nameB:=args[3];\n \+ lname:=args[3];\n elif type(args[3],`&*`(numeric,\{name,symbo l,matrix,array\})) then\n coB:=op(select(type,\{op(args[3])\},nu meric));\n nameB:=op(remove(type,\{op(args[3])\},numeric));\n \+ lname:=args[3]:\n else \n error \"wrong type of third arg ument in RCQ. See ?RCQ for more help.\" \n end if;\nelse\n error \+ \"two or three arguments expected in RCQ. See ?RCQ for more help.\"\ne nd if;\n################################\nSxy:=remove(type,map(op,\{op (x),op(y)\}),cliscalar);\nSxy:=map(op,map(Clifford:-extract,Sxy,'integ ers'));\nsymbxy:=remove(type,Sxy,posint);\nif symbxy <> \{\} then \n \+ return RC(x,y,lname) \nend if;\nm:=max(op(Sxy),1);# 1 is needed when \+ both x and y have maxindex=0\nif type(evalm(lname),matrix) then \n N :=linalg[coldim](evalm(lname)):\n if m>N then \n error \"in put contains index larger than size of bilinear form %1\",lname \n e nd if:\nend if:\nif type(lname,\{name,symbol,array,matrix\}) then\n \+ L:=seq(lname[ii,ii],ii=1..m);\n return RC(x,y,linalg[diag](L))\nelif \n type(lname,`&*`(numeric,\{name,symbol,array,matrix\})) then\n \+ coB:=op(select(type,\{op(lname)\},numeric));\n nameB:=op(select(type ,\{op(lname)\},\{name,symbol,array,matrix\}));\n L:=seq(coB*nameB[ii ,ii],ii=1..m);\n return RC(x,y,linalg[diag](L))\n end if;\nend proc: " }}{PARA 258 "" 0 "" {TEXT -1 19 "\nNo. 22. Procedure " }{TEXT 303 8 "gradeinv" }{TEXT -1 133 " is the grade involution in the Clifford alg ebra,i.e., it reverses signs of odd elements and leaves signs of even \+ elements unchanged." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 48 "Typical use: gradeinv(e1 + e1we2 - 4*e3we4); \n" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 552 "gradeinv:=proc(a1::\{matrix,clisc alar,clibasmon,climon,clipolynom\}) global _CLIENV;\noptions `Copyrigh t (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights r eserved.`;\ndescription `Last revised: November 1, 2002`;\n########### ##################################\nif type(a1,matrix) then return map (procname,a1) end if;\n#if not assigned(_CLIENV) then _CLIENV[_QDEF_PR EFACTOR]:=-1 end if;\nif type(a1,clibasmon) then return (_CLIENV[_QDE F_PREFACTOR])^maxgrade(a1)*a1 \n else return clil inear(a1,procname) \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 23. Define the " }{TEXT 304 5 "wedge" }{TEXT -1 1306 " prod uct of any number of Clifford polynomials. The infix form of this ass ociative multiplication is `&w`. Thus, e1 &w e2 = wedge(e1, e2), etc. Via the procedure 'rmulm' described below, wedge multiplication may \+ be applied to matrices with entries in a Clifford algebra or in an ext erior algebra.\n\nNew feature: When the dimension of the vector space \+ is known, either from the size of the matrix B or from the global para meter dim_V that can be set by the user, the output of the procedure d oes not include terms of grade higher than the dimension of the vector space in case symbolic indices are used. \n\nThe default value of thi s global variable is 9 and it it set by the initialization file when C lifford is loaded.\n\nWhen the procedure is invoked, it checks whether the bilinear form B has been defined. If yes, the procedure checks wh ether the size of B is less than the current value of dim_V. If again \+ yes, a warning message is issued by the procedure and the value of dim _V is reduced. If the size of B is larger than the current value of di m_V, no warning message is issued and the value of dim_V is increased to linalg[coldim](B).\n\nThe warning message can be supressed by addi gn 'false' to a global parameter _warnings_flag whose default value is set to true by the Clifford initialization file." }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 96 "Typical use: wedge(e1 + e2, e4 + e1we2); wedge(e2 + 2*e1, e3, e4); (e2 + 2*e1) &w (e3 + 2* ); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3061 "wedge:=proc(a1::\{clisc alar,clibasmon,climon,clipolynom\},\n a2::\{cliscalar,cliba smon,climon,clipolynom\}) \nlocal ii,kk,wedge2,pi,p1,p2,i1,i2,i12,n12, maxindexflag,expr,maxin;\nglobal dim_V,B,_warnings_flag;\noptions `Cop yright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rig hts reserved.`;\ndescription `Last revised: November 1, 2002`;\n###### #######################################\nkk:='kk':\nif member(0,[args] ) then return 0 \nelif \n remove(type,\{args\},cliscalar)=\{\} then \+ return product(args[kk],kk=1..nargs)\nend if;\nif type(B,matrix) then \n if linalg[coldim](B)<>dim_V then \n if linalg[coldim](B) < d im_V then\n dim_V:=linalg[coldim](B);\n if _warnings_f lag then\nprintf(\"Warning, since B has been (re-)assigned, value of d im_V has been reduced by 'wedge' to %g\\n\",dim_V);\n end if; \n elif linalg[coldim](B)>dim_V then\n dim_V:=linalg[coldim](B) ;\n end if;\n end if;\n end if; \nif not type(dim_V,Range(0,10 )) or \n not type(dim_V,posint) then\n error \"value of dim_V must be a positive integer between 1 and 9, inclusive, but current value o f dim_V is %1\",dim_V\nend if;\n################\ni12:=\{\}:\nfor ii f rom 1 to nargs do\n pi:=args[ii]: \n i12:=i12 union map(op,map(C lifford:-extract,cliterms(pi),'integers')):\nend do;\nn12:= select(mem ber,i12,\{1,2,3,4,5,6,7,8,9\}):\nif not n12=\{\} then\n maxin:=max(o p(n12)); \n maxindexflag:=evalb(maxin > dim_V);\nelse maxindexflag:= false:\nend if:\nif maxindexflag then \n error \"argument(s) contain (s) index larger then current value of dim_V which is now %1. To compl ete computation, increase value of dim_V or assign square matrix of si ze at least %2 by %3 to bilinear form B\",dim_V,maxin,maxin\nend if;\n ################\nwedge2:=proc() local expr,i1,i2,n1,n2,i12,s12,symbin dexflag;global dim_V;\n i1:=\{op(Clifford:-extract(args[1]))\};n1:=nop s(i1):\n i2:=\{op(Clifford:-extract(args[2]))\};n2:=nops(i2):\n if arg s[1]=Id then \n if n2>dim_V then return 0 else return args[2] end i f;\n end if;\n if args[2]=Id then \n if n1>dim_V then return 0 else return args[1] end if;\n end if;\n i1:=\{op(Clifford:-extract(args[ 1]))\};\n i2:=\{op(Clifford:-extract(args[2]))\};\n i12:=i1 union \+ i2;\n s12:= remove(member,i12,\{`1`,`2`,`3`,`4`,`5`,`6`,`7`,`8`,`9` \}):\n symbindexflag:=evalb(not s12=\{\}):\n if i1 intersect i2 <> \{\} then return 0 end if;\n if symbindexflag and nops(i1)+nops(i2) > dim_V then return 0 end if;\nreturn reorder(cat(args[1],\"w\",args[ 2]));\nend proc:\n################\nif nargs=1 then return args\nelif \+ nargs=2 then p1:=displayid(a1):\n p2:=displayid(a2): \n expr:=clibilinear(p1,p2,wedge2);\n \+ if hastype(expr,trig) then \n return clicollect( map(combine,clicollect(expr),trig))\n else \n \+ return reorder(expr)\n end if;\nelse exp r:=procname(procname(a1,a2),args[3..nargs]):\n if hastype(expr,tri g) then \n return clicollect(map(combine,clicollect(expr),trig) )\n else \n return reorder(expr)\n end if;\nend if;\nen d proc:\n" }}{PARA 0 "" 0 "" {TEXT 269 29 "No. 24. Ampersand version o f " }{TEXT 307 5 "wedge" }{TEXT 308 38 ". (Has been moved to Clifford: -setup)\n" }}{PARA 0 "" 0 "" {TEXT 260 18 "No. 25. Procedure " }{TEXT 305 8 "permsign" }{TEXT 306 118 " computes sign of a permutation that \+ sorts a list of indices.\n\nTypical use: permsign([1,3,2]); permsign([ j,1,i,k,2]);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 880 "permsign:=proc(L ::list) local newbas,ss,a,n12,s12,L1,L2,N,f,dummy_set,K,x;\noptions `C opyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All r ights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#### #########################################\nL1:=L:\nN:=nops(L1):\nif N= 1 then return 1 end if:\n################## new\nn12,s12:=selectremove (member,L1,\{1,2,3,4,5,6,7,8,9\});\n#s12:=remove(member,L1,\{1,2,3,4,5 ,6,7,8,9\});\nL2:=[op(sort(n12)),op(sort(s12))];\n################## n ew\nf:=proc() end proc:\nfor ss from 1 to N do\n f(L2[ss]):=L1[ss];\n end do;\ndummy_set:=convert(L1,set);\nK:=0:\nwhile dummy_set <> \{\} d o\n a:=dummy_set[1]:\n dummy_set:=dummy_set[2..-1];\n x:=a:\n whil e f(x)<>a do\n x:=f(x);\n dummy_set:=dummy_set minus \{x\}; \n K:=K+1;\n end do:\nend do;\n#newbas:=cat(e.(op(L2[1..-2])).w, e,L2[-1]):\n#return ((-1)^K*newbas);\nreturn (-1)^K;\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 26. Procedure " }{TEXT 309 7 "cmulN UM" }{TEXT -1 148 " calculates Clifford product between any two Cliffo rd monomials using the recursivelyChevalley's definition of the Cliffo rd product: " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 93 " \+ xu = wedge(x, u) + LC(x, u) = x &w u + LC(x, u) " }}{PARA 258 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 477 "where x is a ve ctor and u is any element in the algebra, wedge(x,u) = x &w u denotes \+ the wedge or exterior product between x and u, and LC(x, u) denotes t he left contraction of u by x. This procedure is now bilinear in both \+ arguments. The infix form is available e.g., e1 &c e2. This procedur e works in Clifford algebras in dimensions up to and including 9. Mul tiplication of matrices with entries in a Clifford algebra can be done with a procedure 'rmulm' described below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 128 "This procedure requires thir d argument of type name or a numeric multiple of a name. Then it compu tes Clifford product in Cl(K)." }}{PARA 258 "" 0 "" {TEXT -1 221 "\nTh is version can take index as a way of passing a parameter. The index \+ could be of type `&*`(numeric,\{name,symbol,array,matrix\}) or of type \{name,symbol,array,matrix\}.\n\nWhen the bilinear form B is symboli c, use cmulRS." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 55 "Typical use: cmulNUM(e1,e3we4,B); cmulNUM(e1,e3we4,-K); " }{TEXT 265 3 " \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2253 "cmulNUM:= proc(a1,a2,lname) \n local L,N,L2,x,x1,x2,S,i,ii,T1,T2,K,p1,p2,coB,na meB,a12;global B:\n options `Copyright (c) 1995-2003 by Rafal Ablamow icz and Bertfried Fauser. All rights reserved.`;\n description `Last \+ revised: November 1, 2002`;\n######################################### ####\n###This is additional code for Maple 6 version:\n############### ##############################\nif hastype(\{a1,a2\},cliprod) then\n \+ a12:=map(Cliplus:-clieval,[a1,a2]);\n return Cliplus:-cliexpand(cli bilinear(a12[1],a12[2],procname,lname))\nend if: \n################### ###################################################################\n# ## old name cmul2B: this procedure computes recursively Clifford produ ct of any two #\n### cliscalars, clibasmons, climons, and clipolynoms \+ in Clifford algebras Cl(lname) #\n################################### ###################################################\n if nargs<>3 the n error \"exactly three arguments are needed\" end if:\n if has(0,map (simplify,[a1,a2])) then return 0 end if;\n if a2=`Id` then return a1 end if:\n if a1=`Id` then return a2 end if:\n L:=Clifford:-extract( a1,'integers');\n N:=nops(L):\n ################\n ##### The follow ing will allow for lname to be -B, for example:\n if type(lname,\{nam e,symbol,array,matrix\}) then\n coB,nameB:=1,lname:\n elif type(l name,`&*`(numeric,\{name,symbol,array,matrix\})) then\n coB:=op(se lect(type,\{op(lname)\},numeric));\n nameB:=op(select(type,\{op(ln ame)\},name));\n else\n error \"third argument is of unexpected t ype\"\n end if;\n ################\n if N=0 then return coeff(a1,Id )*a2\n elif N=1 then\n L2:=Clifford:-extract(a2,'integers'):\n \+ return reorder(simplify(makeclibasmon([L[1],op(L2)])\n +add((-1)^( i-1)*coB*nameB[L[1],L2[i]]*makeclibasmon(subs(L2[i]=NULL,L2)),i=1..nop s(L2))))\n elif N=2 then\n x1:=substring(a1,1..2):x2:=substring(a1 ,4..5);\n p2:=procname(x2,a2,lname):\n S:=clibilinear(x1,p2,proc name,lname);\n return simplify(S-coB*nameB[op(L)]*a2)\n end if;\n \+ x:=cat(e,L[-1]);\n p1:=substring(a1,1..(3*N-4));\n p2:=procname(x,a 2,lname):\n S:=clibilinear(p1,p2,procname,lname)\n -add((-1)^(i) *coB*nameB[L[-i],L[-1]]*\nprocname(makeclibasmon(subs(L[-i]=NULL,L[1.. -2])),a2,lname),i=2..N); \n return reorder(simplify(S))\nend proc:\n " }}{PARA 0 "" 0 "" {TEXT 266 19 "No. 27. Procedure " }{TEXT 310 6 "c mulRS" }{TEXT 311 114 " computes Clifford product using Rota-Stein cli ffordization technique. It can accept now -K in place of the name.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4902 "cmulRS:=proc(a1,a2,lname)\nloca l max_grade,L1,N1,L2,N2,genPS,fun1,fun2,srt,cup,pList1,PN1,\n pLi st2,PN2,pSgn1,pSgn2,a,i,j,m,n,res,pos1,pos2,F1,F2,coB,nameB,a12;\nopti ons `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`; \n#############################################\n###This is additional code for Maple 6 version:\n########################################## ###\nif hastype(\{a1,a2\},cliprod) then\n a12:=map(Cliplus:-clieval, [a1,a2]);\n return Cliplus:-cliexpand(clibilinear(a12[1],a12[2],proc name,lname))\nend if: \n############################################## ############################################\n### This procedure compu tes Clifford product of any two cliscalars, clibasmons, climons, #\n## # and clipolynoms in Clifford algebras Cl(lname) using Rota-Sten cliff ordization #\n### Procedure cmulRS modified by Rafal to accept - K, or -B for lname. #\n############################ ##############################################################\n if n args<>3 then error \"exactly three arguments are needed\" end if:\n i f has(0,map(simplify,[a1,a2])) then return 0 end if;\n if a1 = `Id` t hen return a2 end if;\n if a2 = `Id` then return a1 end if;\n ###### ##########\n ##### The following will allow for lname to be -B, for e xample:\n if type(lname,\{name,symbol,array,matrix\}) then\n coB, nameB:=1,lname:\n elif type(lname,`&*`(numeric,\{name,symbol,array,ma trix\})) then\n coB:=op(select(type,\{op(lname)\},numeric));\n \+ nameB:=op(select(type,\{op(lname)\},name));\n else\n error \"thi rd argument is of unexpected type\"\n end if;\n ################\n \+ L1:=Clifford:-extract(a1,'integers');\n N1:=nops(L1);\n L2:=Clifford :-extract(a2,'integers');\n N2:=nops(L2);\n if N1=1 then \n retur n reorder(simplify(makeclibasmon([L1[1],op(L2)])\n +add((-1)^(i-1)* coB*nameB[L1[1],L2[i]]*makeclibasmon(subs(L2[i]=NULL,L2)),i=1..N2)))\n end if;\n if N2=1 then \n return reorder(simplify(makeclibasmon( [op(L1),L2[1]])\n +add((-1)^(i-1)*coB*nameB[L1[-i],L2[1]]*makecliba smon(subs(L1[-i]=NULL,L1)),i=1..N1)))\n end if;\n#### genPS ; generat e a power set of 1..N, option remember\n genPS:=proc(N)\n local a, i,plst;\n option remember; \n a:=[seq(i,i=1..N)]:\n plst:=[a] :\n for i in a do\n plst:=[op(subs(i=NULL,plst)),op(plst)]:\n \+ end do:\n end proc:\n#### prepare combinatorics for L1:\n fun1:=p roc(a1) a1 end proc:\n for i from 1 to N1 do\n fun1(i):=L1[i];\n \+ end do:\n#### here is the old code for the poweset \n# a:=[seq(i,i=1. .N1)]:\n# pList1:=[a]:\n# for i in a do\n# pList1 := [op(subs(i = NULL,pList1)), op(pList1)]:\n# end do:\n####\npList1:=genPS(N1); \n PN1:=nops(pList1)+1; ## added 1 here\n pList1:=sort(pList1,(a,b)-> evalb(nops(a)<=nops(b)));\n pSgn1 :=[seq((-1)^(add(pList1[i][m]-m,m=1 ..nops(pList1[i]))),i=1..PN1-1)];\n#### prepare combinatorics for L2: \n fun2:=proc(a2) a2 end proc:\n for i from 1 to N2 do\n fun2(i): =L2[i];\n end do:\n#### here is the old code for the poweset \n# a:= [seq(i,i=1..N2)]:\n# pList2:=[a]:\n# for i in a do\n# pList2 := [ op(subs(i = NULL,pList2)), op(pList2)]:\n# end do:\n####\npList2:=gen PS(N2);\n PN2:=nops(pList2)+1; ## added 1 here\n pList2:=sort(pList 2,(a,b)->evalb(nops(a)<=nops(b)));\n pSgn2:=[seq((-1)^(add(pList2[i][ m]-m,m=1..nops(pList2[i]))),i=1..PN2-1)];\n#### cup tangle of the rota -stein sausage tangle\n cup:=proc(lst1,lst2,coB,nameB)\n local i; \n if nops(lst1)<>nops(lst2) then return 0 end if;\n if lst1=[] \+ then return 1 end if;\n if nops(lst1)=1 then return coB*nameB[lst1[ 1],lst2[1]] end if;\n add((-1)^(i-1)*coB*nameB[lst1[-1],lst2[i]]*cu p(lst1[1..-2],subs(lst2[i]=NULL,lst2),coB,nameB)\n \+ ,i=1..nops(lst2))\n end pro c:\n################################################################## ################# \n## Rota-Stein Tangle : cliffordization \+ #\n## compose only such terms which \+ are potentially non zero in the cup(..) tangle #\n#################### ###############################################################\n max _grade:=nops(\{op(L1),op(L2)\}); ## <== new code\n res:=0:\n pos1:= 0:\n for j from 0 to N1 do # for all j-vectors of pList1\n F1:=N1!/ ((N1-j)!*j!);\n pos2:=0:\n for i from 0 to min(N2,max_grade-j) do # for all i-vectors of pList2\n \+ # which do not exceed max_grade (others are zero)\n F2:=N2!/((N2-i) !*i!);\n for n from 1 to F1 do\n for m from 1 to F2 do \n res :=res+\n pSgn1[pos1+n]*pSgn2[pos2+m]*\n cup(map(fun1,p List1[PN1-pos1-n]),map(fun2,pList2[pos2+m]),coB,nameB)*\n make clibasmon([op(map(fun1,pList1[pos1+n])),op(map(fun2,pList2[PN2-pos2-m] ))])\n end do:\n end do:\n pos2:=pos2+F2;\n end do:\n \+ pos1:=pos1+F1;\n end do: \nreturn reorder(res); ## note that cmulRS INCLUDES already reorder !!\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 267 19 "No. 28. Procedure " }{TEXT 312 7 "cmulgen" }{TEXT 313 47 " is just a place holder for a Clifford product." }{TEXT -1 1 "\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "cmulgen:=proc() global _default_Cl ifford_product,_warnings_flag;\noptions `Copyright (c) 1995-2003 by Ra fal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescripti on `Last revised: November 1, 2002`;\n################################ #############\nif _default_Clifford_product <> 'cmulgen' then\n retu rn _default_Clifford_product(args)\nelse \n if _warnings_flag then\n WARNING(\"to assign Clifford product, execute 'useproduct' with arg ument cmulRS, cmulNUM, or cmul_user_defined first\");\n end if;\n \+ return 'cmulgen'(args);\n end if; \nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 268 25 "No. 29. Wrapper function " }{TEXT 314 4 "cmul" }{TEXT 315 90 " for the Clifford product given by cmulNUM, cmulRS, or other p rocedure such as 'cmulgen'.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1378 " cmul:=proc() local lname;\noptions `Copyright (c) 1995-2003 by Rafal A blamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `L ast revised: November 1, 2002`;\n##################################### ########\n if type(op(procname),procedure) then\n lname:=`B`;\n e lse\n lname:=op(procname);\n end if;\n if member(0,[args]) then \+ return 0 end if;\n if nargs <=1 then return args end if;\n if nargs \+ = 2 then\n##########################################################\n ### Speed-wise it makes no difference whether cmulgen or #\n### _defau lt_Clifford_product is used in the following. # ##################### #####################################\n return clicollect(clibilinear (eval(args[1]),eval(args[2]),cmulgen,lname)); \n end if;\n###### <=== do NOT use 'procname' in the next line this will not work\n########## ################################################\n### Speed-wise it ma kes no difference whether cmulgen or #\n### _default_Clifford_product \+ is used in the following. # ######################################### #################\nif not type(_default_Clifford_product,procedure) th en \n error \"global variable _default_Clifford_product must be assi gned a procedure so that 'cmul' could proceed beyond this point. Sorry . For help see ?cmul.\" \nend if;\n return procname(clibilinear(ev al(args[1]),eval(args[2]),cmulgen,lname),args[3..-1]); \nend p roc:\n" }}{PARA 0 "" 0 "" {TEXT 270 29 "No. 30: Ampersand version of \+ " }{TEXT 316 4 "cmul" }{TEXT 317 226 ". This version of `&c` correctly uses -K for index. When K has been assigned a matrix, use\n&c[''K'']( e1,e2) and &c[''-K''](e1,e2). Otherwise, use &c[K](e1,e2), &c[-K](e1,e 2), or &c(e1,e2). (Has been moved to Clifford:-setup).\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2304 "`&m`:=proc() local NP,ARGS,coB,nameB,lname, decindex,flagdec;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowic z and Bertfried Fauser. All rights reserved.`;\ndescription `Last revi sed: November 1, 2002`;\n############################################# \n#######################################\n### Works when &c[''K''] or &c[''-K''] is entered and K is a matrix\n############################ ###########\nflagdec:=true:\nif type(op(procname),procedure) then\n \+ if type([args],listlist) then\n if type(op(args),array) then\n \+ WARNING(\"enclose index in double quotes as in &c[''B''] or &c[' '-B''] when B has been assigned a matrix to avoid the following:\");\n return 'procname(args)';\n end if;\n else coB:=1:\n \+ nameB:=`B`:\n lname:=`B`:\n ARGS:=[args]:\n flagde c:=false:\n end if;\nelse lname:=op(procname);\n ARGS:=[args];\n if type(lname,`&*`(numeric,name)) then\n coB:=op(select(t ype,\{op(lname)\},numeric));\n nameB:=op(select(type,\{op(lnam e)\},name));\n else\n coB:=1:\n nameB:=lname:\n \+ end if;\n flagdec:=false:\n end if;\n########################## #############\ndecindex:=proc() local ARGS,coB,nameB;global B;\nif typ e([args],listlist) then\n if type(op(args),function) then\n ARG S:=op(op(args));\n coB:=1:\n nameB:=eval(op(0,op(args)));\n \+ if type(nameB,`&*`(numeric,name)) then\n coB:=op(select(t ype,\{op(nameB)\},numeric));\n nameB:=op(select(type,\{op(name B)\},name));\n end if;\n elif type(op(args),`&*`(numeric,funct ion)) then\n nameB:=\{op(op(args))\}:\n coB:=op(select(type, nameB,numeric));\n nameB:=op(select(type,nameB,function));\n \+ ARGS:=op(nameB);\n nameB:=op(0,nameB);\n else\n error \"u nable to determine index or wrong index, use name in double quotes as \+ in &c[''B''] or &c[''-B'']\"\n end if;\nelif\n type([args],list) \+ then\n ARGS:=args;\n coB:=1:\n nameB:=`B`; #default name \nelse \n error \"cannot determine arguments and/or index from arguments\" \n end if;\nreturn coB,nameB,[ARGS];\nend proc:\n##################### ################\nif flagdec then \n coB,nameB,ARGS:=decindex(args); \n lname:=coB*nameB;\nend if;\nNP:=nops(ARGS);\nif member(0,ARGS) th en return 0 end if;\nif NP <=1 then return op(ARGS) end if;\nreturn cm ul[eval(lname)](op(ARGS)); \nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 271 18 "No. 31. Procedure " }{TEXT 318 10 "useproduct" }{TEXT 319 80 " tha t allows user to select which procedure is used to compute Clifford pr oduct." }{TEXT 478 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1257 "usepr oduct:=proc(name::\{symbol,name\})\nlocal wstr;\nglobal _default_Cliff ord_product; #,cmulgen;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\n############################################################## #####\n###This procedure uses global variable _default_Clifford_produc t #\n################################################################ ### \nif not member(name,\{cmulRS,cmulNUM,cmulgen,cmul_user_defined\}) then \n WARNING(\"expecting one of the following Clifford products : cmulRS, cmulNUM, cmulgen, or cmul_user_defined\") \nend if;\nif memb er(name,\{cmul_user_defined\}) and not type(name,procedure) then\n W ARNING(\"no computations with cmul can be peformed yet since cmul_user _defined has not been defined as procedure. Select cmulRS, cmulNUM, or a new procedure as argument to useproduct.\");\n _default_Clifford_ product:=name;\nreturn NULL;\nend if;\n############################### #\n_default_Clifford_product:=name; #change value of _default_Clifford _product \n################################\nwstr:=cat(\"cmul will use \",name,\"; for help see pages ?cmul, ?Clifford:-intro, or ?\",name); \nWARNING(wstr);\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 32 . Procedure " }{TEXT 320 5 "cmulQ" }{TEXT -1 20 " and its infix form \+ " }{TEXT 321 3 "&cQ" }{TEXT -1 514 " is a special version of 'cmul' an d '&c'. It gives the Clifford multiplication in the Clifford algebra \+ of the quadratic form Q related to the symmetric part g of B as Q(x) = g(x, x) = B(x, x) where B = g + A (A is the alternating part of B). \+ Like 'cmul', it works now in all dimensions 1 through 9. Via the proc edure 'rmulm' described below in (32), this multiplication can also be applied to matrices with entries in a Clifford algebra.\n\nThis proce dure can now accept an optional index which could be K or -K. " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 78 "Propo sed by Yvon Siret, Universite Joseph Fourier , Grenoble, France. Than ks!" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 158 "Typical use: cmulQ(e1 + e2 + 2*Id, e3we4 + e6); or (e1 + e2) &cQ \+ (2*e2we3 + e4); or &cQ(e1, e2, e3); \n cmulQ(e1 we2+e2,e3+e4,e5-Pi*Id); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1423 "cmulQ:=proc() local ii,N,L,m,Sxy,symbxy,lna me,coB,nameB;global B:\noptions `Copyright (c) 1995-2003 by Rafal Abla mowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n######################################## #####\n####################################\nif type(op(procname),proc edure) then\n lname:=`B`;\nelse\n lname:=op(procname);\nend if; \n####################################\nif member(0,[args]) then retur n 0 end if;\n####################################\nSxy:=map(op,map(cli terms,\{args\}));\nSxy:=map(op,map(Clifford:-extract,Sxy,'integers')); \nsymbxy:=remove(type,Sxy,posint);\nif symbxy <> \{\} then \n return cmul[lname](args) \nend if;\nm:=max(op(Sxy),1);# 1 is needed when bot h x and y have maxindex=0\nif type(evalm(lname),matrix) then \n N:=l inalg[coldim](evalm(lname)):\n if m>N then \n error \"input con tains index larger than size of bilinear form %1\",lname \n end if: \nend if:\n################################\nif type(lname,\{name,symb ol,array,matrix\}) then\n L:=seq(lname[ii,ii],ii=1..m);\n return c mul[linalg[diag](L)](args);\nelif \n type(lname,`&*`(numeric,\{name, symbol,array,matrix\})) then\n coB:=op(select(type,\{op(lname)\},num eric));\n nameB:=op(select(type,\{op(lname)\},\{name,symbol,array,ma trix\}));\n L:=seq(coB*nameB[ii,ii],ii=1..m);\n return cmul[linalg [diag](L)](args); \nelse\n error \"index of unexpected type has bee n found in cmulQ\"\nend if;\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 272 29 "No. 33. Ampersand version of " }{TEXT 322 5 "cmulQ" }{TEXT 323 222 ". This version can accept index B and -B. When B has been defined as matrix, use\n&cQ[''B''](e1,e2) and &cQ[''-B''](e1,e2) . Otherwise, use &cQ[B](e1,e2), &cQ[-B](e1,e2) or &cQ(e1,e2). \n(Has been moved to Clifford:-setup).\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 34. Procedu re " }{TEXT 324 10 "scalarpart" }{TEXT -1 137 " computes the scalar pa rt of the given Clifford polynomial. For example, scalarpart(e1 + e2 we3) = 0 but scalarpart(2*Id + e2we3) = 2. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 46 "Typical use: scalarpart (2*Id + e1 + e1we2); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 375 "scalar part:=proc(a::\{cliscalar,clibasmon,climon,clipolynom\}) local a1,p; \+ \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried F auser. All rights reserved.`;\ndescription `Last revised: November 1, \+ 2002`;\n#############################################\na1:=simplify(a) :\nif type(a1,cliscalar) then return a1 end if;\np:=clicollect(a1):\nr eturn coeff(p,Id);\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. \+ 35. Procedure " }{TEXT 325 10 "vectorpart" }{TEXT -1 353 " computes th e k-vector part of the given Clifford polynomial u where k is a nonneg ative integer. For example, vectorpart(e1 + 3*e2we3, 2) = 3*e2we3. Wh en k = 0 then the procedure returns the scalar part of u times 'Id', e .g., vectorpart(2*Id + 3*e2we3, 0) = 2*Id. Note that vectorpart(2*Id \+ + e1we2, 0) equals 2*Id while scalarpart(2*Id + e1we2) = 2. " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 47 "Typic al use: vectorpart(e1 + e2we3 + e3, 1); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 570 "vectorpart:=proc(a::\{cliscalar,clibasmon,climon,cli polynom\},a2::nonnegint) \nlocal a1,p,K;\noptions `Copyright (c) 1995- 2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`; \ndescription `Last revised: November 1, 2002`;\n##################### ########################\na1:=simplify(a):\nif maxgrade(a1) < a2 then \+ return 0 end if;\n K:=proc() if maxgrade(args[1])=a2 then true else \+ false end if end proc:\nif type(a1,`+`) then p:=select(K,a1) elif\n \+ maxgrade(a1)<>a2 then p:=NULL else \n p:=a1 \nend if;\nif p=NULL the n return 0 else return p end if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 36. Procedure " }{TEXT 326 4 "cexp" }{TEXT -1 236 " c omputes Clifford exponential of a Clifford number in Cl(B) up to the o rder specified by the second argument which is a nonnegative integer \+ n. It n = 0 then this procedure returns 'Id'. It can accept another ar gument such as B or -B. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 185 "Typical use: cexp(e1we2*t, 3);cexp(e1we2*t, \+ 3,K);\n cexp((e1 + e1we2)*t, 4); cexp((e1 + e1we2) *t, 4,-K); \n cexp(e1we2, 3); cexp(e1 + e1we2, 4,K );\n" }}{PARA 258 "> " 0 "" {MPLTEXT 1 0 1359 "cexp:=proc(p::\{numeric ,cliscalar,clibasmon,climon,clipolynom\},N::nonnegint) \nlocal pp,k,an s,ans1,ans2,lname,coB,nameB;\noptions `Copyright (c) 1995-2003 by Rafa l Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n################################## ###########\nif nargs=2 then\n coB:=1:\n nameB:=`B`: \n lname :=`B`: \nelif nargs=3 then\n if type(args[3],\{name,symbol,matrix,a rray\}) then\n coB:=1:\n nameB:=args[3];\n lname:=ar gs[3];\n elif type(args[3],`&*`(numeric,\{name,symbol,matrix,array \})) then\n coB:=op(select(type,\{op(args[3])\},numeric));\n \+ nameB:=op(remove(type,\{op(args[3])\},numeric));\n lname:=arg s[3]:\n else \n error \"wrong type of third argument in cexp. See ?cexp for more help.\" \n end if;\nelse\n error \"two or thr ee arguments expected in cexp. See ?cexp for more help.\"\nend if;\n## ##############################\nk:='k':\nif type(p,\{numeric,cliscalar \}) then return (add(p^k/k!,k=0..N)) end if;\nif evalb(vectorpart(p,0) =p) then \n pp:=scalarpart(p);\n return ((add(pp^k/k!,k=0..N)*Id)) \nend if;\npp:=clisort(displayid(p)):\nif N=0 then return Id \n eli f N=1 then return Id+pp; \n else \n ans1:=cexp(pp,N-1,lname); \n ans2:=cexp(pp,N-2,lname);\n ans:=ans1+cmul[lname](((ans 1-ans2)*(N-1)!),pp)/N!;\n return ans;\nend if;\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 37. Procedure " }{TEXT 327 5 "cexpQ " }{TEXT -1 257 " computes Clifford exponential of a Clifford number i n Cl(Q) up to the order specified by the second argument which is a n onnegative integer n. It n = 0 then this procedure returns 'Id'. This procedure can also accept an optional argument such as B or -B." }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 210 "Typi cal use: cexpQ(e1we2*t, 3); or cexpQ((e1 + 2*e1we2)*t, 4);\n \+ cexpQ(e1we2*t, 3,K); or cexpQ((e1 + 2*e1we2)*t, 4,K);\n \+ cexpQ(Id+2*e1we3,4); or cexpQ(e1 + 2*e1we2, 4,-K);\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1373 "cexpQ:=proc(p::\{numeric,clis calar,clibasmon,climon,clipolynom\},N::nonnegint) \nlocal pp,k,ans,ans 1,ans2,lname,coB,nameB;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\nif nargs=2 then\n coB:=1:\n nameB:=`B`: \n lname:=`B` : \nelif nargs=3 then\n if type(args[3],\{name,symbol,matrix,array \}) then\n coB:=1:\n nameB:=args[3];\n lname:=args[3 ];\n elif type(args[3],`&*`(numeric,\{name,symbol,matrix,array\})) \+ then\n coB:=op(select(type,\{op(args[3])\},numeric));\n na meB:=op(remove(type,\{op(args[3])\},numeric));\n lname:=args[3]: \n else \n error \"wrong type of third argument in cexpQ. See ?cexpQ for more help.\" \n end if;\nelse\n error \"two or three \+ arguments expected in cexpQ. See ?cexpQ for more help.\"\nend if;\n### #############################\nk:='k':\nif type(p,\{numeric,cliscalar \}) then return (add(p^k/k!,k=0..N)) end if;\nif evalb(vectorpart(p,0) =p) then \n pp:=scalarpart(p);\n return add(pp^k/k!,k=0..N)*Id \ne nd if;\npp:=clisort(displayid(p)):\nif N=0 then return Id \n elif N= 1 then return Id+pp; \n else \n ans1:=cexpQ(pp,N-1,lname); \n ans2:=cexpQ(pp,N-2,lname);\n ans:=ans1+cmulQ[lname] (((ans1-ans2)*(N-1)!),pp)/N!;\n return ans;\nend if;\nend proc :\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 38. Procedure " }{TEXT 328 4 "wexp" }{TEXT -1 168 " computes exterior exponential of a Clifford n umber u up to the order specified by the second argument which is a n onnegative integer n. It returns 'Id' when n = 0. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 38 "Typical use: wexp( e1we2 + e3we4, 5); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 610 "wexp:= pr oc(p::\{cliscalar,clibasmon,climon,clipolynom\},N::nonnegative) \nloca l pp,power,cu,i;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revis ed: November 1, 2002`;\n############################################# \n if nargs<>2 then error \"two parameters are needed in 'wexp'\" end if;\n pp:=expand(p);\n if N=0 then return 1 elif\n N=1 then ret urn 1+clisort(pp) end if;\n power:=pp;\n cu:=1+pp;\n for i from 2 t o N do\n power:=wedge(power,pp);\n cu:=cu + power/i!;\n end d o;\n return subs(Id=1,clicollect(clisort(cu)));\n end proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 39. Procedure " }{TEXT 329 9 "rever sion" }{TEXT -1 411 " calculates reversion in the Clifford algebra. It is linear in its argument and it is always a Clifford algebra anti-au tomorphism. When the antisymmetric part of B is not zero, 'reversion' does not preserve the multilinear structure of the algebra because it mixes grades, i.e., it does not preserve the gradation of the exterio r algebra. This procedure can now take a third optional argument such as B or -B." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 53 "Typical use: reversion(2*e1we2 + 4*Id - e3we4we5); \n" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 2639 "reversion:=proc(a1::\{cliscalar, clibasmon,climon,clipolynom,matrix\}) \n local ind,expr,wtp, ptw,lname,flagindexed;\n global _scalartypes,B;\noptions `Co pyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All ri ghts reserved.`;\ndescription `Last revised: November 1, 2002`;\n##### ########################################\nif hastype([args[1]],cliprod ) then \n error \"in order to handle 'type/cliprod', load in package Cliplus\" \n end if;\n############################\nif type(a1,clisca lar) then return a1 end if;\n############################\nif nargs=1 \+ then\n lname:=`B`;\n flagindexed:=false:\nelif nargs=2 and type( args[2],\{symbol,name,array,matrix,`&*`(algebraic,name)\}) then\n l name:=args[2];\n flagindexed:=true:\nelse error \"only one or two a rguments are expected\"\nend if;\n############################\n### Au xiliary function that converts wedges to Clifford products: wedge ->> \+ Clifford product\n############################\nwtp:=proc(a1,lname) lo cal ind,i,arg,rdmon,eq1,ans; global _scalartypes; \nif type(a1,\{`+` ,`*`\}) then return (map(wtp,a1,lname)) \n elif type(a1,_scalartypes ) then return a1\n elif type(a1,symbol) and SearchText(w,a1)=0 then \+ return a1\n elif type(a1,symbol) and not member(length(a1),\{5,8,11, 14,17,20,23,26\}) \n then return a1 \nend if;\nrdmon:=reorder(a 1):\nind:=Clifford:-extract(a1,'integers'):\ni:='i':\narg:=[seq(cat(e, op(ind[i])),i=1..nops(ind))];\neq1:=cat(op(arg))=simplify(eval(cmul[ln ame](op(arg))));\nif a1=rdmon then ans:=simplify(solve(eq1,a1)) \n \+ else ans:=-simplify(solve(-eq1,-rdmon)) \nend if;\nif nops( ind) < 4 then return ans else return wtp(ans,lname) end if;\nend proc: \n############################\n### Auxiliary function that converts C lifford products to wedge: Clifford products ->> wedge\n############## ##############\nptw:=proc(a1,lname) local i,arg,revarg; global _scalar types; \nif type(a1,\{`+`,`*`\}) then return (map(ptw,a1,lname)) \n \+ elif type(a1,_scalartypes) then return a1 \n elif type(a1,symbol) an d SearchText(e,a1)=0 then return a1 \n elif type(a1,symbol) and leng th(a1)=2 then return a1 \n elif type(a1,symbol) and not member(lengt h(a1),\{2,4,6,8,10,12,14,16,18\})\n then return a1 \n end if;\n i:='i':\narg:=[seq(cat(e,substring(a1,2*i..2*i)),i=1..(length(a1)/2))] ;\nrevarg:=[seq(arg[nops(arg)-i],i=0..(nops(arg)-1))];\nreturn expand( eval(cmul[lname](op(revarg))))\nend proc:\n########################### ###\n### Now the actual function:\n##############################\nif \+ type(a1,matrix) then return map(reversion,a1,lname) end if;\nexpr:=ptw (expand(wtp(a1,lname)),lname);\nexpr:=expand(displayid(expr)):\nreturn clisort(expr)\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 40. \+ Procedure " }{TEXT 330 11 "conjugation" }{TEXT -1 317 " calculates con jugation in the Clifford algebra. It is linear in its argument. Note \+ that 'conjugation' is defined as a composition of 'reversion' and 'gra deinv'. Hence, it does not preserve the multivector gradation when th e antisymmetric part of B is non-zero. It can now accept optional arg ument such as B or -B." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 41 "Typical use: conjugation(e1 + 4*e2we3); " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 823 "co njugation:=proc(a1::algebraic) local lname;global B;\noptions `Copyrig ht (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights \+ reserved.`;\ndescription `Last revised: November 1, 2002`;\n########## ###################################\nif nargs=1 then\n lname:=`B`; \nelif nargs=2 and type(args[2],\n \{symbol,name,array,matrix,`&*` (numeric,\{symbol,name,array,matrix\})\}) then\n lname:=args[2];\ne lse error \"only one or two arguments are expected\"\nend if;\n####### ####################\nif type(a1,matrix) then return map(procname,a1,l name) elif\n type(a1,cliscalar) then return a1 elif\n type(a1,\{cl ibasmon,climon,clipolynom\}) then\n return eval(gradeinv(revers ion(a1,lname)))\nelse \n error \"wrong input type: input must be of \+ type cliscalar, clibasmon, climon, clipolynom, or 'matrix'\" \nend if; \nend proc:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 41. Procedure " }{TEXT 331 8 "c_conjug" }{TEXT -1 72 " calculates complex conjugate in a complexified Clifford algebra; thu s, " }}{PARA 258 "" 0 "" {TEXT -1 80 " \+ c_conjug(u) = c_conjug(a + I*b) = a - I*b " }}{PARA 258 "" 0 "" {TEXT -1 140 "where a and b are in the real Clifford algebra and `I` i s the imaginary unit, i.e., I = sqrt(-1). This procedure is linear in \+ its argument. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 51 "Typical use: c_conjug((1 + 2*I)*e1 - 3*I*e1we2); \n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 697 "c_conjug:=proc(a1::algebraic) loca l ba,co,terms,t,i;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowi cz and Bertfried Fauser. All rights reserved.`;\ndescription `Last rev ised: November 1, 2002`;\n############################################ #\nif type(a1,matrix) then return map(procname,a1) elif\n type(a1,cl iscalar) then return conjugate(a1) elif\n type(a1,\{clibasmon,climon ,clipolynom\}) then\n t:='t':\n ba:=cliterms(a1);\n \+ co:=[coeffs(a1,ba,'t')];\n terms:=[t];i:='i':\n retur n clisort(add(conjugate(co[i])*terms[i],i=1..nops(co)))\n else \nerr or \"wrong input type: input must be of type cliscalar, clibasmon, cli mon, clipolynom, or 'matrix'\" \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 42. Procedure " }{TEXT 332 6 "buildm" }{TEXT -1 863 " builds a matrix for the given element u of the Clifford algebra \+ Cl(B) in the left- or right-regular representation, or under Lie or au tomorphism action with respect to an ordered basis specified by the us er. The element p is entered as the first argument and the basis in t he form of a list is specified as the second argument, e.g., buildm(u, basis). It is also possible to specify options 'left', 'right', 'Lie ', 'auto', 'false, and 'true'. For example, one can find the left-regu lar representation of the algebra on itself or, when Cl(B) is simple a nd isomorphic to a ring of real matrices, one can find matrices repres enting Clifford polynomials in a real basis of a minimal ideal. Howev er, there are new procedures below specifically designed for finding s pinor representations of Clifford algebras in terms of real, complex, \+ and quaternionic matrices. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 270 "Typical use: \n\nbuildm(e1, [Id, e1, e 2, e1we2]); buildm(e1, [Id, e1, e2, e1we2], 'right'); buildm(e1, [Id, \+ e1, e2, e1we2], 'Lie');\nbuildm(e2, [Id, e1, e2, e1we2],'false'); buil dm(e1we2+e2, [Id, e1, e2, e1we2], 'true'); buildm(e1, [Id, e1, e2, e1w e2], 'Lie','false'); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2967 "bui ldm:=proc(a1::\{cliscalar,clibasmon,climon,clipolynom\},\n \+ a2::list(\{cliscalar,clibasmon,climon,clipolynom\}))\nlocal A,L,N,a11 ,xm,i,j,Lbasis,neq,vars,sys,sol,nontrivial,a33,flag;\noptions `Copyrig ht (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights \+ reserved.`;\ndescription `Last revised: November 1, 2002`;\n########## ###################################\nflag:=true:\nif nargs=2 then a33: ='left' end if;\nif nargs=3 then \n if member(args[3],\{'true','fals e'\}) then flag:=args[3];\n \+ a33:='left';\n elif member(args[3],\{'left','right','Lie','auto'\} ) \n then a33:=args[3]\n else \+ error \"third optional argument must be 'left', 'right', 'Lie', 'auto' , 'true', 'false'\"\n end if; \nend if;\nif nargs=4 then\n if memb er(args[3],\{'left','right','Lie','auto'\}) and member(args[4],\{'fals e','true'\}) then\n a33:=args[3]; \n flag:=args[4];\n else \n error \"third optional argument must be 'left', 'right ', 'Lie', 'auto', and the fourth optional argument must be 'false' or \+ 'true'\"\n end if;\nend if;\nif nargs>4 then error \"too many argume nts. See ?buildm for more help.\" end if;\n########################### ######################\nif flag then \nA:=linalg[genmatrix](args[2],cb asis(maxindex(args[2])));\nif linalg[rank](A) < nops(args[2]) then \n \+ error \"elements of the list %1 are linearly dependent. Apply 'findb asis' to this list first.\",a2 \nend if;\nend if;\n###local procedure \nnontrivial:=proc(S::\{set(\{relation,algebraic\}),list(\{relation,al gebraic\})\}) \nlocal istrivial;\nprintlevel:=2:\nistrivial:=proc(x) i f type(x,relation) then evalb(x) else evalb(x=0) end if end;\nremove(i strivial,S)\nend proc:\n### \nL:=a2:N:=nops(L):xm:=array(1..N,1..N):\n if a33='left' then \n for i from 1 to N do \n eq||i:=clicoll ect(expand(cmul(a1,L[i])-add(xm[j,i]*L[j],j=1..N))) \n end do;\neli f a33='right' then \n for i from 1 to N do \n \+ eq||i:=clicollect(expand(cmul(L[i],a1)-add(xm[j,i]*L[j],j=1..N)))\n \+ end do;\nelif a33='Lie' then\n for i from 1 to N do\n e q||i:=clicollect(expand(cmul(L[i],a1)-cmul(a1,L[i])-add(xm[j,i]*L[j],j =1..N)))\n end do;\nelif a33='auto' then\n a11:=cinv(a1):\n \+ for i from 1 to N do \n eq||i:=clicollect(ex pand(cmul(cmul(a1,L[i]),a11)-add(xm[j,i]*L[j],j=1..N)))\n end do; \nelse error \"third optional argument must be 'left', 'right', 'Lie', or 'auto'\"\nend if;\n############################################### ###########\nLbasis:=[op(`union` (seq(cliterms(L[i]),i=1..N)))];\nfor \+ i from 1 to N do \n for j from 1 to nops(Lbasis) do \n neq[i ,j]:=coeff(eq||i,Lbasis[j])=0 \nend do;\nend do;\nvars:=convert(evalm( xm),set):sys:=map(op,\{entries(neq)\});\nsys:=nontrivial(sys): #elimin ate trivial equations\nsol:=solve(sys,vars);\nif sol=NULL then \n er ror \"no matrix represents %1 in the basis %2 under the %3 action\",a1 ,a2,a33; \nend if;\nassign(sol);\nreturn evalm(xm);\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 43. Procedure " }{TEXT 333 9 "findb asis" }{TEXT -1 680 " finds a basis in a linear vector space spanned b y a set of Clifford polynomials entered as a list. The procedure is u sed, for example, when finding a basis for a spinor space S considere d as a minimal left or right ideal in Cl(B) generated by a primitive i dempotent f. To speed up computations, it is advisable to a standard C lifford basis for Cl(B) in the form of a list of basis monomials as th e second argument. If only one list is specified, 'findbasis' determi nes a suitable Clifford basis itself but it takes twice as much time t hen since it creates a Clifford basis by using 'cbasis(maxindex)' wher e 'maxindex' is the maximum index found among the elements of the list ." }}{PARA 258 "" 0 "" {TEXT -1 69 "\nTypical use: findbasis([2*e1+e2, e2+e1we2,e1we2],[Id,e1,e2,e1we2]);\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1428 "findbasis:=proc(a1,a2) local L,clibasis,M,i,m,r,v,S; \nglobal \+ _prolevel;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and B ertfried Fauser. All rights reserved.`;\ndescription `Last revised: No vember 1, 2002`;\n#############################################\nif ev alb(_prolevel=false) then\n if nargs=1 and not (type(a1,list(\{cliba smon,climon,clipolynom\})) or \n type(a1,set(\{c libasmon,climon,clipolynom\}))) then\nerror \"argument of type list/se t(\{clibasmon,climon, or clipolynom\}) was expected\"\n elif nargs=2 and \n not ((type(a1,list(\{clibasmon,climon,clipolynom\})) or \+ \n type(a1, set(\{clibasmon,climon,clipolynom\}))) and \n \+ (type(a2,list(clibasmon)) or type(a2,set(clibasmon)))) or nar gs>2 then\nerror \"arguments of type list/set(\{clibasmon,climon,clipo lynom\}) and list/set(clibasmon) were expected\" \nend if;\nend if;\ni f nops(a1)=1 then return a1 end if;\nL:=sort(map(displayid,convert(a1, list)),bygrade):\nif nargs=2 then clibasis:=sort(convert(a2,list),bygr ade) else \n clibasis:=sort(convert(`union`(op(map(cliterms,L))),lis t),bygrade);\nend if;\nM:=linalg[genmatrix](L,clibasis);\nr:=linalg[ra nk](M):m:=linalg[rowdim](M):\nfor i from 1 to m do v[i]:=linalg[row](M ,i) end do;\nS:=[v[1]]:\nfor i from 2 to m while nops(S) < r do \n \+ if linalg[rank](linalg[stackmatrix](op(S),v[i]))=nops(S)+1 \n th en S:=[op(S),v[i]] \n end if\nend do;\nreturn [seq(L[i],i=map(op,S) )]\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 44. Procedure " }{TEXT 334 12 "minimalideal" }{TEXT -1 143 " calculates a real basis f or a left S=Cl(B)f or right S=fCl(B) minimal ideal in the algebra Cl(B ) where f is a primitive idempotent in Cl(B). " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 151 "The first argument of \+ the procedure is an ordered list of basis monomials sorted bygrade, e. g., a Clifford basis generated by the procedure 'cbasis'. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 198 "Note: to s ort a list L by grade one may use sort(L, bygrade) where 'bygrade' i s a new procedure in this package described below. The output from th e procedure 'cbasis' is already sorted that way." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 392 "The second argument is the idempotent f. If the idempotent f is the same as the one stored \+ under clidata()[4] then 'minimalideal' uses the generators of S stor ed under clidata()[5] to generate the real basis and it returns the st ored list clidata()[5] as the second list in its ouput. If f does n ot equal clidata()[4] then complete computations are performed but th ey may take longer. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 129 "It is assumed that the numerical values of B have been specified.\n\nThe procedure returns a list consisting of two ord ered lists: " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 91 "(1) the first list contains the real basis of S written \+ as expanded Clifford polynomials; " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 106 "(2) the second list contains basis m onomials from the standard basis in Cl(B) which generate the \+ " }}{PARA 258 "" 0 "" {TEXT -1 108 " first list by multiplying f o n the left or on the right depending whether S=Cl(B)f or S=fCl(B). \+ " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 260 "There is a one-to-one correspodence between the two ordered lists .\n\nTypical use: minimalideal([Id,e1,e2,e3,e1we2,e1we3,e2we3,e1we2we3 ],(1/2)*(Id+e3),'left');\n minimalideal([Id,e1, e2,e3,e1we2,e1we3,e2we3,e1we2we3],(1/2)*(Id+e3),'right');\n" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2246 "minimalideal: =proc(a1,a2,a3) \nlocal L,gens,m,flag1,f,flag_left,data,SB,g,SBgens,pq ,p,q,l,ni,realdim,dimoverK,cb,N,bel; \nglobal B,_shortcut_in_minimalid eal,_prolevel;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz a nd Bertfried Fauser. All rights reserved.`;\ndescription `Last revised : November 1, 2002`;\n#############################################\ni f not type(B,diagmatrix) then \n error \"bilinear form B has not bee n assigned a matrix or is not diagonal\" \nend if; \nif not _prolevel \+ then\n if not type(a1,list(\{clibasmon,climon,clipolynom\})) then\n \+ error \"first argument must of type list(\{clibasmon,climon,c lipolynom\})\" \n elif not type(a2,'primitiveidemp') then \n \+ error \"second argument must be a primitive idempotent\" \n \+ elif not member(a3,\{'left','right',\"left\",\"right\"\}) th en\n error \"third argument must be 'left', or 'rig ht'\" \n end if;\n end if;\nf:=displayid(eval(a2)):\nif member(a3,\{'l eft',\"left\"\}) then flag_left:=true else flag_left:=false end if;\ng :='g':\nL:=sort(a1,bygrade):\nif _shortcut_in_minimalideal then\n m: =maxindex(L):\n flag1:=evalb(L=cbasis(m)): \n if flag1 then\n \+ data:=clidata():\n if eval(eval(data[4]))=eval(f) or eval(eval( data[4]))=gradeinv(f) then\n SBgens:=data[5]:\n if fla g_left then SB:=[seq(cmulQ(g,f),g=SBgens)] else \n \+ SB:=[seq(cmulQ(f,g),g=SBgens)] \n end if;\n retu rn [SB,SBgens,a3];\n end if;\n end if;\nend if; \n#If can't \+ use the shortcut, perform necessary computations.\npq:=Bsignature():\n p:=pq[1]:q:=pq[2]:\nl:=floor((p+q)/2);ni:=2^(l-1);\nif member((p-q) mo d 8,\{0,1,2\}) then \n realdim:=2*ni; \n dimoverK:=2*ni; \ne lif member((p-q) mod 8,\{3,7\}) then \n realdim:=4*ni; \n di moverK:=2*ni; \nelse\n realdim:=4*ni; \n dimoverK:=ni \nend \+ if;\ngens:=clidata()[5]: #put elements from clidata()[5] first in L\nL :=remove(member,L,gens):\nL:=[op(gens),op(L)]:\nSB:=[f]:SBgens:=[Id]:c b:=remove(member,L,[Id]); \nfor g in cb while nops(SB) < realdim do\n \+ N:=nops(SB):\n if flag_left then bel:=cmulQ(g,f) else bel:=cmulQ (f,g) end if; \n SB:=findbasis([op(SB),bel]); \n if nops(SB)>N \+ then SBgens:=[op(SBgens),g] end if;\nend do:\nreturn [SB,SBgens,a3];\n end proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 48. Procedure " } {TEXT 335 6 "Kfield" }{TEXT -1 340 " computes a basis for a field K. \+ The field K is the field of the spinor space S = Cl(B)f or S = fCl(B) \+ of the given Clifford algebra Cl(B). It is isomorphic to the reals, \+ or to the complexes, or to the quaternions according to whether (p-q) mod 8 is 0, 1, 2, or 3, 7, or 4, 5, 6, respectively (here [p,q] is the signature of B). " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 205 "Assuming that the bilinear form B has been d efined, the first argument of the procedure is expected to be the same as the output from the procedure 'minimalideal'. The second argument is the idempotent f." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 225 "The procedure eliminates from the list of basis e lements in the real ideal space nilpotent elements and leaves only tho se whose square modulo f is either +1 or -1. It returns those element s as the first list in its output. " }}{PARA 258 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 200 "If the primitive idempotent f is \+ the same as the one stored under clidata()[4] and if the generators of the real basis in the minimal ideal S match those stored under clidat a()[5], then the procedure" }}{PARA 258 "" 0 "" {TEXT -1 99 "uses gene rators of K stored under clidata()[6] and returns them as the second \+ list in its ouput. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 178 "Thus, the second list in the output contains gene rators (Clifford basis monomials) of the elements in the first list. \+ Elements of the two lists are in one-to-one relationship. " }}{PARA 258 "" 0 "" {TEXT -1 204 "\nTypical use: dim:=2:B:=linalg[diag](1,-1): clibasis:=cbasis(dim):data:=clidata(B):f:=data[4]:\n \+ sbasis:=minimalideal(clibasis,f,'left'); \n \+ Kfield(sbasis,f);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4296 "Kfiel d:=proc(a1::list(\{list,string,symbol\}),a2::clipolynom) \nlocal SB,ge ns,f,ff,k,n,fg,f_from_data,field,flag3,side,expr,i,ijk,g,dimen,Kbasis, Kgens,Kdim,data,T4: \nglobal B,_shortcut_in_Kfield,_prolevel;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. Al l rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n# ############################################\n#### Local procedure nee ded only in 'Kfield' ###\nT4:=proc() \nlocal gens,Kbasis,f,mi,clibas,c libas2,x,y,z; global B;\nKbasis:=args[1];f:=Kbasis[1];mi:=max(op(map(m axindex,Kbasis)));\nclibas:=subsop(1=NULL,cbasis(mi));\nif type(B,matr ix) then gens:=subsop(1=NULL,clidata()[6]);\n clib as:=remove(member,clibas,gens):\n clibas:=[op(gens ),op(clibas)];\nend if;\nclibas2:=[]:\nfor x in clibas do \n if eva lb(cmul(x,x) = -Id) then clibas2:=[op(clibas2),x] end if; \nend do:\nf or x in clibas2 do \nfor y in remove(member,clibas2,[x]) do\nfor z in \+ remove(member,clibas2,[x,y]) do\n if member(cmul(x,f),\{Kbasis[2] ,-Kbasis[2]\}) then \n if member(cmul(y,f),\{Kbasis[3],-Kbasis [3]\}) then\n if member(cmul(z,f),\{Kbasis[4],-Kbasis[4]\}) then \n if type([x,y,z],'purequatbasis') then return [x ,y,z]\n end if;\n end if;\n end if;\n end if;\nend do;\nend do;\ne nd do;\nend proc:\n##############################################\nif \+ not _prolevel then\n if not type(a2,'primitiveidemp') then \n e rror \"second argument must be a primitive idempotent\"\n end if;\ne nd if;\nSB:=a1[1]:gens:=a1[2]:side:=a1[3]:f:=eval(a2):i:='i':g:='g':\n if not member(f,SB) then \n error \"idempotent entered %1 is not a m ember of the first list\",f \nend if;\n###new line here instead of >>> not assigned(B)<<<\nif not type(B,matrix) then \n error \"matrix mus t be assigned to B\" \nend if;\nif side='left' then flag3:=true else f lag3:=false end if;\ndata:=clidata():\nfield:=data[1]:\nif field = 're al' then return [[f],[Id]] \nelif field = 'complex' then \n if _shortcut_in_Kfield then\n f_from_data:=eval(eval(data[4])) :\n fg:=gradeinv(f): \n if member(f_from_data,\{f ,-f,fg,-fg\}) and gens=data[5] then \+ Kgens:=data[6];\nif flag3 then Kbasis:=[f,seq (cmul(Kgens[i],f),i=2..nops(Kgens))]\n else Kbasis:=[f,seq(cmu l(f,Kgens[i]),i=2..nops(Kgens))] \nend if;\nreturn ([Kbasis,Kgens]) \n end if;\nend if;\n#Do this when shortcut can't be used when field = 'c omplex'\nKdim:=2: \nKbasis:=[f]:Kgens:=[Id]:ff:=[op(data[4])]:n:=nops( ff);\nfor i from 1 to nops(SB) while nops(Kbasis) < Kdim do\n if c mul(gens[i],gens[i])=-Id then\n expr:=gens[i]:\n for k f rom 1 to n while expr<>0 do\n expr:=cmul(ff[n-k+1],expr,ff[ k]);\n end do; \n if expr<>0 then Kbasis:=[op(Kbasis),SB[ i]];\n Kgens:=[op(Kgens),gens[i]] \n end if;\n \+ end if;\nend do;\nreturn [Kbasis,Kgens]\nelif field = 'quaternionic ' then \n dimen:=linalg[coldim](B):\n if dimen=2 then Kbasis:= [op(SB)];\n Kgens:=[op(gens)];\n \+ return [Kbasis,Kgens]\n elif member(dimen,\{3,4,5,6,7,8,9\}) th en\n if _shortcut_in_Kfield then\n f_from_data:= eval(eval(data[4])):\n fg:=gradeinv(f): \n \+ if member(f_from_data,\{f,-f,fg,-fg\}) and gens=data[5] then \+ Kgens:=data[6];\nif \+ flag3 then Kbasis:=[f,seq(cmul(Kgens[i],f),i=2..nops(Kgens))]\n \+ else \n Kbasis:=[f,seq(cmul(f,Kgens[i]),i= 2..nops(Kgens))] \nend if;\nreturn [Kbasis,Kgens] \nend if;\nend if;\n end if;\n#Do this when shortcut can't be used and field = 'quaternioni c'\nKdim:=4:\nKbasis:=[f]:Kgens:=[Id]:ff:=[op(data[4])]:n:=nops(ff);\n for i from 1 to nops(SB) while nops(Kbasis) < Kdim do\n if cmul(ge ns[i],gens[i])=-Id then\n expr:=gens[i]:\n for k from 1 \+ to n while expr<>0 do\n expr:=cmul(ff[n-k+1],expr,ff[k]);\n end do; \n if expr<>0 then Kbasis:=[op(Kbasis),SB[i]] e nd if;\nend if;\nend do;\n ijk:=T4(Kbasis);\n Kgens:=[Id,op(ij k)]:\nif flag3 then Kbasis:=[f,seq(cmul(g,f),g=ijk)] else \n \+ Kbasis:=[f,seq(cmul(f,g),g=ijk)]\nend if;\nreturn [Kbasis,Kgens]\n else error \"wrong name of the field. See ?Kfield for more help.\" \n end if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 46. Procedu re " }{TEXT 336 12 "spinorKbasis" }{TEXT -1 263 " finds a spinor basis for S=Cl(B)f or S=fCl(B) over a field K where K is isomorphic to the reals, or to the complexes, or to the quaternions according to whethe r (p-q) mod 8 is 0, 1, 2, or 3, 7, or 4, 5, 6, respectively (here [p,q] is the signature of B). " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 276 "The first argument is an ordered list \+ SBgens containing generators of a real basis in a minimal ideal Cl(B)f or fCl(B) (it doesn't matter whether the ideal was left or right). T hese generators are found by the procedure 'minimalideal' and are retu rned by it as a second list." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 104 "The second argument is the primitive i dempotent f used to generate the minimal ideal Cl(B)f or fCl(B)." }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 150 "The \+ third argument is a list FBgens of generators that generate the field \+ K; these generators are returned as a second list by the procedure 'Kf ield'." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 143 "The fourth argument is either 'left' or 'right' depending whet her we deal with the left minimal ideal Cl(B)f or the right minimal id eal Cl(B)f." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 434 "If the first three arguments in the input match respecti vely clidata()[5], clidata()[4], and clidata()[6] in that order, i.e ., SBgens=clidata()[5], f=clidata()[4], and FBgens=clidata()[6], the n the procedure finds previously computed generators of S over K which are stored as clidata()[7]. These generators are then used to comput e the K-basis for S=Cl(B)f or S=fCl(B) depending whether the fourth ar gument is 'left' or 'right'." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 47 "The procedure returns a list of three e lements:" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 163 "(1) the first list is an ordered list of Clifford polynomials \+ which give a basis in Cl(B)f or fCl(B) (depending on what was the fou rth argument in the procedure);" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 178 "(2) the second list is an ordered list of generators over f which give the elements in the first list. Ther e is a one-to-one correspodence between the elements of the two lists. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 254 " (3) the third element in the output is either 'left' or 'right' and it matches the fourth argument in the input to the procedure. That elem ent is to remind the user that the basis returned as the first list is for the left or right ideal respectively. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 334 "Typical use: dim:=2:B: =linalg[diag](1,-1):clibasis:=cbasis(dim):data:=clidata(B):f:=data[4]: \n sbasis:=minimalideal(clibasis,f,'left');\n \+ fbasis:=Kfield(sbasis,f);\n \+ SBgens:=sbasis[2];FBgens:=fbasis[2];\n s pinorKbasis(SBgens,f,FBgens,'left')\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2864 "spinorKbasis:=\nproc(a1::list,a2::\{clibasmon,climon,clipolyno m\},a3::list,a4::\{string,symbol\}) \nlocal flag,flag_left,Kdim,f,SBge ns,SB,FBgens,g,SBKbasis,SBKgens,data,i,poss,m,p; \nglobal B,_shortcut_ in_spinorKbasis,_prolevel;\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription ` Last revised: November 1, 2002`;\n#################################### #########\nif not type(B,matrix) then \n error \"matrix must be assi gned to B\" \nend if;\nif not _prolevel then\n if not type(a2,'idemp otent') then \n error \"second argument must be an idempotent\" e lif\n not member(a4,\{'left','right',\"left\",\"right\"\}) then \n \+ error \"the fourth argument must be 'left', or 'right'\"\n end i f;\nend if;\nSBgens:=a1:f:=eval(a2):FBgens:=a3:\nif SBgens=FBgens then return [[f],[Id],a4] end if;\nif a4='left' or a4=\"left\" then flag_l eft:=true else flag_left:=false end if;\ndata:=clidata():\nif _shortcu t_in_spinorKbasis then\n if eval(f)=eval(data[4]) and SBgens=data [5] and FBgens=data[6] then\n SBKgens:=data[7];\n SBKbasis:= []:\n g:='g':\n if flag_left then SBKbasis:=[seq(cmulQ(g,f), g=SBKgens)]\n else SBKbasis:=[seq(cmulQ(f,g),g=SBKge ns)]\n end if; \n return [SBKbasis,SBKgens,a4];\n end \+ if;\nend if; \nKdim:=nops(FBgens):SB:=[]:\ng:='g':\nif flag_left then \+ SB:=[seq(cmulQ(g,f),g=SBgens)] \n else SB:=[seq(cmulQ(f,g) ,g=SBgens)]\nend if;\nif Kdim=1 then return [SB,SBgens,a4] end if;\nm: =max(op(map(maxindex,SBgens)));\nposs:=cbasis(m);\nSBKgens:=[Id]:\ng:= 'g':\nif flag_left then SB:=remove(member,SB,[seq(cmul(f,g),g=FBgens)] )\n else SB:=remove(member,SB,[seq(cmul(g,f),g=FBgens)])\n end if;\nposs:=remove(member,poss,FBgens);\nfor g in poss while nops(S B)>0 do\n if flag_left then \n for i from 1 to Kdim do p[i]:= cmul(g,f,FBgens[i]) end do;\n else \n for i from 1 to Kdim do p[i]:=cmul(FBgens[i],f,g) end do;\n end if; \n for i from 1 to Kdim do\n flag[1,i]:=member(p[i],SB): \n flag[2,i ]:=member(-p[i],SB):\n end do;\n if Kdim=2 then \n if ( flag[1,1] or flag[2,1]) and (flag[1,2] or flag[2,2]) then\n S B:=remove(member,SB,[p[1],-p[1],p[2],-p[2]]):\n SBKgens:=[op( SBKgens),g]\n end if:\n else\n if (flag[1,1] or flag[2,1 ]) and \n (flag[1,2] or flag[2,2]) and\n (flag[1,3] or flag[2,3]) and\n (flag[1,4] or flag[2,4])\n then\n \+ SB:=remove(member,SB,[p[1],-p[1],p[2],-p[2],p[3],-p[3],p[4],-p[4]]) :\n SBKgens:=[op(SBKgens),g]\n end if:\n end if;\n \+ if flag[1,1] then SBKbasis:=[op(SBKbasis),p[1]] else\n \+ SBKbasis:=[op(SBKbasis),-p[1]] \n end if;\n end do;\ng:='g ':\nif flag_left then SBKbasis:=[seq(cmul(g,f),g=SBKgens)] else\n \+ SBKbasis:=[seq(cmul(f,g),g=SBKgens)]\nend if;\nreturn [SB Kbasis,SBKgens,a4]\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. \+ 47. Procedure " }{TEXT 337 10 "squaremodf" }{TEXT -1 390 " computes th e square of a basis element u in a left or right minimal ideal Cl(B)f \+ or fCl(B) entered as the first argument modulo a primitive idempotent f entered as the second argument. The procedure doesn't check wheth er f is primitive or not. Thus, the procedure returns 1 or -1 dependi ng whether cmul(u,u) = f or cmul(u,u) = -f. The procedure returns 0 \+ if u is a nilpotent element." }}{PARA 258 "" 0 "" {TEXT -1 115 "\nThis procedure is needed to identify/verify squares of the basis elements \+ in the field K of the spinor ideal S. \n" }}{PARA 258 "" 0 "" {TEXT -1 54 "Typical use: squaremodf((1/2)*(Id+e1),(1/2)*(Id+e1);\n " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 783 "squaremodf:=proc(a1::\{clibasmon,c limon,clipolynom\},a2::idempotent) \nlocal p;global B;\noptions `Copyr ight (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All right s reserved.`;\ndescription `Last revised: November 1, 2002`;\n######## #####################################\nif nargs<>2 then \n error \"t wo arguments needed of type clibasmon, or climon, or clipolynom, and ' idempotent'\" \nend if;\nif a1=a2 then return 1 elif\n not type(B,ma trix) then error \"matrix must be assigned to B\" \nend if;\np:=cmul(a 1,a1):\nif expand(p-a2)=0 then return 1 elif\n expand(p+a2)=0 then r eturn -1 elif\n (p=0 or type(a1,nilpotent)) then return 0 else \+ \n error \"either element %1 is not a basis element or it does not belong to the spinor space Cl(Q)f (or fCl(Q))\",a1 \nend if;\nend pr oc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 48. Procedure " }{TEXT 338 8 "RHnumber" }{TEXT -1 76 " gives the Radon-Hurwitz number for any integer.\n\nTypical use: RHnumber(2);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 504 "RHnumber:=proc(a1::integer)\noptions `Copyright (c) \+ 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserve d.`;\ndescription `Last revised: November 1, 2002`;\n################# ############################\nif member(a1,\{0,1,2\}) then return a1 e lif\n a1=3 then return 2 elif\n member(a1,\{4,5,6,7\}) then return 3 elif\n a1>=8 then return RHnumber(a1-8)+4 elif\n a1<0 then retu rn RHnumber(a1+8)-4 else\n error \"wrong value of the argument. See \+ ?RHnumber for more help.\" \nend if;\nend proc:\n" }}{PARA 258 "" 0 " " {TEXT -1 19 "No. 49. Procedure " }{TEXT 339 7 "clidata" }{TEXT -1 304 " returns a list containing basic information about the orthogonal Clifford algebra Cl(Q) of the given bilinear form B (assumed to have \+ been diagonalized). The procedure must be called with B, or with a si gnature of B given as a list [p,q], or simply as clidata() (currently \+ defined B will then be used)." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 47 "It returns a list with the following e lements:" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 187 "(a) the first entry is the string 'real', 'complex', or 'quat ernionic' depending whether the spinor representation of Cl(Q) is over the field K of the reals, complexes, or quaternions;\n" }}{PARA 258 "" 0 "" {TEXT -1 305 "(b) the second entry is the dimension of the spi nor representation over the field K;\n\n(c) the third entry is 'simple ' or 'semisimple' depending on the structure of the algebra;\n\n(d) th e fourth entry is a primitive idempotent f which may be used to gene rate a left or right minimal ideal in the algebra." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 574 "NOTE: the idempoten ts are stored here in an unevaluated form so that they could be easily recognized as Clifford products of simpler projection operators. The number of factors in these products is determined by the value of q - RHnumber(q-p).\n\n(e) the fifth entry is a list of basis monomials \+ ordered by grade which generate Cl(Q)f and fCl(Q).\n\n(f) the sixth en try is a list of basis monomials ordered by grade which give a basis f or K (this is in terms of these monomials that matrices representing C lifford polynomials will be written by the procedure 'spinorKrepr').\n " }}{PARA 258 "" 0 "" {TEXT -1 92 "(g) the seventh entry is a list of \+ basis monomials ordered by grade which generate S over K." }}{PARA 258 "" 0 "" {TEXT -1 139 "\nIf the procedure is called as 'clidata()' \+ then it returns information about the Clifford algebra of the currentl y defined bilinear form B." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 81 "Typical use: clidata(); clidata([2,3]); clida ta(B);clidata(linalg[diag](1,1,1));\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 473 "clidata:=proc() local a1,clidata2;global B;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights re served.`;\ndescription `Last revised: November 1, 2002`;\n############ #################################\nif nargs=0 then a1:=`B` else a1:=ar gs end if:\nif not type(a1,\{list(nonnegint),matrix\}) then\n WARNIN G(\"to find out about Clifford algebra Cl_\{p,q\} try clidata([p,q]) o r enter ?clidata for more help\");\n return ('procname(args)')\nend \+ if;\n" }}{PARA 258 "" 0 "" {TEXT -1 76 "This is a data file that is re ad in when needed by the procedure 'clidata'.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "clidata2" }{TEXT -1 0 "" }{MPLTEXT 1 0 16600 ":=proc(a 1::\{list(nonnegint),matrix\})\nlocal SBgens,FBgens,SBKgens,p,q,l,ni,K ,dimoverK,dimoverR,numfact,struct,primidemp;\nglobal B;\noptions `Copy right (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All righ ts reserved.`,remember;\ndescription `Last revised: November 1, 2002`; \n#############################################\n#K = field of spinor \+ repesentation, it is R, C, or H depending on [p,q]\n#dimoverK = dimens ion of spinor representation over the field K\n#dimoverR = dimension o f spinor representation over the reals R\n#numfact = number of idempot ent factors in any primitive idempotent\n#SBgens = basis monomials gen erating Cl(Q)f and fCl(Q) over R\n#FBgens = basis monomials providing \+ a basis for K\n#SBKgens = basis monomials generating Cl(Q)f and fCl(Q) over K \n#p = number of +1 in the diagonal form Q of B\n#q = number o f -1 in the diagonal form Q of B\n#struct = structure of Cl(Q) is 'sim ple' or 'semisimple'\n#primidemp = primitive idempotent f to generate \+ Cl(B)f or fCl(B)\nif nargs=0 then\n###new line instead of >>>not assig ned(B)<<<\nif not type(B,matrix) then \n error \"matrix must be assi gned to B\" else\n return clidata(B)\nend if;\nend if; \nif type( args[1],list(nonnegint)) then p:=args[1][1]:q:=args[1][2]: \n elif t ype(args[1],matrix) then \n p:=Bsignature(args)[1]; q:=Bsignatu re(args)[2] \n else \n error \"wrong argument types in 'clida ta'\" \n end if;\nif type(args[1],list(nonnegint)) and (p>9 or q>9) \+ then\n error \"p and q must satisfy 0 <= p,q <= 9\" \nend if;\nl:=fl oor((p+q)/2);ni:=2^(l-1);\nif member((p-q) mod 8,\{0,1,2\}) then \n \+ K:='real'; dimoverR:=2*ni; dimoverK:=2*ni; \nelif member((p-q) mod \+ 8,\{3,7\}) then \n K:='complex'; dimoverR:=2*2*ni; dimoverK:=2*ni ; else\n K:='quaternionic'; dimoverR:=4*ni; dimoverK:=ni \nend if ;\nnumfact:=q-RHnumber(q-p);\nif modp((p-q) = 1,4) then struct:='semis imple' \n else struct:='simple' \nend if;\nprimidemp:=table():SBgens :=table():FBgens:=table():SBKgens:=table():\n######################### >>>DATA<<<#################################\n#Real, simple (13 cases) \nprimidemp[[0,0]]:=Id; #real numbers\nSBgens[[0,0]]:=[Id];\nFBgens[[ 0,0]]:=[Id];\nSBKgens[[0,0]]:=SBgens[[0,0]];\n\nprimidemp[[1,1]]:=(1/2 )*(Id+e1we2);\nSBgens[[1,1]]:=[Id,e1];\nFBgens[[1,1]]:=[Id];\nSBKgens[ [1,1]]:=SBgens[[1,1]];\n\nprimidemp[[2,0]]:=(1/2)*(Id+e1);\nSBgens[[2, 0]]:=[Id,e2];\nFBgens[[2,0]]:=[Id];\nSBKgens[[2,0]]:=SBgens[[2,0]];\n \nprimidemp[[2,2]]:=\n''cmulQ''((1/2)*(Id+e1we3),(1/2)*(Id+e2we4));\nS Bgens[[2,2]]:=[Id,e1,e2,e1we2];\nFBgens[[2,2]]:=[Id];\nSBKgens[[2,2]]: =SBgens[[2,2]];\n\nprimidemp[[3,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*( Id+e3we4));\nSBgens[[3,1]]:=[Id,e2,e3,e2we3];\nFBgens[[3,1]]:=[Id];\nS BKgens[[3,1]]:=SBgens[[3,1]];\n\nprimidemp[[0,6]]:=\n''cmulQ''((1/2)*( Id+e1we2we3),(1/2)*(Id+e3we4we5),(1/2)*(Id+e1we4we6));\nSBgens[[0,6]]: =[Id,e1,e2,e3,e4,e5,e6,e1we5];\nFBgens[[0,6]]:=[Id];\nSBKgens[[0,6]]:= SBgens[[0,6]];\n\nprimidemp[[3,3]]:=\n''cmulQ''((1/2)*(Id+e1we4),(1/2) *(Id+e2we5),(1/2)*(Id+e3we6));\nSBgens[[3,3]]:=[Id,e1,e2,e3,e1we2,e1we 3,e2we3,e1we2we3];\nFBgens[[3,3]]:=[Id];\nSBKgens[[3,3]]:=SBgens[[3,3] ];\n\nprimidemp[[4,2]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e3we5),(1/ 2)*(Id+e4we6));\nSBgens[[4,2]]:=[Id,e2,e3,e4,e2we3,e2we4,e3we4,e2we3we 4];\nFBgens[[4,2]]:=[Id];\nSBKgens[[4,2]]:=SBgens[[4,2]];\n\nprimidemp [[4,4]]:=\n''cmulQ''((1/2)*(Id+e1we5),(1/2)*(Id+e2we6),(1/2)*(Id+e3we7 ),(1/2)*(Id+e4we8));\nSBgens[[4,4]]:=[Id,e1,e2,e3,e4,e1we2,e1we3,e1we4 ,e2we3,e2we4,e3we4,e1we2we3,\n e1we2we4,e1we3we4,e2we3we4,e1we2we3we4] ;\nFBgens[[4,4]]:=[Id];\nSBKgens[[4,4]]:=SBgens[[4,4]];\n\nprimidemp[[ 5,3]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e3we6),(1/2)*(Id+e4we7),(1/ 2)*(Id+e5we8));\nSBgens[[5,3]]:=[Id,e2,e3,e4,e5,e2we3,e2we4,e2we5,e3we 4,e3we5,e4we5,e2we3we4,\ne2we3we5,e2we4we5,e3we4we5,e2we3we4we5];\nFBg ens[[5,3]]:=[Id];\nSBKgens[[5,3]]:=SBgens[[5,3]];\n\nprimidemp[[8,0]]: =\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),(1/2)*(Id+e4we5we6we 7),\n (1/2)*(Id+e2we4we6we8));\nSBgens[[8,0]]:=[Id,e2,e3,e4,e 5,e6,e7,e8,e2we3,e2we4,e2we5,e2we6,e2we7,\ne2we8,e3we8,e2we3we8];\nFBg ens[[8,0]]:=[Id];\nSBKgens[[8,0]]:=SBgens[[8,0]];\n\nprimidemp[[1,7]]: =\n''cmulQ''((1/2)*(Id+e2we3we4),(1/2)*(Id+e4we5we6),(1/2)*(Id+e2we5we 7),\n (1/2)*(Id+e1we8));\nSBgens[[1,7]]:=[Id,e1,e2,e3,e4,e5,e 6,e7,e1we2,e1we3,e1we4,e1we5,e1we6,\ne1we7,e2we6,e1we2we6];\nFBgens[[1 ,7]]:=[Id];\nSBKgens[[1,7]]:=SBgens[[1,7]];\n\nprimidemp[[0,8]]:=\n''c mulQ''((1/2)*(Id+e1we2we3),(1/2)*(Id+e3we4we5),(1/2)*(Id+e1we4we6),\n \+ (1/2)*(Id+e3we6we7));\nSBgens[[0,8]]:=\n[Id,e1,e2,e3,e4,e5,e6 ,e7,e8,e1we8,e2we8,e3we8,e4we8,e5we8,e6we8,e7we8];\nFBgens[[0,8]]:=[Id ];\nSBKgens[[0,8]]:=SBgens[[0,8]];\n\n#Complex, simple (15 cases)\npri midemp[[0,1]]:=Id; #complex numbers\nSBgens[[0,1]]:=[Id,e1];\nFBgens[ [0,1]]:=[Id,e1];\nSBKgens[[0,1]]:=[Id,e1];\n\nprimidemp[[1,2]]:=(1/2)* (Id+e1we3);\nSBgens[[1,2]]:=[Id,e1,e2,e1we2];\nFBgens[[1,2]]:=[Id,e2]; \nSBKgens[[1,2]]:=[Id,e1];\n\nprimidemp[[3,0]]:=(1/2)*(Id+e1);\nSBgens [[3,0]]:=[Id,e2,e3,e2we3];\nFBgens[[3,0]]:=[Id,e2we3];\nSBKgens[[3,0]] :=[Id,e2];\n\nprimidemp[[0,5]]:=\n''cmulQ''((1/2)*(Id+e1we2we3),(1/2)* (Id+e3we4we5));\nSBgens[[0,5]]:=[Id,e1,e2,e3,e4,e5,e1we4,e1we5];\nFBge ns[[0,5]]:=[Id,e3];\nSBKgens[[0,5]]:=[Id,e1,e4,e1we4];\n\nprimidemp[[2 ,3]]:=\n''cmulQ''((1/2)*(Id+e1we4),(1/2)*(Id+e2we5));\nSBgens[[2,3]]:= [Id,e1,e2,e3,e1we2,e1we3,e2we3,e1we2we3];\nFBgens[[2,3]]:=[Id,e3];\nSB Kgens[[2,3]]:=[Id,e1,e2,e1we2];\n\nprimidemp[[4,1]]:=\n''cmulQ''((1/2) *(Id+e1),(1/2)*(Id+e4we5));\nSBgens[[4,1]]:=[Id,e2,e3,e4,e2we3,e2we4,e 3we4,e2we3we4];\nFBgens[[4,1]]:=[Id,e2we3];\nSBKgens[[4,1]]:=[Id,e2,e4 ,e2we4];\n\nprimidemp[[1,6]]:=\n''cmulQ''((1/2)*(Id+e2we3we4),(1/2)*(I d+e4we5we6),(1/2)*(Id+e1we7));\nSBgens[[1,6]]:=[Id,e1,e2,e3,e4,e5,e6,e 1we2,e1we3,e1we4,e1we5,e1we6,e2we5, \+ e2we6,e1we2we5,e1we2we6]; \nFBgens[[1,6]]:=[Id,e4];\nSBKg ens[[1,6]]:=[Id,e1,e2,e5,e1we2,e1we5,e2we5,e1we2we5];\n\nprimidemp[[3, 4]]:=\n''cmulQ''((1/2)*(Id+e1we5),(1/2)*(Id+e2we6),(1/2)*(Id+e3we7)); \nSBgens[[3,4]]:=[Id,e1,e2,e3,e4,e1we2,e1we3,e1we4,e2we3,e2we4,e3we4, \n e1we2we3,e1we2we4,e1we3we4,e2we3we4,e1we2we3we4]; \n FBgens[[3,4]]:=[Id,e4];\nSBKgens[[3,4]]:=[Id,e1,e2,e3,e1we2,e1we3,e2we 3,e1we2we3];\n\nprimidemp[[5,2]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+ e4we6),(1/2)*(Id+e5we7));\nSBgens[[5,2]]:=[Id,e2,e3,e4,e5,e2we3,e2we4, e2we5,e3we4,e3we5,e4we5,\n e2we3we4,e2we3we5,e2we4we5,e 3we4we5,e2we3we4we5]; \nFBgens[[5,2]]:=[Id,e2we3];\nSBKgens[[5,2]]:=[I d,e2,e4,e5,e2we4,e2we5,e4we5,e2we4we5];\n\nprimidemp[[7,0]]:=\n''cmulQ ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),(1/2)*(Id+e4we5we6we7));\nSBge ns[[7,0]]:=[Id,e2,e3,e4,e5,e6,e7,e2we3,e2we4,e2we5,e2we6,e2we7,\n \+ e4we6,e4we7,e2we4we6,e2we4we7]; \nFBgens[[7,0]]:=[Id,e2we3] ;\nSBKgens[[7,0]]:=[Id,e2,e4,e6,e2we4,e2we6,e4we6,e2we4we6];\n\nprimid emp[[0,9]]:=\n''cmulQ''((1/2)*(Id+e1we2we3),(1/2)*(Id+e3we4we5),(1/2)* (Id+e1we4we6),\n (1/2)*(Id+e3we6we7));\nSBgens[[0,9]]:=\n[Id, e1,e2,e3,e4,e5,e6,e7,e8,e9,e1we8,e1we9,e2we8,e2we9,e3we8,e3we9,\n e4we 8,e4we9,e5we8,e5we9,e6we8,e6we9,e7we8,e7we9,e8we9,e1we8we9,\n e2we8we9 ,e3we8we9,e4we8we9,e5we8we9,e6we8we9,e7we8we9];\nFBgens[[0,9]]:=[Id,e8 we9];\nSBKgens[[0,9]]:=[Id,e1,e2,e3,e4,e5,e6,e7,e8,e1we8,e2we8,e3we8,e 4we8,\n e5we8,e6we8,e7we8];\n\nprimidemp[[2,7]]:=\n''c mulQ''((1/2)*(Id+e3we4we5),(1/2)*(Id+e5we6we7),(1/2)*(Id+e1we8),\n \+ (1/2)*(Id+e2we9));\nSBgens[[2,7]]:=\n[Id,e1,e2,e3,e4,e5,e6,e7,e1 we2,e1we3,e1we4,e1we5,e1we6,e1we7,e2we3,\n e2we4,e2we5,e2we6,e2we7,e3w e6,e3we7,e1we2we3,e1we2we4,e1we2we5,\n e1we2we6,e1we2we7,e1we3we6,e1we 3we7,e2we3we6,e2we3we7,e1we2we3we6,\n e1we2we3we7];\nFBgens[[2,7]]:=[I d,e5];\nSBKgens[[2,7]]:=\n[Id,e1,e2,e3,e6,e1we2,e1we3,e1we6,e2we3,e2we 6,e3we6,e1we2we3,e1we2we6,e1we3we6,\n e2we3we6,e1we2we3we6];\n\nprimid emp[[4,5]]:=\n''cmulQ''((1/2)*(Id+e1we6),(1/2)*(Id+e2we7),(1/2)*(Id+e3 we8),(1/2)*(Id+e4we9));\nSBgens[[4,5]]:=\n[Id,e1,e2,e3,e4,e5,e1we2,e1w e3,e1we4,e1we5,e2we3,e2we4,e2we5,e3we4,\n e3we5,e4we5,e1we2we3,e1we2we 4,e1we2we5,e1we3we4,e1we3we5,e1we4we5,\n e2we3we4,e2we3we5,e2we4we5,e3 we4we5,e1we2we3we4,e1we2we3we5,\n e1we2we4we5,e1we3we4we5,e2we3we4we5, e1we2we3we4we5];\nFBgens[[4,5]]:=[Id,e5];\nSBKgens[[4,5]]:=\n[Id,e1,e2 ,e3,e4,e1we2,e1we3,e1we4,e2we3,e2we4,e3we4,e1we2we3,e1we2we4,\n e1we3w e4,e2we3we4,e1we2we3we4];\n\nprimidemp[[6,3]]:=\n''cmulQ''((1/2)*(Id+e 1),(1/2)*(Id+e4we7),(1/2)*(Id+e5we8),(1/2)*(Id+e6we9));\nSBgens[[6,3]] :=\n[Id,e2,e3,e4,e5,e6,e2we3,e2we4,e2we5,e2we6,e3we4,e3we5,e3we6,e4we5 ,\n e4we6,e5we6,e2we3we4,e2we3we5,e2we3we6,e2we4we5,e2we4we6,e2we5we6, \n e3we4we5,e3we4we6,e3we5we6,e4we5we6,e2we3we4we5,e2we3we4we6,\n e2we 3we5we6,e2we4we5we6,e3we4we5we6,e2we3we4we5we6];\nFBgens[[6,3]]:=[Id,e 2we3];\nSBKgens[[6,3]]:=\n[Id,e2,e4,e5,e6,e2we4,e2we5,e2we6,e4we5,e4we 6,e5we6,e2we4we5,e2we4we6,\n e2we5we6,e4we5we6,e2we4we5we6];\n\nprimid emp[[8,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),(1/2)*(Id +e4we5we6we7),\n (1/2)*(Id+e8we9));\nSBgens[[8,1]]:=\n[Id,e2, e3,e4,e5,e6,e7,e8,e2we3,e2we4,e2we5,e2we6,e2we7,e2we8,e3we8,\n e4we6,e 4we7,e4we8,e5we8,e6we8,e7we8,e2we3we8,e2we4we6,e2we4we7,\n e2we4we8,e2 we5we8,e2we6we8,e2we7we8,e4we6we8,e4we7we8,e2we4we6we8,\n e2we4we7we8] ;\nFBgens[[8,1]]:=[Id,e2we3];\nSBKgens[[8,1]]:=\n[Id,e2,e4,e6,e8,e2we4 ,e2we6,e2we8,e4we6,e4we8,e6we8,e2we4we6,e2we4we8,\n e2we6we8,e4we6we8, e2we4we6we8];\n\n#Quaternionic, simple (12 cases)\nprimidemp[[0,2]]:=I d; #quaternions\nSBgens[[0,2]]:=[Id,e1,e2,e1we2];\nFBgens[[0,2]]:=[Id, e1,e2,e1we2];\nSBKgens[[0,2]]:=[Id];\n\nprimidemp[[0,4]]:=(1/2)*(Id+e1 we2we3);\nSBgens[[0,4]]:=[Id,e1,e2,e3,e4,e1we4,e2we4,e3we4];\nFBgens[[ 0,4]]:=[Id,e1,e1we3,e3];\nSBKgens[[0,4]]:=[Id,e4];\n\nprimidemp[[1,3]] :=(1/2)*(Id+e1we4);\nSBgens[[1,3]]:=[Id,e1,e2,e3,e1we2,e1we3,e2we3,e1w e2we3];\nFBgens[[1,3]]:=[Id,e2,e3,e2we3];\nSBKgens[[1,3]]:=[Id,e1];\n \nprimidemp[[4,0]]:=(1/2)*(Id+e1);\nSBgens[[4,0]]:=[Id,e2,e3,e4,e2we3, e2we4,e3we4,e2we3we4];\nFBgens[[4,0]]:=[Id,e2we3,e2we4,e3we4];\nSBKgen s[[4,0]]:=[Id,e2];\n\nprimidemp[[1,5]]:=\n''cmulQ''((1/2)*(Id+e2we3we4 ),(1/2)*(Id+e1we6));\nSBgens[[1,5]]:=[Id,e1,e2,e3,e4,e5,e1we2,e1we3,e1 we4,e1we5,e2we5,e3we5,\n e4we5,e1we2we5,e1we3we5,e1we4w e5];\nFBgens[[1,5]]:=[Id,e2,e2we4,e4];\nSBKgens[[1,5]]:=[Id,e1,e5,e1we 5];\n\nprimidemp[[2,4]]:=\n''cmulQ''((1/2)*(Id+e1we5),(1/2)*(Id+e2we6) );\nSBgens[[2,4]]:=[Id,e1,e2,e3,e4,e1we2,e1we3,e1we4,e2we3,e2we4,e3we4 ,\n e1we2we3,e1we2we4,e1we3we4,e2we3we4,e1we2we3we4];\n FBgens[[2,4]]:=[Id,e3,e4,e3we4];\nSBKgens[[2,4]]:=[Id,e1,e2,e1we2];\n \nprimidemp[[5,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e5we6));\nSBge ns[[5,1]]:=[Id,e2,e3,e4,e5,e2we3,e2we4,e2we5,e3we4,e3we5,e4we5,\n \+ e2we3we4,e2we3we5,e2we4we5,e3we4we5,e2we3we4we5];\nFBgens[[ 5,1]]:=[Id,e2we3,e2we4,e3we4];\nSBKgens[[5,1]]:=[Id,e2,e5,e2we5];\n\np rimidemp[[6,0]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5));\nS Bgens[[6,0]]:=[Id,e2,e3,e4,e5,e6,e2we3,e2we4,e2we5,e2we6,e3we6,e4we6, \n e5we6,e2we3we6,e2we4we6,e2we5we6];\nFBgens[[6,0]]:=[ Id,e2we3,e3we5,e2we5];\nSBKgens[[6,0]]:=[Id,e2,e6,e2we6];\n\nprimidemp [[2,6]]:=\n''cmulQ''((1/2)*(Id+e3we4we5),(1/2)*(Id+e1we7),(1/2)*(Id+e2 we8));\nSBgens[[2,6]]:=\n[Id,e1,e2,e3,e4,e5,e6,e1we2,e1we3,e1we4,e1we5 ,e1we6,e2we3,e2we4,e2we5,\n e2we6,e3we6,e4we6,e5we6,e1we2we3,e1we2we4, e1we2we5,e1we2we6,e1we3we6,\n e1we4we6,e1we5we6,e2we3we6,e2we4we6,e2we 5we6,e1we2we3we6,e1we2we4we6,\n e1we2we5we6];\nFBgens[[2,6]]:=[Id,e3,e 3we5,e5];\nSBKgens[[2,6]]:=[Id,e1,e2,e6,e1we2,e1we6,e2we6,e1we2we6];\n \nprimidemp[[3,5]]:=\n''cmulQ''((1/2)*(Id+e1we6),(1/2)*(Id+e2we7),(1/2 )*(Id+e3we8));\nSBgens[[3,5]]:=\n[Id,e1,e2,e3,e4,e5,e1we2,e1we3,e1we4, e1we5,e2we3,e2we4,e2we5,e3we4,\n e3we5,e4we5,e1we2we3,e1we2we4,e1we2we 5,e1we3we4,e1we3we5,e1we4we5,\n e2we3we4,e2we3we5,e2we4we5,e3we4we5,e1 we2we3we4,e1we2we3we5,\n e1we2we4we5,e1we3we4we5,e2we3we4we5,e1we2we3w e4we5];\nFBgens[[3,5]]:=[Id,e4,e5,e4we5];\nSBKgens[[3,5]]:=[Id,e1,e2,e 3,e1we2,e1we3,e2we3,e1we2we3];\n\nprimidemp[[6,2]]:=\n''cmulQ''((1/2)* (Id+e1),(1/2)*(Id+e5we7),(1/2)*(Id+e6we8));\nSBgens[[6,2]]:=\n[Id,e2,e 3,e4,e5,e6,e2we3,e2we4,e2we5,e2we6,e3we4,e3we5,e3we6,e4we5,\n e4we6,e5 we6,e2we3we4,e2we3we5,e2we3we6,e2we4we5,e2we4we6,e2we5we6,\n e3we4we5, e3we4we6,e3we5we6,e4we5we6,e2we3we4we5,e2we3we4we6,\n e2we3we5we6,e2we 4we5we6,e3we4we5we6,e2we3we4we5we6];\nFBgens[[6,2]]:=[Id,e2we3,e2we4,e 3we4];\nSBKgens[[6,2]]:=[Id,e2,e5,e6,e2we5,e2we6,e5we6,e2we5we6];\n\np rimidemp[[7,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),(1/2 )*(Id+e7we8));\nSBgens[[7,1]]:=\n[Id,e2,e3,e4,e5,e6,e7,e2we3,e2we4,e2w e5,e2we6,e2we7,e3we6,e3we7,e4we6,\n e4we7,e5we6,e5we7,e6we7,e2we3we6,e 2we3we7,e2we4we6,e2we4we7,e2we5we6,\n e2we5we7,e2we6we7,e3we6we7,e4we6 we7,e5we6we7,e2we3we6we7,e2we4we6we7,\n e2we5we6we7];\nFBgens[[7,1]]:= [Id,e2we3,e3we5,e2we5];\nSBKgens[[7,1]]:=[Id,e2,e6,e7,e2we6,e2we7,e6we 7,e2we6we7];\n\n#Real, semi-simple (8 cases)\nprimidemp[[1,0]]:=(1/2)* (Id+e1);\nSBgens[[1,0]]:=[Id];\nFBgens[[1,0]]:=[Id];\nSBKgens[[1,0]]:= SBgens[[1,0]];\n\nprimidemp[[2,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(I d+e2we3));\nSBgens[[2,1]]:=[Id,e2];\nFBgens[[2,1]]:=[Id];\nSBKgens[[2, 1]]:=SBgens[[2,1]];\n\nprimidemp[[3,2]]:=\n''cmulQ''((1/2)*(Id+e1),(1/ 2)*(Id+e2we4),(1/2)*(Id+e3we5));\nSBgens[[3,2]]:=[Id,e2,e3,e2we3];\nFB gens[[3,2]]:=[Id];\nSBKgens[[3,2]]:=SBgens[[3,2]];\n\nprimidemp[[0,7]] := ''cmulQ''((1/2)*(Id+e1we2we3),(1/2)*(Id+e3we4we5),(1/2)*(Id+e1we4we 6),\n (1/2)*(Id+e3we6we7));\nSBgens[[0,7]]:=[Id,e1,e2,e3,e4,e 5,e6,e7];\nFBgens[[0,7]]:=[Id];\nSBKgens[[0,7]]:=SBgens[[0,7]];\n\npri midemp[[4,3]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we5),(1/2)*(Id+e3 we6),\n (1/2)*(Id+e4we7));\nSBgens[[4,3]]:=[Id,e2,e3,e4,e2we3 ,e2we4,e3we4,e2we3we4];\nFBgens[[4,3]]:=[Id];\nSBKgens[[4,3]]:=SBgens[ [4,3]];\n\nprimidemp[[9,0]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3 we4we5),1/2*(Id+e2we3we6we7),\n (1/2)*(Id+e2we3we8we9),(1/2)* (Id+e2we4we6we8));\nSBgens[[9,0]]:=\n[Id,e2,e3,e4,e5,e6,e7,e8,e9,e2we3 ,e2we4,e2we5,e2we6,e2we7,e2we8,e2we9];\nFBgens[[9,0]]:=[Id];\nSBKgens[ [9,0]]:=SBgens[[9,0]];\n\nprimidemp[[5,4]]:=\n''cmulQ''((1/2)*(Id+e1), (1/2)*(Id+e2we6),(1/2)*(Id+e3we7),\n (1/2)*(Id+e4we8),(1/2)*( Id+e5we9));\nSBgens[[5,4]]:=[Id,e2,e3,e4,e5,e2we3,e2we4,e2we5,e3we4,e3 we5,e4we5,e2we3we4, e2we3we5,e2we4we5,e3we4we5,e2we3we4we5];\nFBgens[[ 5,4]]:=[Id];\nSBKgens[[5,4]]:=SBgens[[5,4]];\n\nprimidemp[[1,8]]:=\n'' cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),1/2*(Id+e2we3we6we7),\n \+ (1/2)*(Id+e2we3we8we9),(1/2)*(Id+e2we4we6we8));\nSBgens[[1,8]] :=[Id,e2,e3,e4,e5,e6,e7,e8,e9,e2we3,e2we4,e2we5,e2we6,e2we7,e2we8,e2we 9];\nFBgens[[1,8]]:=[Id];\nSBKgens[[1,8]]:=SBgens[[1,8]];\n\n#Complex, semi-simple - none\n\n#Quaternionic, semi-simple (5 cases)\nprimidemp [[0,3]]:=(1/2)*(Id+e1we2we3);\nSBgens[[0,3]]:=[Id,e1,e2,e3];\nFBgens[[ 0,3]]:=[Id,e1,e2,e1we2];\nSBKgens[[0,3]]:=[Id];\n\nprimidemp[[1,4]]:= \n''cmulQ''((1/2)*(Id+e2we3we4),(1/2)*(Id+e1we5));\nSBgens[[1,4]]:=[Id ,e1,e2,e3,e4,e1we2,e1we3,e1we4];\nFBgens[[1,4]]:=[Id,e2,e3,e2we3];\nSB Kgens[[1,4]]:=[Id,e1];\n\nprimidemp[[5,0]]:=\n''cmulQ''((1/2)*(Id+e1), (1/2)*(Id+e2we3we4we5));\nSBgens[[5,0]]:=[Id,e2,e3,e4,e5,e2we3,e2we4,e 2we5];\nFBgens[[5,0]]:=[Id,e2we3,e3we5,e2we5];\nSBKgens[[5,0]]:=[Id,e2 ];\n\nprimidemp[[2,5]]:=\n''cmulQ''((1/2)*(Id+e3we4we5),(1/2)*(Id+e1we 6),(1/2)*(Id+e2we7));\nSBgens[[2,5]]:=[Id,e1,e2,e3,e4,e5,e1we2,e1we3,e 1we4,e1we5,\n e2we3,e2we4,e2we5,e1we2we3,e1we2we4,e1we2 we5];\nFBgens[[2,5]]:=[Id,e3,e3we5,e5];\nSBKgens[[2,5]]:=[Id,e1,e2,e1w e2];\n\nprimidemp[[6,1]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we3we4 we5),(1/2)*(Id+e6we7));\nSBgens[[6,1]]:=[Id,e2,e3,e4,e5,e6,e2we3,e2we4 ,e2we5,e2we6,e3we6,\n e4we6,e5we6,e2we3we6,e2we4we6,e2 we5we6];\nFBgens[[6,1]]:=[Id,e2we3,e3we5,e2we5];\nSBKgens[[6,1]]:=[Id, e2,e6,e2we6];\n\nprimidemp[[7,2]]:=''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e 2we8),\n (1/2)*(Id+e3we9),(1/2)*(Id+e4we5we6w e7));\nSBgens[[7,2]]:=[Id,e2,e3,e4,e5,e6,e7,e2we3,e2we4,e2we5,e2we6,e2 we7,\ne3we4,e3we5,e3we6,e3we7,e4we5,e4we6,e4we7,e2we3we4,e2we3we5,e2we 3we6,\ne2we3we7,e2we4we5,e2we4we6,e2we4we7,e3we4we5,e3we4we6,e3we4we7, \ne2we3we4we5,e2we3we4we6,e2we3we4we7];\nFBgens[[7,2]]:=[Id,e4we5,e5we 7,e4we7];\nSBKgens[[7,2]]:=[Id,e2,e3,e4,e2we3,e2we4,e3we4,e2we3we4];\n \nprimidemp[[3,6]]:=\n''cmulQ''((1/2)*(Id+e1),(1/2)*(Id+e2we4),\n \+ (1/2)*(Id+e3we5),(1/2)*(Id+e6we7we8we9));\nSBgens[[3,6]]:=[Id,e2, e3,e6,e7,e8,e9,e2we3,e2we6,e2we7,e2we8,e2we9,e3we6,e3we7,\ne3we8,e3we9 ,e6we7,e6we8,e6we9,e2we3we6,e2we3we7,e2we3we8,e2we3we9,e2we6we7,\ne2we 6we8,e2we6we9,e3we6we7,e3we6we8,e3we6we9,e2we3we6we7,e2we3we6we8,\ne2w e3we6we9];\nFBgens[[3,6]]:=[Id,e6we7,e7we9,e6we9];\nSBKgens[[3,6]]:=[I d,e2,e3,e6,e2we3,e2we6,e3we6,e2we3we6];\n\nreturn ([K,dimoverK,struct, primidemp[[p,q]],\n SBgens[[p,q]],FBgens[[p,q]],SBKgens[[p,q]]] );\nend proc:\n##################\nreturn clidata2(a1); #### <<< Retur n from 'clidata'\nend proc: #### <<< End of 'clidata'\n" }}{PARA 258 " " 0 "" {TEXT -1 18 "No. 53. Procedure " }{TEXT 340 10 "Bsignature" } {TEXT -1 313 " finds the signature of the form B assuming that B is a diagonal matrix or a symmetric matrix. It returns a list L with two o r three integers depending on whether B is non-degenerate or degenerat e, that is, L=[p,q] or L=[p,q,d]. Here d = dim(rad B), and p (q) denot es number of +1 (-1) in the diagonal form of B." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 42 "Typical use: Bsignature (); Bsignature(B);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1289 "Bsignatur e:=proc() local curB,Bdiag,pos,neg,deg,i,L;global B;\noptions `Copyrig ht (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights \+ reserved.`;\ndescription `Last revised: November 1, 2002`;\n########## ###################################\nif nargs=0 then\n if not type( B,matrix) then\n error \"square matric should be assigned to B f irst\"\n else curB:=B \n end if;\nelif nargs=1 then\n if not \+ type(evalm(args[1]),matrix) then\n error \"argument entered is n ot a matrix\"\n else curB:=evalm(args[1]) \n end if;\nelse erro r \"wrong number of arguments. See ?Bsignature for more help.\" \nend \+ if;\nBdiag:=diagonalize(evalm(curB-(curB-linalg[transpose](curB))/2)); \nif not type(Bdiag,diagmatrix) then \n error \"unable to diagonaliz e symmetric part of the input\"\nend if;\nL:=map(signum,[seq(Bdiag[i,i ],i=1..linalg[coldim](Bdiag))]):\nif not type(L,list(integer)) then\n \+ error \"unable to determine signs of expressions %1\",L\nend if;\npo s:=0:neg:=0:deg:=0:\nfor i from 1 to linalg[coldim](Bdiag) do\nif L[i] <>0 then\n if evalf(L[i])>0 then pos:=pos+1 elif\n evalf(L[i])< 0 then neg:=neg+1 else\n error \"unable to determine sign of %1\" ,Bdiag[i,i]\n end if;\nelse deg:=deg+1;\nend if;\nend do;\nif deg=0 \+ then return [pos,neg] else return [pos,neg,deg] end if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 157 "No. 51. Spinor representation of Cl( Q) in S=Cl(Q)f and S=fCl(Q) over the field K of the reals, complexes, \+ or quaternions when Cl(Q) is simple.\nThe procedure " }{TEXT 341 11 "s pinorKrepr" }{TEXT -1 183 " finds matrix representation of any Cliffor d polynomial in a minimal left or right ideal in Cl(Q) generated by a \+ primitive idempotent f. The procedure is invoked with four arguments: " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 69 "( 1) the first argument is an algebraic expression of type clipolynom;" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 210 "(2 ) the second argument is a list of generators of the minimal ideal S c onsidered as a K-vector space. For standard f equal to clidata()[4] t hese generators are stored under clidata()[6] for the given form B; " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 241 "(3 ) the third argument is a list of basis elements spanning K. For stan dard f equal to clidata()[4] these generators are stored under clidata ()[5]. Matrices computed by 'spinorKrepr' will be expressed in terms \+ of these basis elements of K;" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 111 "(4) the fourth argument is a one of th e strings 'left' or 'right' depending whether the ideal is left or rig ht." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 562 "When standard input is used, i.e., the second argument equals cli data()[7] and the third argument equals clidata()[5], the procedure tr ies to use previously computed matrices representing 1-vectors. These matrices are stored as .m files with the names 'matrealL.m', 'matcomp L.m', 'matquatL.m' for real, complex, and quaternionic matrices in the left-regular spinor representation. If the first argument entered bel ongs to Cl(Q) whose 1-vector matrices have been previously computed, t he procedure calls 'matKrepr' which makes use of these pre-computed ma trices." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 470 "Typical use: dim:=4:B:=linalg[diag](1,-1,-1,-1):clibasis:=cbas is(dim):data:=clidata():\n f:=data[4]:\n \+ sbasis:=minimalideal(clibasis,f,'left');\n \+ fbasis:=Kfield(sbasis,f);\n Kb asis:=spinorKbasis(sbasis[2],f,fbasis[2],'left');\n \+ spinorKrepr(e1,Kbasis[1],fbasis[2],'left');\n \+ spinorKrepr(2*e1+Id-3*e1we2we3,Kbasis[1],fbasis[2],'left');\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 5376 "spinorKrepr:=proc(a1::\{clibasmon ,climon,clipolynom,numeric\},\n a2::list(\{clibasmon, climon,clipolynom\}),\n a3::list(\{clibasmon,climon,c lipolynom\}),\n a4::\{string,symbol\})\nlocal i,j,k,r eprdim,r,a,FBgens,eq,hbasis,g,terms,sys,vars,sol,M,pqsig,pq,\n fl ag_left,data,Kbasis,f,v,pqmod8,n,expr,flag_simple;\nglobal B,_prolevel ,_shortcut_in_spinorKrepr,matrealL,matrealR,matcompL,matcompR,matquatL ,matquatR;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and B ertfried Fauser. All rights reserved.`;\ndescription `Last revised: No vember 1, 2002`;\n#############################################\nif no t type(B,diagmatrix) then \n error \"bilinear form B must be defined as diagonal matrix\" \nelse pq:=Bsignature() \nend if;\n############# #####################\nif pq[1]-pq[2]=1 mod 4 then flag_simple:=false \+ else flag_simple:=true end if;\n##################################\nif maxindex(a1) > linalg[coldim](B) then\n error \"maximum index %1 fo und in input is greater than the size %2 of the current bilinear form \+ B\", maxindex(a1),linalg[coldim](B) \nend if;\n####################### ###########\nhbasis:=a2:FBgens:=a3:reprdim:=nops(hbasis):n:=nops(FBgen s):\n##################################\nif member(a4,\{'left',\"left \"\}) then flag_left:=true elif\n member(a4,\{'right',\"right\"\}) t hen flag_left:=false else\n error \"last argument expected to be 'le ft' or 'right' but received %1 instead\",a4\nend if; \n############### #########################################################\n#This proce dure gives faithful representations when Cl(p,q) is simple\n#and unfai thful when Cl(p,q) is semi-simple. In order to get faithful\n#represen tations in this last case, use 'matKrepr' or use this procedure\n#as s hown in examples.\n################################################### #####################\n#if flag_simple then\nif a1=Id then return lin alg[diag](1$reprdim) elif\n a1=-Id then return linalg[diag](-1$reprd im) elif\n type(a1,numeric) then return linalg[diag](a1$reprdim) \ne nd if;\n#else\n#if a1=Id then return cdfmatrix([linalg[diag](1$reprdi m)$2]) elif\n# a1=-Id then return cdfmatrix([linalg[diag](-1$reprdim )$2]) elif\n# type(a1,numeric) then return cdfmatrix([linalg[diag](a 1$reprdim)$2]) \n# end if;\n# end if;\n#when _shortcut_in_spinorKrepr \+ is false, 'matKrepr' is not used\nif _shortcut_in_spinorKrepr then\n \+ pqmod8:=(pq[1]-pq[2]) mod 8:\n if member(pqmod8,\{0,1,2\}) and flag _left then \n #if not assigned(matrealL) then readlib(matrealL) \+ end if;\n pqsig:=map(op,[indices(matrealL)]) \n elif member(pqm od8,\{0,1,2\}) and not flag_left then\n #if not assigned(matrealR ) then readlib(matrealR) end if;\n pqsig:=map(op,[indices(matrea lR)]) \n elif member(pqmod8,\{3,7\}) and flag_left then \n #if \+ not assigned(matcompL) then readlib(matcompL) end if;\n pqsig:=m ap(op,[indices(matcompL)]) \n elif member(pqmod8,\{3,7\}) and not fl ag_left then\n #if not assigned(matcompR) then readlib(matcompR) \+ end if;\n pqsig:=map(op,[indices(matcompR)]) \n elif member(pq mod8,\{4,5,6\}) and flag_left then \n #if not assigned(matquatL) \+ then readlib(matquatL) end if;\n pqsig:=map(op,[indices(matquatL )]) \n elif member(pqmod8,\{4,5,6\}) and not flag_left then\n # if not assigned(matquatR) then readlib(matquatR) end if;\n pqsig :=map(op,[indices(matquatR)]) \n end if;\n########################## ###########\n if member(pq,pqsig) then \n data:=clidata(pq ):f:=eval(eval(data[4])):\n g:='g': \n if flag_left the n Kbasis:=[seq(cmulQ(g,f),g=data[7])] \n else Kbasi s:=[seq(cmulQ(f,g),g=data[7])] \n end if; \n if hbasis=Kbasi s then\n if FBgens=data[6] then return matKrepr(a1,a4) end if; \n end if;\n end if;\nend if;\n########################### ##########\n#Continue finding the matrix\na:='a':j:='j':k:='k':\nif fl ag_left then\n expr:=add(add(a[j,k]*cmulQ(hbasis[j],FBgens[k]),j=1.. reprdim),k=1..n);\n for j from 1 to reprdim do r[j]:=add(a[j,k] * FB gens[k],k=1..n) end do; \n for i from 1 to reprdim do\n eq:= expand(cmulQ(a1,hbasis[i])-expr);\n terms:=cliterms(eq);\n \+ eq:=clicollect(eq,terms);\n sys:=\{coeffs(eq,terms)\}:\n \+ vars:=\{seq(seq(a[j,k],k=1..n),j=1..reprdim)\};\n sol:=solve(sys ,vars);\n if sol=NULL then \nerror \"unable to find matrix due i nput error: check if the last argument matches the one previously used in 'spinorKbasis'\"\n end if; \n v[i]:=convert([seq(subs( sol,r[j]),j=1..reprdim)],vector);\n end do:\nM:=linalg[transpose](li nalg[stackmatrix](seq(eval(v[i]),i=1..reprdim)));\nreturn subs(Id=1,ev alm(M));\nelse \n expr:=add(add(a[j,k]*cmulQ(FBgens[k],hbasis[j] ),j=1..reprdim),k=1..n);\n for j from 1 to reprdim do r[j]:=add(a[j, k] * FBgens[k],k=1..n) end do; \n for i from 1 to reprdim do \+ \n eq:=expand(cmulQ(hbasis[i],a1)-expr);\n terms:=cliterm s(eq);\n eq:=clicollect(eq,terms);\n sys:=\{coeffs(eq,term s)\}:\n vars:=\{seq(seq(a[j,k],k=1..n),j=1..reprdim)\};\n \+ sol:=solve(sys,vars);\n if sol=NULL then \nerror \"unable to fin d matrix due to input error: check if the last argument matches the on e previously used in 'spinorKbasis'\"\n end if; \n v[i]:=c onvert([seq(subs(sol,r[j]),j=1..reprdim)],vector);\n end do:\nM:=lin alg[transpose](linalg[stackmatrix](seq(eval(v[i]),i=1..reprdim)));\nre turn subs(Id=1,evalm(M));\nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 20 "No. 52. Procedure " }{TEXT 342 5 "rmulm" }{TEXT -1 110 " extends the following multiplications to matrix entries: cmul, cmulQ , wedge, omul, `&r`, `&*`\n " }}{PARA 258 "" 0 "" {TEXT -1 271 "In this last case, the commutative multiplication `*` is appl ied to the matrix entries. It takes three arguments or four arguments . If the fourth argument is used, it is either of type name/symbol/arr ay/matrix or a numeric multiple of such type, for example, K or -K. \+ " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 90 "T o apply Clifford multiplication 'cmul[B]' to matrix entries enter one \+ of the following: " }}{PARA 258 "" 0 "" {TEXT -1 143 "rmulm(M1, M2, c mul); rmulm(M1,M2,cmul,B);rmulm(M1,M2,cmul,K);rmulm(M1,M2,cmul,-K);\n& cm(M1, M2); &cm[B](M1,M2);&cm[K](M1,M2);&cm[-K](M1,M2); \n" }}{PARA 258 "" 0 "" {TEXT -1 89 "To apply Clifford multiplication 'cmulQ[B]' t o matrix entries enter one of the following:" }}{PARA 258 "" 0 "" {TEXT -1 235 "rmulm(M1, M2, cmulQ); rmulm(M1,M2,cmulQ,B);rmulm(M1,M2,c mulQ,K);rmulm(M1,M2,cmulQ,-K);\n&cQm(M1, M2); &cQm[B](M1,M2);&cQm[K](M 1,M2);&cQm[-K](M1,M2); \n\nTo apply wedge multiplication 'wedge' to \+ matrix entries enter one of the following:" }}{PARA 258 "" 0 "" {TEXT -1 60 "rmulm(M1, M2, `&w`); M1 &wm M2; rmulm(M1, M2, wedge); " } }{PARA 258 "" 0 "" {TEXT -1 113 "\nTo apply some generic possibly non- commutative multiplication `&r` to matrix entries enter one of the fol lowing:" }}{PARA 258 "" 0 "" {TEXT -1 37 "rmulm(M1, M2, `&r`); M1 & rm M2; " }}{PARA 258 "" 0 "" {TEXT -1 98 "\nTo apply standard commut ative scalar multiplication to matrix entries enter one of the followi ng:" }}{PARA 258 "" 0 "" {TEXT -1 39 "rmulm(M1, M2, `&*`); M1 &* \+ M2; " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 70 "Similarly for matrices with quaternionic entries we have as fol lows: " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 89 "To apply quaternionic multiplication 'qmul' to matrix entries e nter one of the following:" }}{PARA 258 "" 0 "" {TEXT -1 72 "rmulm(M1, M2, `&q`); M1 &qm M2; rmulm(M1,M2,qmul);\n\nTypical use: " }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 73 "M1 := linalg[matrix](2, 2, [Id + e1we2, e2 + e3, e1 - e2, Id + e2we3]); " }}{PARA 258 "" 0 "" {TEXT -1 137 "M2 := linalg[matrix](2, 2, [Id + e2w e3, e3 + e4, e1 - e2, Id + e1we3]); \n\nM1 := linalg[matrix](2, 2, [I d + 2*qi + 3*qj, qi, qi + qj]); " }}{PARA 258 "" 0 "" {TEXT -1 58 "M2 := linalg[matrix](2, 2, [Id + qi, qj, qk, Id - qi]); \n" }}{PARA 258 "> " 0 "" {MPLTEXT 1 0 6995 "rmulm:=proc(a1::\{list(matrix),dfmatr ix,matrix,clipolynom,cliscalar,clibasmon,climon\},\n a2::\{ list(matrix),dfmatrix,matrix,clipolynom,cliscalar,clibasmon,climon\}, \n a3::\{name,function,procedure,symbol\}) \nlocal ar1,ar2, L,newL,m1,m2,r1,r2,c1,c2,i,j,k,M,reset_prolevel,coB,nameB,lname,tail,o ut;\nglobal _prolevel, `&r`;\noptions `Copyright (c) 1995-2003 by Rafa l Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n################################## ###########\n################################\nif has(0,map(simplify,[ a1,a2])) then return 0 end if;\n################################ \nif nargs=3 then\n coB:=1:\n nameB:=`B`: \n lname:=`B`: \nelif n args=4 then\n if type(eval(args[4]),\{name,symbol,matrix,array\}) t hen\n coB:=1:\n nameB:=args[4];\n lname:=args[4];\n \+ elif type(eval(args[4]),`&*`(numeric,\{name,symbol,matrix,array\})) then\n coB:=op(select(type,\{op(args[4])\},numeric));\n n ameB:=op(remove(type,\{op(args[4])\},numeric));\n lname:=args[4] :\n else \n error \"wrong type of fourth argument %1 in rmulm \",args[4] \n end if;\nelse\n error \"three or four arguments exp ected in rmulm\"\nend if;\n################################\ntail:=op( subsop(1=NULL,subsop(1=NULL,[args])));\n############################## ##\n#return (a1,a2,a3,coB,nameB,lname,tail);\n######################## ########\nif _prolevel then reset_prolevel:=true:\n \+ _prolevel:=false:\n else reset_prolevel:=false\nend if ; \n################################\nif type(a1,matrix ) and not type(a1,\{dfmatrix,climatrix,cliscalar\}) and \n type(a2 ,matrix) and not type(a2,\{dfmatrix,climatrix,cliscalar\})\nthen \n \+ _prolevel:=reset_prolevel:\n return evalm(a1 &* a2) \n end if;\n ################################\nif type(a1,list(matrix)) and type(a2 ,list(matrix)) then \n if nops(a1)<>nops(a2) then error \"received l ists of unequal lengths\" \n else\n i:='i':\n _prolevel:=r eset_prolevel:\n return [seq(procname(a1[i],a2[i],tail),i=1..nops (a1))]\n end if;\nend if;\n################################\nif type (a1,dfmatrix) and type(a2,dfmatrix) then\n return cdfmatrix(procname (ddfmatrix(a1),ddfmatrix(a2),tail))\nend if;\n######################## ########\nif type(a1,\{clipolynom,cliscalar,clibasmon,climon\}) then \+ \n if type(a2,list(matrix)) then return (map2(procname,args)) \n e lif type(a2,dfmatrix) then \n return subs(Id=1,convert(map2(pro cname,a1,ddfmatrix(a2),tail),dfmatrix))\n end if\nend if;\n######### #######################\nif type(a2,\{clipolynom,cliscalar,clibasmon,c limon\}) then \n if type(a1,list(matrix)) then return map(procname,a rgs) \n elif type(a1,dfmatrix) then \n return subs(Id=1,conve rt(map(procname,ddfmatrix(a1),a2,tail),dfmatrix))\n end if\nend if; \n################################\nif not member(a3,\{`&*`,`&r`,Clipl us:-climul,cmul,cmulQ,wedge,qmul,Octonion:-omul\}) then \n error \"t hird argument must be one of the following: cmul, cmulQ, wedge, qmul, \+ omul, &*, &r but received %1 instead\",a3 \nend if;\n################# ###############\nif member(a3,\{`&*`\}) and \n (type(a1,\{clibasmon, climon,clipolynom,climatrix\}) or\n type(a2,\{clibasmon,climon,clip olynom,climatrix\})) then\nerror \"it makes no sense to apply commutat ive multiplication &* to non-commuting elements %1 and %2\",a1,a2 \nen d if;\n################################\nar1:=evalm(a1):ar2:=evalm(a2) :\nif not type(a1,matrix) and type(ar1,matrix) then \n _prolevel :=reset_prolevel: \n return procname(ar1,a2,tail) \nend if;\ni f not type(a2,matrix) and type(ar2,matrix) then \n _prolevel:=re set_prolevel:\n return procname(a1,ar2,tail) \nend if;\n######## ###################################################################### ######\n##If both inputs are of type clipolynom, climon, or clibasmon \+ do the following:\n################################################### #################################\nif (type(evalm(a1),\{clibasmon,clim on,clipolynom\}) \n and \n type(evalm(a2),\{clibasmon,climon,cli polynom\}))\nthen \n if member(a3,\{Cliplus:-climul,cmul,cmulQ\}) t hen\n _prolevel:=reset_prolevel: \n return simplify(reorde r(a3[lname](a1,a2)))\n elif \n member(a3,\{wedge,qmul,omul\}) then\n _prolevel:=reset_prolevel:\n if _warnings_flag and n args=4 then\n WARNING(sprintf(\"ignoring fourth argument %a\", lname))\n end if; \n return eval('simplify'('reorder'(a3 (a1,a2))));\n else\n _prolevel:=reset_prolevel: \n retur n simplify(a3[lname](a1,a2)) \n end if;\nend if; \n################ ###########################\n##If m1 is a polynomial and m2 is a matri x:\n###########################################\nif type(evalm(a1),\{c libasmon,climon,clipolynom,cliscalar\}) \n and \n type(a2,matrix) \n then \n if member(a3,\{qmul\}) then \n m2:=map(eval,a2 ) \n else \n m2:=a2 \n end if;\n L:=map(displayid,co nvert(m2,'mlist'));\n newL:=[]:\n for i from 1 to nops(L) do newL: =[op(newL),a3[lname](a1,L[i])] end do;\n if not member(a3,\{qmul\}) \+ then\n _prolevel:=reset_prolevel: \n return map(displayid, map(simplify,linalg[matrix](linalg[rowdim](a2),linalg[coldim](a2),newL )))\n else \n _prolevel:=reset_prolevel: \n return map(s implify,linalg[matrix](linalg[rowdim](a2),linalg[coldim](a2),newL))\ne nd if:\nend if: \n#######################################\n#a2 is a po lynomial and a1 is a matrix\n#######################################\n if type(evalm(a2),\{clibasmon,climon,clipolynom,cliscalar\}) \nand \n \+ type(a1,matrix) \n then \n if member(a3,\{qmul\}) then \n \+ m1:=map(eval,a1) \n else \n m1:=a1 \n end if;\n \+ L:=map(displayid,convert(m1,'mlist'));\n newL:=[]:\nfor i from 1 t o nops(L) do newL:=[op(newL),a3[lname](L[i],a2)] end do;\nif not membe r(a3,\{qmul\}) then\n _prolevel:=reset_prolevel:\n return map(simp lify,linalg[matrix](linalg[rowdim](a1),linalg[coldim](a1),newL))\nelse \n _prolevel:=reset_prolevel: \n return map(simplify,linalg[matrix ](linalg[rowdim](a1),linalg[coldim](a1),newL))\nend if:\nend if: \n### ###################################################\n##If both inputs \+ are of type matrix, do the following:\n############################### #######################\nif member(a3,\{qmul\}) then \n m1:=evalm(map (eval,a1));m2:=evalm(map(eval,a2))\nelse \n m1:=evalm(a1);m2:=evalm(a 2); \nend if;\nm1:=displayid(m1):m2:=displayid(m2):\nr1:=linalg[rowdim ](m1):r2:=linalg[rowdim](m2):\nc1:=linalg[coldim](m1):c2:=linalg[coldi m](m2):\nif c1 <> r2 then \n error \"matrices have incompatible dime nsions and cannot be multiplied\" \nend if;\nM:=linalg[matrix](r1,c2,[ ]);\nk:='k':\nfor i from 1 to r1 do\nfor j from 1 to c2 do\nif a3=`&*` then \n M[i,j]:=sum(m1[i,k] * m2[k,j],k=1..c1) \nelse \n M[i,j]:= map(simplify,add(a3[lname](m1[i,k],m2[k,j]),k=1..c1)) \nend if;\nod en d do;\n_prolevel:=reset_prolevel:\nif member(a3,\{Cliplus:-climul,cmul ,cmulQ,wedge\}) then \n return subs(Id=1,map(reorder,map(simplify,ev alm(M)))) else\n return subs(Id=1,map(simplify,evalm(M))) \nend if; \nend proc:" }}{PARA 0 "" 0 "" {TEXT 261 9 "\nNo. 53: " }{TEXT 343 5 " `&cm`" }{TEXT 344 333 " denotes multiplication of matrices when Cliffo rd product of Cl(B) is applied to matrix entries. One can use index as in &cm[K](p1,p2), &cm[-K](p1,p2), or &cm(p1,p2), &cm(M1,M2. However, \+ when K has been assigned a matrix, put K between double quotes as in & cm[''K''](p1,p2), &cm[''-K''](p1,p2).\n(Has been moved to Clifford:-se tup).\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 8 "No. 54: " } {TEXT 345 6 "`&cQm`" }{TEXT 346 416 " denotes multiplication of matric es when Clifford product of Cl(Q) is applied to matrix entries. One ca n use index as in &cQm[K](p1,p2), or &cQm[-K](p1,p2) provided index ha s not been assigned a matrix. If K has been assigned a matrix, put K b etween double quotes as in &cQm[''K''](p1,p2), or &cQm[''-K''](p1,p2). Procedure can also be used withouht the index as in &cQm(p1,p2).\n(Ha s been moved to Clifford:-setup).\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 274 8 "No. 55: " }{TEXT 347 5 "`&wm`" }{TEXT 348 131 " denotes m ultiplication of matrices when wedge/exterior product is applied to ma trix entries:\n(Has been moved to Clifford:-setup).\n" }}{PARA 0 "" 0 "" {TEXT 262 8 "No. 56: " }{TEXT 349 5 "`&qm`" }{TEXT 350 127 " denote s multiplication of matrices when quaternion product is applied to mat rix entries:\n(Has been moved to Clifford:-setup).\n" }}{PARA 0 "" 0 " " {TEXT 275 8 "No. 57: " }{TEXT 351 5 "`&om`" }{TEXT 352 154 " denotes multiplication of matrices when non-associative octonionic multiplica tion is applied to the matrix entries.\n(Has been moved to Clifford:-s etup).\n" }}{PARA 0 "" 0 "" {TEXT 263 8 "No. 58: " }{TEXT 353 5 "`&rm` " }{TEXT 354 217 " denotes multiplication of matrices when a generic a ssociative but possibly not commutative `&r` product is applied to mat rix entries. It can take index. User needs to define procedue `&r` in \+ a similar mannet to `&c`." }{TEXT -1 1 "\n" }{TEXT 479 37 "(Has been m oved to Clifford:-setup).\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 59. \+ Procedure " }{TEXT 355 8 "matKrepr" }{TEXT -1 261 " uses previously co mputed matrices of basis 1-vectors to find a matrix representation in \+ a minimal left or right ideal of any Clifford polynomial in the given \+ Clifford algebra Cl(Q). Depending on the signature [p,q] of the quadr atic form Q, these matrices are " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 119 "real if (p - q) mod 8 is 0, 1, 2; \n complex if (p - q) mod 8 is 3 or 7; \nquaternionic if (p - q) mod 8 i s 4, 5, or 6." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 311 "The real matrices of 1-vectors in dimensions from 2 to 8 have been computed with the procedure 'spinorKrepr' in minimal left i deals and stored in a form of a table called 'matrealL' in Maple libra ry. The indices of the table are given by the signature [p,q]. To see \+ matrices in a specific signature [p,q], enter" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 17 ">matrealL([p,q]);" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 69 "(assu ming, of course, that the matrices for this signature are real)." }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 359 "Simi larly for complex matrices in dimensions from 3 to 7 which are stored \+ in the file 'matcompL.m' and for quaternionic matrices in dimensions f rom 2 to 8 which are stored in the file 'matquatL.m'.\n\nSimilarly for matrices representing basis 1-vectors in right minimal ideals; in thi s case corresponding files are: 'matrealR.m', 'matcompR.m', and 'matqu atR.m'." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 316 "Matrices representing Clifford polynomials are generally compu ted with 'matKrepr' much faster than with 'spinorKrepr' because the fo rmer is a linear procedure that uses matrix multiplication 'rmulm' to \+ compute matrices representing basis monomials.\n\nNOTE: This procedure can now handle semi-simple Clifford algebras." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 13 "Typical use: " }}{PARA 258 "" 0 "" {TEXT -1 92 "to see matrices representing 1-vectors in a l eft minimal ideal for the current form B enter:" }}{PARA 258 "" 0 "" {TEXT -1 12 ">matKrepr();" }}{PARA 258 "" 0 "" {TEXT -1 4 " " }} {PARA 258 "" 0 "" {TEXT -1 103 "to find a matrix representing a Cliffo rd polynomial p for the current B in a left minimal ideal enter:\n" }} {PARA 258 "" 0 "" {TEXT -1 36 ">matKrepr(p); \n>matKrepr(p,'left');\n " }}{PARA 0 "" 0 "" {TEXT 256 313 "to find a matrix representing a Cli fford polynomial p for the current B in a right minimal ideal enter:\n \n>matKrepr(p,'right');\n\nto see matrices representing 1-vectors in a minimal left or right ideal when Q has the signature [p,q], enter:\n \n>matKrepr([p,q]);\n>matKrepr([p,q],'left');\n\nor\n\n>matKrepr([p,q] ,'right');" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4866 "matKrepr:=proc() \nlocal mindex,Bsize,dim,ind,pq,pq sig,matdata,i,a1,a2,dimrepr,ans,pqmod8,pqmod4,matdatatable,\n m,f lag_simple,k,L,t,co,x,reprmulm;\nglobal B,matrealL,matcompL,matquatL,m atrealR,matcompR,matquatR:\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription ` Last revised: November 1, 2002`;\n#################################### #########\n#Checking argument types\nif not member(nargs,\{0,1,2\}) th en \n error \"wrong number of arguments: expects 0, 1, or 2 argument (s)\" \nend if;\nif member(nargs,\{1,2\}) and not type(args[1],\{list, clibasmon,climon,clipolynom\}) then\n error \"first argument must be of type 'list', clibasmon, climon, or clipolynom but received one of \+ type %1\",whattype(args[1]) \nend if;\nif nargs=2 and not member(args [2],\{'left','right'\}) then \n error \"second argument, when used, \+ must be 'left' or 'right', but received %1\",args[2] \nend if;\nif na rgs<>0 then a1:=args[1] end if;\nif nargs=0 or type(a1,\{clibasmon,cli mon,clipolynom\}) then\n if not type(B,matrix) then \n error \+ \"matrix must be assigned to B\"\n elif not type(B,'diagmatrix') th en\n error \"bilinear form B must be diagonal\"\n else \n \+ pq:=Bsignature();\n pqmod8:=(pq[1]-pq[2]) mod 8;\n pqmo d4:=(pq[1]-pq[2]) mod 4;\n flag_simple:=evalb(pqmod4<>1);\n e nd if;\nelif type(a1,list) then pq:=a1:pqmod8:=(pq[1]-pq[2]) mod 8 \ne lse error \"wrong argument(s)\"\nend if;\n############################ ##################\nif type(a1,\{clibasmon,climon,clipolynom\}) then\n mindex:=maxindex(a1):Bsize:=linalg[coldim](B):\n if mindex > Bsiz e then\n error \"input error: maximum index in your input %1 is g reater than the size %2 of the currently defined bilinear form B\",min dex,Bsize \n end if;\nend if;\nif nargs=1 or nargs=0 then a2:='left' else a2:=args[2] end if;\n#read in appropriate data file: \nif member (pqmod8,\{0,1,2\}) then\n if a2='left' then \n #if not as signed(matrealL) then readlib(matrealL) end if;\n matdatatabl e:=matrealL:\n else\n #if not assigned(matrealR) then rea dlib(matrealR) end if;\n matdatatable:=matrealR:\n end i f;\nelif member(pqmod8,\{3,7\}) then\n if a2='left' then\n \+ #if not assigned(matcompL) then readlib(matcompL) end if;\n \+ matdatatable:=matcompL:\n else \n #if not assigned(matco mpR) then readlib(matcompR) end if;\n matdatatable:=matcompR: \n end if;\nelif member(pqmod8, \{4,5,6\}) then\n if a2='lef t' then\n #if not assigned(matquatL) then readlib(matquatL) e nd if;\n matdatatable:=matquatL:\n else\n #if not assigned(matquatR) then readlib(matquatR) end if;\n matdatat able:=matquatR:\n end if; \n else error \"wrong value of pqmod8: \+ %1\",pqmod8 \nend if;\n#######################################\npqsig: =map(op,[indices(matdatatable)]);\nif not member(pq,pqsig) then\n er ror \"matrices for signature %1 in %2 minimal ideal have not been comp uted yet\",pq,a2 \nend if;\n#######################################\n matdata:=matdatatable[pq]:\nif nargs=0 or type(a1,list) then \n retu rn matdata\nend if;\n#Continue if the first element is a polynomial\nd im:=linalg[coldim](B):dimrepr:=linalg[coldim](rhs(matdata[1]));\nif di m<>nops(matdata) then \n error \"size of B is different from the num ber of 1-matrices\"\nend if;\n######################################## \nreprmulm:=proc() \n if nargs=1 then return args \n elif nargs=2 \+ then return subs(Id=1,rmulm(args,`cmulQ`)) \n else return subs(Id=1, reprmulm(args[1..(nargs-2)],rmulm(args[nargs-1],args[nargs],`cmulQ`))) \n end if;\nend proc:\n########################################\nm :=array(1..nops(matdata)):\nfor i from 1 to nops(matdata) do m[i]:=rhs (matdata[i]) end do;\nif type(a1,clibasmon) then\n ind:=Clifford:-ex tract(a1,'integers'): \n if a1='Id' then \n if flag_simple then \n return linalg[diag](1$dimrepr) \n else \n r eturn convert([linalg[diag](1$dimrepr)$2],'dfmatrix') \n end if; \n end if; \n if nops(ind)=1 then ind:=op(ind):\n return \+ subs(Id=1,evalm(m[ind])) \n else return subs(Id=1,reprmulm(seq(evalm (m[ind[i]]),i=1..nops(ind)))) \n end if:\nend if;\n################# ########################\nans:=clilinear(a1,'K'):\nif flag_simple then \n return subs(Id=1,evalm(eval(subs(K=procname,ans)))) \nend if;\na ns:=eval(subs(K=procname,ans));\nif type(ans,`+`) then ans:=[op(ans)] \+ elif\n type(ans,`*`) then ans:=[ans] else\n error \"unexpected typ e in matKrepr\" \nend if;\nL:=select(type,ans,matrix);\nans:=remove(ty pe,ans,matrix);\nk:='k':x:='x':\nfor t in ans do\n m:=ddfmatrix(op( select(type,[op(t)],matrix)));\n co:=mul(x,x=remove(type,[op(t)],ma trix));\n L:=[op(L),convert([seq(evalm(co*m[k]),k=1..2)],'dfmatrix' )]\nend do:\nif nops(L)=1 then return L[1] end if;\nans:=L[1]:\nfor k \+ from 2 to nops(L) do\nans:=adfmatrix(ans,L[k]) end do:\nreturn evalm(a ns);\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 25 "No. 60. Sorting fu nction " }{TEXT 376 7 "bygrade" }{TEXT -1 789 ": it sorts a list of Cl ifford basis monomials, Clifford monomials, or Clifford polynomials. B asis monomials and Clifford monomials are sorted by grade; in case of \+ a tie it sorts by lexicographic order based on the basis monomials. Ho wever, basis monomials are put before Clifford monomials. If any of th e elements is a Clifford polynomial, then ties are resolved by sorting by the weight of each element (defined as the sum of the grades of al l terms) and then by then number of Clifford basis monomials in each e xpression. It returns true or false in each case, and can be used in s orting a list of basis monomials, Clifford monomials, and Clifford pol ynomials in the construction sort(L, bygrade).\n\nUse: bygrade(p1,p2) \+ where p1 and p2 are of type 'clibasmon', 'climon', or 'clipolynom';\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1985 "bygrade:=proc(a1::\{clibasmon ,climon,clipolynom\},\n a2::\{clibasmon,climon,clipolynom \}) \nlocal flag1,flag2,flag11,flag22,p1,p2,n1,n2,c1,c2,x,w1,w2;\nopti ons `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`; \n#############################################\nif type(a1,clibasmon) then p1:=a1;\n flag1:=true:\n \+ flag11:=true:\n n1:=Clifford:-e xtract(p1): \n elif type(a1,climon) then p1:=op(cliterms(a1));\n \+ flag1:=true:\n flag1 1:=false:\n n1:=Clifford:-extract(p1): \n \+ else p1:=a1;\n flag1:=false:\nend if;\nif type(a2,clibasmon) th en p2:=a2;\n flag2:=true:\n \+ flag22:=true:\n n2:=Clifford:-extr act(p2): \n elif type(a2,climon) then p2:=op(cliterms(a2));\n \+ flag2:=true:\n flag22:= false:\n n2:=Clifford:-extract(p2): \n els e p2:=a2;\n flag2:=false:\nend if;\nx:='x':\nif flag1 and flag2 then\n if nops(n1)nops (n2) then return false\n else \n if evalb(flag11 and flag22) th en return lexorder(p1,p2)\n elif evalb(flag11 and not flag22) \+ then return lexorder(p1,p2)\n elif evalb(not flag11 and flag22 ) then return not lexorder(p2,p1);\n else return true\n e nd if;\n end if; \nelse \n n1:=maxgrade(p1):\n c1:=cliterms(p1 ):\n w1:=add(maxgrade(x),x=c1):\n n2:=maxgrade(p2):\n c2:=cliter ms(p2):\n w2:=add(maxgrade(x),x=c2):\n if n1=n2 then\n if w1= w2 then \n if nops(c1)<=nops(c2) then return true else return \+ false end if;\n else if w1 " 0 "" {MPLTEXT 1 0 2121 "commutingelements:=proc(a1::list( clibasmon)) \nlocal g,groupgens,L,L2,numfact,f,flag1,flag2,flag3,gen,p ,q,i;\nglobal B;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revis ed: November 1, 2002`;\n############################################# \nif not type(B,matrix) then \n error \"matrix must be assigned to B \"\nend if;\nif not type(B,'diagmatrix') then \n error \"the bilinea r form B is not diagonal as expected\" \nend if;\np:=Bsignature(B)[1]: q:=Bsignature(B)[2]:\nnumfact:=q-RHnumber(q-p):\nflag1:=member(Id,a1): \nL:=remove(member,a1,[Id]):\n#return a1 if it was [Id]\nif L=[] then \+ return args end if; \n#return a1 if had one element of square 1 or [] \+ if the square <>1 \nif nops(L)=1 then\n if cmul(L[1],L[1])=Id then r eturn L\n else return [] \nend if;\nend if;\n#First, sort the lis t\nL:=sort(L,bygrade):\n#Find first element of square 1 mod Id\nflag2: =false:L2:=[]:groupgens:=[]:\nfor g in L while not flag2 do \n if e valb(cmul(g,g)=Id) then groupgens:=[g];flag2:=true\n else L2:=[op(L 2),g] fi end do:\nL:=remove(member,L,[op(L2),op(groupgens)]);\nif L=[] then \n if flag1 then \n return [Id] else return groupgens \n \+ end if;\nend if; \nif nops(groupgens)=numfact then \n return (sort (groupgens,bygrade)) end if;\n#Find commuting elements with square 1 m od Id in the specified list of basis monomials\nfor g in L while nops( groupgens)0)) \n then groupgens:=[op(groupgens),g] \+ \n end if;\nend if:\nend do:\nif groupgens=[] then return args el se return sort(groupgens,bygrade) end if;\nend proc:\n" }}{PARA 258 " " 0 "" {TEXT -1 19 "No. 62. Procedure " }{TEXT 378 16 "factoridempote nt" }{TEXT -1 369 " can factor the given idempotent e into a product o f N elements of the type (1/2)*(Id+e[i]), i=1..N, where \{e[i],i=1..N \} is a set of commuting basis monomials with square 1 mod Id in the s tandard (canonical) basis of Cl(Q). It is known that when N = q - RHn umber(q-p) then e is primitive. \n\nTypical use: factoridempotent(f); #here f is expected to be an idempotent\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1736 "factoridempotent:=proc(a1::idempotent) \nlocal T,ee ,i,L,flag,flag1,flag2,b1b2,b1,b2,ans;\nglobal B;\noptions `Copyright ( c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights rese rved.`;\ndescription `Last revised: November 1, 2002`;\n############## ###############################\nif a1=Id then return Id end if;\nif n ot type(B,matrix) then \n error \"matrix must be assigned to B\"\nen d if;\nif not type(B,'diagmatrix') then \n error \"the bilinear form B is not diagonal as expected\" \nend if;\nee:=eval(a1):\nL:=sort(rem ove(member,convert(cliterms(ee),list),[Id]),bygrade):\nif nops(L)=1 th en \n ans:=(1/2)*(Id+L[1]);\n if displayid(a1-ans)=0 then return a ns else return a1 end if;\nend if;\nflag1:=true:\nwhile flag1 do\nflag 2:=true:\nL:=sort(L,bygrade);\nfor b1 in L while flag2 do\nfor b2 in r emove(member,L,[b1]) while flag2 do\n b1b2:=cmulQ(b1,b2):\n if m ember(b1b2,L) then flag2:=false;\n L:=remove (member,L,[b1b2]) end if;\n if member(-b1b2,L) then flag2:=false;\n L:=remove(member,L,[-b1b2]) end if;\n if flag2 then flag1:=false end if;\nod od end do: \nL:=commutingelements (L);\nif nops(L)=1 then \n ans:=(1/2)*(Id+L[1]);\n if displayid(a1 -ans)=0 then return ans else return a1 end if;\nend if;\nL:=sort(L,byg rade);\ni:='i':\nans:='cmulQ'(seq((1/2)*(Id+L[i]),i=1..nops(L)));\nif \+ eval(ans)-a1=0 then return (ans) end if;\n#try another sign permutatio n\nfor i from 1 to nops(L) do\n L||i:=[L[i],-L[i]]\nend do:\nT:=com binat[cartprod]([seq(L||i,i=1..nops(L))]):\nflag:=false:\nwhile not T[ finished] and not flag do \nL:=T[nextvalue]();\nans:='cmulQ'(seq((1/2) *(Id+L[i]),i=1..nops(L)));\nif eval(ans)-a1=0 then flag:=true:return a ns end if;\nend do:\n#return unfactored\nreturn a1;\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 19 "No. 63. Procedure " }{TEXT 379 11 "mak ealiases" }{TEXT -1 996 " allows the user to alias basis monomials in \+ a Clifford algebra Cl(V), e.g., to alias e1we2 as e12, or e2we1 as e21 . The procedure accepts a positive integer p>1 where p denotes the dim ension of the vector space V. A practical limitation on p is of cours e the amount of memory Maple will allocate to store these aliases sinc e every basis monomial, not necessarily written in the standard order, will be aliased. This procedure is intended to be used when p < 5 al though it can be used also when p < 10. Remember that to unalias e12 \+ one needs to either restart Maple or simply assign e12:='e12'.\n\nAs a memory saving feature, option 'ordered' (or \"ordered\") may be enter ed as a second parameter. If the second parameter is used, aliases are created only for monomials with ordered indices, for example, e12 wil l be an alias for e1we2.\n\nThe procedure returns a list of aliases to be defined so they can bee seen by the user. In order to finish the \+ definition process, use 'eval' as shown below.\n" }}{PARA 258 "" 0 "" {TEXT -1 139 "Once basis elements have been aliased, Clifford multipli cation can be done using these aliases.\n\nTypical use: \n\n>makealias es(3);\n>eval(%);\n" }}{PARA 258 "" 0 "" {TEXT -1 41 "or\n\n>makealias es(3,'ordered');\n>eval(%);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 802 "m akealiases:=proc(a1::posint,a2::\{symbol,string\}) \nlocal L,i,k,l,K,s ;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried \+ Fauser. All rights reserved.`,remember;\ndescription `Last revised: No vember 1, 2002`;\n#############################################\nif no t a1>1 then \n error \"first parameter must be a positive integer la rger than one\" \nend if;\nif nargs=2 and not member(a2,\{'ordered',\" ordered\"\}) then\n error \"second optional parameter, when used, mu st be 'ordered'\" \nend if;\nk:='k':l:='l':i:='i':\nL:=[seq(op(combina t[choose]([seq(i,i=1..a1)],k)),k=2..a1)];\nif nargs=1 then \n K:=[se q(op(combinat[permute](l)),l=L)];\n s:=seq(cat(e,op(K[i]))=makecliba smon(K[i]),i=1..nops(K))\nelse\n s:=seq(cat(e,op(L[i]))=makeclibasmo n(L[i]),i=1..nops(L))\nend if;\nreturn 'alias'(s)\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 64. Procedure " }{TEXT 380 4 "cinv " }{TEXT -1 1285 " calculates a symbolic inverse of any Clifford polyn omial p in the given Clifford algebra Cl(B) or in its subalgebra. The procedure determines a basis for the smallest subalgebra of Cl(B) in \+ which the inverse might exist. For example, if the polynomial p conta ins only even grades, then the inverse is sought in an even subalgebra of Cl(B); otherwise, the inverse is sought in a Clifford algebra over a vector space V whose dimension equals tha maximum index in p. \n\n If the bilinear form B is not assigned then every Clifford polynomial \+ in Cl(B) has a symbolic inverse. If the bilinear form B is assigned th en not every element in Cl(B) has the inverse. For example, nilpotent and non-trivial idempotent elements have no inverses. Elements p suc h that p &c p = a*p for some 'cliscalar' also have no inverses (these elements are called here 'almost idempotent').\n\nThus, if B is assig ned and the inverse does not exist, the procedure tries to identify if p is one of the above types and if so, it returns an appropriate erro r message. Otherwise it returns 'NULL'.\n\nThis procedure can be used with a second optional argument K of type symbol, name, matrix , or a rray. In that case, it computes the inverse in Cl(K). The seconf argum ent can also be -K, or any numeric multiple of K." }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 70 "Typical use: cinv(e1 \+ + 2*e2);cinv(e1 + 2*e2,K); cinv(e1 + 2*e2,-K); \n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4208 "cinv:=proc(a1::\{cliscalar,clibasmon,climon,clipo lynom\}) \nlocal p,pp,pinv,mindex,cinv11,s,aaa,flagB,flagBdiag,S,lname ,flagindexed;\nglobal B,_warnings_flag;\noptions `Copyright (c) 1995-2 003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`,re member;\ndescription `Last revised: November 1, 2002`;\n############## ###############################\nif nargs=1 then\n lname:=`B`;\n \+ flagindexed:=false:\nelif nargs=2 and type(args[2],\{symbol,name,arra y,matrix,`&*`(algebraic,name)\}) then\n lname:=args[2];\n flagin dexed:=true:\nelse error \"only one or two arguments are expected\"\ne nd if;\n############################\ncinv11:=proc(a1,lname)\nlocal i, d,dbasis,N,u,xm,v,uv,vu,vars,sys,L1,v1,nontrivial;\nglobal evenelement ;\n nontrivial:=proc(S::\{set(\{relation,algebraic\}),list(\{relatio n,algebraic\})\}) \n local istrivial;\n is trivial:=proc(x) \n if type(x,relation) then \+ evalb(x) else evalb(x=0) end if; \n end proc;\n \+ remove(istrivial,S)\n end proc: \ni:='i':\nd:=maxi ndex(a1):\nif type(a1,'evenelement') then dbasis:=cbasis(d,'even')\n \+ else dbasis:=cbasis(d) \nend if:\nN:=nops(dbas is):\nu:=clicollect(reorder(a1)):\nxm:=array(1..N):\nv:=sum(xm[i]*dbas is[i],'i'=1..N);\nuv:=collect(cmul[lname](u,v)-Id,dbasis);\nvu:=collec t(cmul[lname](v,u)-Id,dbasis);\nvars:=\{coeffs(v,dbasis)\};\nsys:=\{co effs(uv,dbasis),coeffs(vu,dbasis)\};\nsys:=nontrivial(sys); #eliminate trivial equations\nL1:=solve(sys,vars);\nif L1=NULL then return (NULL ) else \nv1:=subs(L1,v);\nv1:=reorder(v1):\nv1:=clicollect(v1):\nv1:=m ap(normal,v1);\nreturn (eval(v1)): \nend if;\nend proc:\n############# ########################\nif type(a1,cliscalar) then\n if a1<>0 then return 1/a1 else error \"0 has no inverse\" end if;\nend if;\nmindex: =maxindex(a1);\nif mindex=0 then return Id/scalarpart(a1) end if;\np:= simplify(reorder(a1)):\np:=displayid(p):\npinv:=cinv11(p,lname);\nif e valb(pinv<>NULL) then return pinv end if; \n########################## ###########\nflagB:=type(evalm(lname),matrix):\nif not flagB then retu rn \"unable to find inverse of %1\",a1 end if;\n###################### ###############\nif _warnings_flag then\n WARNING(`testing why enter ed argument has no inverse...`)\nend if;\n#Checking these special case s only when lname is assigned a matrix:\ns:='s':aaa:='aaa':\nflagBdiag :=type(evalm(lname),diagmatrix):\n#################################### ###\n###Checking if element a1 is nilpotent\n######################### ##############\nif type([p,lname],nilpotent) then\n if flagBdiag the n \n error \"element %1 is nilpotent in signature %2 and as such \+ it has no inverse\",a1,Bsignature(lname) \n else\n error \"elem ent %1 is nilpotent in current %2 and as such it has no inverse\",a1,l name \n end if;\nend if;\n#######################################\n# ##Checking if element a1 is idempotent\n############################## #########\nif not member(p,\{Id\}) and type([p,lname],idempotent) then \n if flagBdiag then \nerror \"element %1 is an idempotent in signat ure %2 and as such it has no inverse\",a1,Bsignature(lname)\n else \+ \nerror \"element %1 is an idempotent in current %2 and as such it has no inverse\",a1,lname\n end if;\nend if;\n######################### ##############\n###Checking if a1 is almost idempotent\n############## ######################### \npp:=cmul[lname](p,p):\nif match(pp=aaa*p,c literms(p),'s') then \n if flagBdiag then \n error \"element 'p'=% 1 is almost an idempotent since %2 and as such it has no inverse in si gnature %3\", a1,subs(s,'cmul'('p','p')=aaa*'p'),Bsignature(lname)\n \+ else \n error \"element 'p'=%1 is almost an idempotent since %2 and as such it has no inverse in current %3\", a1,subs(s,'cmul'('p','p')= aaa*'p'),lname\n end if;\nend if;\n################################# ######\nS:=\{solve(pp-s*p,s)\}:\nif not evalb(S=\{\}) then \n if fla gBdiag then \n error \"element 'p'=%1 is almost an idempotent since \+ %2 and as such it has no inverse in signature %3\", a1,subs(aaa=op(S), 'cmul'('p','p')=aaa*'p'),Bsignature(lname)\n else \n error \"eleme nt 'p'=%1 is almost an idempotent since %2 and as such it has no inver se in current\", a1,subs(aaa=op(S),'cmul'('p','p')=aaa*'p'),lname\n \+ end if;\nend if;\nreturn NULL\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 65. Procedure " }{TEXT 381 9 "pseudodet" }{TEXT -1 87 " com putes pseudodeterminant of a 2 x 2 matrix with entries in a Clifford a lgebra Cl(B)." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 68 "Typical use: M := linalg[matrix](2, 2, [Id, e1 + e2, e3, \+ e4we3]); " }}{PARA 258 "" 0 "" {TEXT -1 37 " \+ pseudodet(M);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 532 "pseudodet:=proc(a1::\{climatrix,matrix\}) local M,a, b,c,d;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertf ried Fauser. All rights reserved.`;\ndescription `Last revised: Novemb er 1, 2002`;\n#############################################\nM:=map(di splayid,evalm(a1)):\nif linalg[rowdim](M) <> 2 or linalg[coldim](M) <> 2 then \n error \"matrix must be 2 x 2\" \nend if;\na:=simplify(M[1 ,1]): b:=simplify(M[1,2]):\nc:=simplify(M[2,1]): d:=simplify(M[2,2]) :\nreturn simplify(cmul(a,reversion(d)) - cmul(b,reversion(c)))\nend p roc:\n" }}{PARA 258 "" 0 "" {TEXT -1 45 "No. 66. Defining quaternionic mutliplication " }{TEXT 382 4 "qmul" }{TEXT -1 687 ". Quaternions ar e defined as the even elements in Cl(3) (or the para-bivectors in Cl(3 )). Thus, a quaternion basis is [Id, e3we2,e1we3,e2we1] and it is avai lable as the first component of global variable '_quatbasis' defined a t the initialization time (type _quatbasis or _quatbasis[1] at the Map le prompt to see it). See P. Lounesto, \"Clifford Algebras and Spinor s\", page 49, for more information on quaternions. Any element that b elongs to this vector space is now of type 'quaternion'. The infix for m of this multiplication is `&q`. Via the procedure 'rmulm', the qua ternionic multiplication may also be applied to matrices with quaterni onic entries and is then denoted by `&qm`." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 121 "NOTE: in order to see answer s displayed in terms of the basis \{Id, qi, qj, qk\}, apply 'qdisplay' to the result of 'qmul'." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 105 "Typical use: qmul(Id + e1we2, e1we3); or (I d + 2*e1we2) &q (e2we3 + e1we2); or (Id + qi) &q (qj + qk); \n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1298 "qmul:=proc() local q1,q2,q3,step1 ,repqmul; \n global B,qi,qj,qk,_default_Clifford_p roduct;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bert fried Fauser. All rights reserved.`;\ndescription `Last revised: Novem ber 1, 2002`;\n#############################################\nif membe r(0,[args]) then return 0 end if;\nif nargs=1 then return qdisplay(arg s) end if;\n repqmul:=proc() \n if nargs=1 then return args elif \n nargs=2 then return 'qmul'(args) else\n return repqmu l(args[1..(nargs-2)],'qmul'(args[nargs-1],args[nargs])) \n end if; \n end proc:\nif nargs>2 then \n q3:=eval(repqmul(args)):\n retu rn qdisplay(map(combine,q3,trig)) \nend if;\n_default_Clifford_product :='cmulNUM':\nq1:=eval(args[1]):q2:=eval(args[2]):\nif type(q1,`^`) or type(q2,`^`) then \n error \"illegal expression found: use 'qinv' f or the quaternionic inverse\" \nend if;\nif type(q1,cliscalar) or type (q2,cliscalar) then \n return qdisplay(q1*q2) \nend if;\nif q1=Id th en return qdisplay(q2) end if;\nif q2=Id then return qdisplay(q1) end \+ if;\nif not type(q1,quaternion) or not type(q2,quaternion) then\n er ror \"wrong input type: input must be of type 'cliscalar' or 'quaterni on'\" \nend if;\nstep1:=reorder(cmul(q1,q2));\nreturn qdisplay(map(com bine,clicollect(step1),trig))\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 375 23 "No. 67. Ampersand form " }{TEXT 383 4 "`&q`" }{TEXT 384 4 " of " }{TEXT 385 4 "qmul" }{TEXT 386 39 ".\n(Has been moved to Clifford:- setup).\n" }}{PARA 258 "" 0 "" {TEXT -1 42 "No. 68. Defining quaternio nic conjugation " }{TEXT 387 8 "q_conjug" }{TEXT -1 112 ". Recall tha t complex conjugation was named 'c_conjug' while the Clifford conjugat ion was just 'conjugation'. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 68 "Typical use: q_conjug(Id + 2*e1we2); or q_conjug(Id + 2*qi + qk); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 557 " q_conjug:=proc(q::algebraic) local q1; global qi,qj,qk;\noptions `Copy right (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All righ ts reserved.`;\ndescription `Last revised: November 1, 2002`;\n####### ######################################\nif type(q,matrix) then return \+ map(procname,q) elif\n type(q,\{cliscalar,quaternion\}) then\nq1:=ev al(q):\nif type(q1,cliscalar) then return q1 \nelse\n return qdispl ay(2*scalarpart(q1)-q1)\nend if;\nelse\n error \"wrong input types: \+ input must be of type 'cliscalar', 'quaternion', or 'matrix' \" \nend \+ if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 26 "No. 69. Quaternioni c norm " }{TEXT 388 5 "qnorm" }{TEXT -1 24 " is defined as follows: " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 65 " Ty pical use: qnorm(Id + 2*e1we2); or qnorm(Id + qi + qj + qk); " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "qn orm:=proc(q::\{cliscalar,quaternion\}) local q1,n,co; global qi,qj,qk; \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried F auser. All rights reserved.`;\ndescription `Last revised: November 1, \+ 2002`;\n#############################################\nq1:=expand(eval (q));\nif type(q1,cliscalar) then return abs(q1) \nelse\n n:=0:for c o in [coeffs(q1,cliterms(q1))] do n:=n+co^2 end do;\n return combine (sqrt(n),trig) \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 38 "No. 70. Quaternionic inverse is named " }{TEXT 389 4 "qinv" } {TEXT -1 141 ". Recall that the inverse of a Clifford polynomial can \+ be calculated with 'cinv' and that quaternions form a noncommutative d ivision ring. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 " " {TEXT -1 66 "Typical use: qinv(Id + 2*e1we2); or qinv(Id + 2*qi + 3* qj + qk); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "qinv:=proc(q::\{cliscalar,quaternion\}) local q1,q2; \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried \+ Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#############################################\nq1:=eval(q):\n if q1=0 then error \"zero quaternion has no inverse\"\nelif type(q1,cl iscalar) and q1<>0 then return 1/q1\nelse q2:=q_conjug(q1)/(qnorm(q1)) ^2:\n return qdisplay(map(combine,q2,trig))\nend if;\nend proc:\n " }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 71. Procedure " }{TEXT 390 8 "q display" }{TEXT -1 101 " displays quaternions or matrices with quatern ionic entries in terms of the basis \{Id, qi, qj, qk\}. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 360 93 "Typical use: qdisplay(e1we2 + 2*Id); map(qdisplay, matrix(2, 2, [Id, e1we2, e2we3, e1we3])); " }{TEXT -1 2 " \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "q display:=proc(a1::\{algebraic,array\}) local q; global qi,qj,qk;\nopti ons `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`; \n#############################################\nif type(a1,matrix) th en\n if not type(a1,climatrix) then \n return evalm(a1) else \n return map(qdisplay,a1) \n end if;\nend if;\nq:=eval(a1):\nif \+ type(q,cliscalar) then return q end if;\nif type(q,quaternion) then\nq :=map(combine,clicollect(reorder(q)),trig);\nreturn coeff(q,Id)-coeff( q,e1we2)*'qk'+coeff(q,e1we3)*'qj'-coeff(q,e2we3)*'qi'\nelse \nerror \" wrong input type: input must be of type 'cliscalar', 'quaternion', or \+ 'matrix' \" \nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "N o. 72. Procedure " }{TEXT 391 5 "rot3d" }{TEXT -1 161 " rotates a vect or in 3-dimensional Euclidean space V using the quaternion multiplica tion. Namely, any vector v is transformed according to the following \+ law: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 84 " v - > q &c v &c qinv(q) " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 459 "where q is a quaternion given in the basis [Id, e 1we2, e1we3, e2we3]. The first entry should be a vector (or any elemen t of the Clifford algebra) while the second element is a quaternion. \+ Type '_quatbasis' to see how quaternions are defined here. Elements ' qi', 'qj', 'qk' are defined at the time of initialization and denote t he pure-quaternion basis elements. It is assumed that the user has de fined a bilinear form B as the 3 x 3 identify matrix as in:\n" }} {PARA 258 "" 0 "" {TEXT -1 28 " >B := linalg[diag](1$3); \n" }}{PARA 258 "" 0 "" {TEXT -1 108 "before using 'rot3d'. Of course, 'rot3d' wi ll also work if the first argument were any element in Cl(3). \n" }} {PARA 258 "" 0 "" {TEXT -1 296 "NOTE: traditionally one uses \{1, i, j , k\} to denote a quaternion basis. Here, we are using symbol 'qi' fo r 'i', 'qj' for 'j', and 'qk' for 'k'. Symbol 'Id' denotes, as usual, the unit element in all Clifford algebras as well as the unit element in reals, complexes, quaternions, and octonions. " }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 55 "Typical use: rot3d(e1 + e2, Id + 2*qi - 3*qj + 2*qk); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 838 "rot3d:=proc(a1::\{cliscalar,clibas mon,climon,clipolynom\},\n a2::quaternion) \nlocal q2,q2inv ; global B,qi,qj,qk; \noptions `Copyright (c) 1995-2003 by Rafal Ablam owicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last \+ revised: November 1, 2002`;\n######################################### ####\nif not assigned(B) or not type(B,matrix) then \n error \"bilin ear form B has not been assigned yet. It must be defined as the identi ty 3 x 3 matrix.\"\nend if:\nif not linalg[equal](B,linalg[diag](1$3)) then \n error \"the identity 3 x 3 matrix must be assigned to B\" \+ \nend if;\nif nargs <> 2 then \n error \"two arguments needed of typ e algebraic and quaternion\" \nend if; \nq2:=clisort(map(combine,a2,tr ig)); \nq2inv:=clisort(map(combine,qinv(q2),trig)); \nreturn clicollec t(clisort(map(combine,cmulQ(q2,a1,q2inv),trig))) \nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 73. Procedure " }{TEXT 392 9 "ispro duct" }{TEXT -1 238 " can determine whether the given Clifford polynom ial, e.g. p := Id + 4*e1we2 + e3we4, is a product of 1-vectors in the given Clifford algebra. It can be used with two options `all`, or `an y`, or can be used without any option as follows:" }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 12 "Typical use:" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 48 "ispro duct(p); answers true or false;" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 151 "isproduct(p, 'any'); \+ answers true or false, and gives a list of n vectors [v1, v2, ..., vn] such that the Clifford product v1 &c v2 &c ... &c vn = p;" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 161 "isproduct( p, 'all'); answers true or false, and gives a list of general vecto rs [v1, v2, ..., vn] such that the Clifford product v1 &c v2 &c ... &c vn = p;\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4894 "isproduct:=proc( p::\{cliscalar,clibasmon,climon,clipolynom\},\n s::\{st ring,symbol\}) \nlocal M,maxg,T,co,vv,x,cf,pnew,p1,L,v,j,S,S2,i,v1v2,e xpr,t,sys,\nvars,sol,ventries,flag,flagB,flagtB,param,flagsol,eq,P1,P2 ,die,parvalues;\nglobal _MaxSols,B;\noptions `Copyright (c) 1995-2003 \+ by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndesc ription `Last revised: November 1, 2002`;\n########################### ##################\nif not member(nargs,\{1,2\}) then\n error \"one or two arguments needed of type 'cliscalar', 'clibasmon', 'climon', 'cli polynom', and 'symbol'\"\nend if;\nif nargs=2 and not member(s,\{'all' ,'any'\}) then\n error \"second (optional) argument must be 'all' or 'any'\"\nend if;\nif not type(B,diagmatrix) then \n error \"diagona l matrix must be assigned to B\" end if;\nmaxg:=maxgrade(p);\n######## #############################################\n#An element of grade 0 \+ is always factorable in Cl(B):\n###################################### ###############\nif maxg=0 then \n if nargs=1 then return true end i f;\n flag:=false:\n for i from 1 to linalg[coldim](B) while not fl ag do\n if B[i,i]<>0 then flag:=true;\n return [true,[( scalarpart(p)/B[i,i])*e||i,e||i]] \n end if;\n end do;\nerror \+ \"none of the basis 1-vectors has a square equal to 1 or -1\" \nend if ;\n#####################################################\n#Any 1-vecto r is already factored:\n############################################## #######\nif vectorpart(p,1)-p=0 then \n if nargs=1 then return true \n else return [true,[p]] \n end if;\nend if;\n######## #############################################\n#Any basis monomial is \+ already factored:\n################################################### ##\nflagB:=type(B,diagmatrix):\np1:=factor(reorder(displayid(p))):\nfl agtB:=evalb(type(p1,\{clibasmon,climon\}) and flagB):\nif flagtB then \+ \n S:=op(Clifford:-extract(p1,'integers'));\n if nargs=1 the n return true else \n v:=[e||S];\n if not remove(hastype,p1, clibasmon)=NULL\n then v[1]:=remove(hastype,p1,clibasmon)*v[1 ] \n end if;\n return [true,v] \n end if; \nend if;\n #########################################################\n#If p does \+ not belong to any of the special cases above,\n#find common indices to all monomial terms in p, if any,\n#and then simplify p by factoring o ut the common factors:\n############################################## ###########\nT:=cliterms(p):\nco:=`intersect`(op(map(convert,map(Cliff ord:-extract,T,'integers'),set)));\nx:='x':\nif nops(co)<>0 then\n c o:=sort(convert(co,list));\n vv:=[seq(cat(e,x),x=co)];\n cf:=cmul( op(vv));\n pnew:=cmul(p,cf,cf,cf);\n if nargs=1 then M:=procname(p new) \n elif\n nargs=2 then L:=procname(pnew,s);\n \+ M:=[L[1],[op(L[2]),op(vv)]]; \n end if;\n return M\nend i f; \n#####################################################\n#This is \+ the most general case when no common indices\n#in monomial terms are f ound:\n#####################################################\nS2:=map( Clifford:-extract,cliterms(p),'integers');\nS:=\{op(map(op,S2))\}; \nv :=table([]):\nfor j from 1 to maxg do\nv[j]:=0:\nfor i in S do v[j]:=v [j]+cat(x,j,i)*cat(e,i) \nend do;\nend do;\nv1v2:=cmul(seq(v[j],j=1..m axg));\nexpr:=clicollect(simplify(reorder(p-v1v2))):\nt:=cliterms(expr );sys:=\{\}:\nfor i from 1 to nops(t) do sys:=\{op(sys),coeff(expr,op( i,t))=0\} end do:\nvars:=sort([op(indets(sys))],lexorder); \n_MaxSols: =1: #setting maximum number of solutions to one\nvars:=convert(vars, set):\nsol:=[solve(sys,vars)]:\nif nops(sol)=0 then return false end i f;\nventries:=[seq(v[j],j=1..maxg)];\n################################ #######################\n#Finally, we need to return result in appropr iate form.\n#By now, if p were not factorable, 'false' should have\n#b een returned:\n####################################################### \nif nargs=1 then return true end if; \nif nargs=2 and s='all' then re turn [true,subs(sol[1],ventries)] end if; \n########################## ###############################\n#If the second parameter is 'any', as sign random values\n#to the parameters showing up in the answer. These random\n#values will change with each execution of the program:\n#### #####################################################\nif nargs=2 and \+ s='any' then \nparam:=proc(a1::\{`=`\}) \n if lhs(a1)=rhs(a1) or rhs (a1)=0 then true else false end if;\nend proc:\nflagsol:=false:\nfor i from 1 to 2 while not flagsol do\nS2:=\{\}:P1:=\{\}:P2:=\{\}:\nS2:=\{ op(sol[1])\};\nparvalues:=[1,-1,1/2,-1/2,1/3,-1/3];\ndie := rand(1..6) :\nfor eq in select(param,S2) do \n if rhs(eq)=0 then P1:=P1 union \{ eq\}\n else P1:=P1 union \{lhs(eq)=parvalues[die()]\};\n end if;\nend do;\nP2:=remove(param,S2):\nL:=map(op,subs(P2,ventries) );\nif not member(0,subs(P1,map(denom,L))) then flagsol:=true end if; \nend do:\nif flagsol then return [true,subs(P1,subs(P2,ventries))]\n \+ else return [true,subs(sol[1],ventries)]\nend if;\nend if;\n end proc:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 74. Procedure " }{TEXT 393 14 "isVahlenmatrix" }{TEXT -1 258 " determines if the given 2 x 2 matrix is Vahlen matrix as defined in P. Lounesto, \"Clical and counter-examples\", in eds. R. Ablamowi cz, P. Lounesto, and J. Parra, `Clifford algebras with symbolic and nu meric computations`, Birkhauser, Boston, 1996, page 19." }}{PARA 258 " " 0 "" {TEXT -1 349 "\nVahlen matrix V is a 2 x 2 matrix with entries \+ in a Clifford algebra Cl(p, q) such that if \n\n V := matrix(2, 2, [a, b, c, d]); \+ \+ \nand a,b,c,d are elements in Cl(p, q), then the following conditions must be met:" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 38 "1. a, b, c, d are products of vectors;" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 74 "2. th e pseudodeterminant of V is +1 or -1 (or Id or -Id in the algebra); " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 98 "3 . a &c reversion(b), reversion(b) &c d, d &c reversion(c), and reversi on(c) &c a are all vectors." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 31 "Typical use: isVahlenmatrix(V);" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 120 "V := matrix(2, 2, [Id - e1we4, -e1 + e4, e1 + e4, Id + e1we4]) (this exam ple of Vahlen matrix is due to Johannes Maks)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1212 "isVahlenmatrix:=p roc(cm::\{matrix,climatrix\}) \nlocal expr1,expr2,a,b,c,d,m; global B; \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried F auser. All rights reserved.`;\ndescription `Last revised: November 1, \+ 2002`;\n#############################################\nif not type(B,m atrix) then \n error \"square matrix must be assigned to B\" \nend i f;\nif linalg[rowdim](cm)<>2 or linalg[coldim](cm)<>2 then \n error \+ \"to calculate pseudodeterminant matrix must be 2 x 2\" \nend if;\nm:= displayid(cm):\na:=simplify(m[1,1]):b:=simplify(m[1,2]):\nc:=simplify( m[2,1]):d:=simplify(m[2,2]):\n######################################## ##\n### Condition 1:\n##########################################\nif a <>0 then if not isproduct(a) then return false fi end if;\nif b<>0 the n if not isproduct(b) then return false fi end if;\nif c<>0 then if no t isproduct(c) then return false fi end if;\nif d<>0 then if not ispro duct(d) then return false fi end if;\n################################ ##########\n### Condition 2:\n######################################## ##\nif not member(pseudodet(m),\{1,-1,Id,-Id\}) then return false end \+ if;\n##########################################\n### Condition 3:\n### #######################################\n" }{TEXT 359 0 "" }{MPLTEXT 1 0 585 "expr1:=simplify(cmul(a,reversion(b)));\nexpr2:=simplify(vecto rpart(expr1,1));\nif not evalb(simplify(expr1-expr2)=0) then return fa lse end if;\nexpr1:=simplify(cmul(reversion(b),d));\nexpr2:=simplify(v ectorpart(expr1,1));\nif not evalb(simplify(expr1-expr2)=0) then retur n false end if;\nexpr1:=simplify(cmul(d,reversion(c)));\nexpr2:=simpli fy(vectorpart(expr1,1));\nif not evalb(simplify(expr1-expr2)=0) then r eturn false end if;\nexpr1:=simplify(cmul(reversion(c),a));\nexpr2:=si mplify(vectorpart(expr1,1));\nif not evalb(simplify(expr1-expr2)=0) th en return false end if;\nreturn true\nend proc:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. \+ 75. Procedure " }{TEXT 394 10 "climinpoly" }{TEXT -1 407 " finds the m inimal polynomial of any Clifford polynomial p. It may be used with an optional second argument 'powers' in which case it returns a list of \+ consecutive powers p^k of p which are linearly independent, k=1..(n-1) where n = degree of the minimal polynomial of p. If the second option al argument is 'horner' then polynomial is returned in 'horner' form. \+ This procedure can accept now optional index." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 83 "Typical use: climinpoly (p);climinpoly[K](p);\n climinpoly(p,'s');" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1406 "c liminpoly:=proc(p::\{cliscalar,clibasmon,climon,clipolynom\})\nlocal d p,L,flag,pp,expr,a,k,eq,sys,vars,sol,poly,lname;\noptions `Copyright ( c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights rese rved.`;\ndescription `Last revised: November 1, 2002`;\n############## ###############################\nif type(op(procname),procedure) then \n lname:=`B`;\n else\n lname:=op(procname);\nend if;\ndp:=disp layid(p):\nif maxgrade(dp)=0 then L:=[Id] else L:=[Id,dp] end if;\nfla g:=false:k:='k':a:='a':\nwhile not flag do\npp:=cmul[lname](L[nops(L)] ,dp):\nexpr:=expand(add(a[k]*L[k],k=1..nops(L)));\neq:=clicollect(pp-e xpr);\nsys:=\{coeffs(eq,cliterms(eq))\};\nvars:=\{seq(a[k],k=1..nops(L ))\};\nsol:=solve(sys,vars):\nif sol<>NULL then flag:=true else L:=[op (L),pp] end if;\nend do;\npoly:='x'^nops(L)-add(a[k]*'x'^(k-1),k=1..no ps(L));\npoly:=sort(subs(sol,poly)); \nif nargs=1 then return poly\nel if nargs=2 then\n if args[2]='powers' then return [poly,L]\n \+ elif args[2]='horner' then return convert(poly,horner)\n else e rror \"second (optional) argument must be 'powers' or 'horner' \"\n \+ end if;\nelif nargs=3 then\n if member(args[2],\{'powers','horne r'\}) and\n member(args[3],\{'powers','horner'\}) then\n \+ return ([convert(poly,horner),L])\n else error \"wrong argu ments\"\n end if;\nelse error \"wrong number of arguments: one, tw o, or three arguments are needed only\"\nend if;\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 18 "No. 76. Procedure " }{TEXT 395 15 "subs _climinpoly" }{TEXT -1 283 " substitutes any Clifford polynomial p int o any polynomial pol in one variable. It may be used with an optional \+ third argument in which case it returns unevaluated polynomial pol in \+ 'horner' form. For example, one can use this procedure to verify that \+ the given Clifford polynomial p" }{TEXT 356 1 " " }{TEXT -1 37 "satisf ies its own minimal polynomial." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 197 "Typical use: subs_climinpoly(p,pol);\n subs_climinpoly(p,pol, 'horner');\n \+ subs_climinpoly(p,pol, \"horner\");\n subs_cli minpoly(p,pol, horner);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1332 "subs _clipolynom:=proc(clinumber::\{symbol,cliscalar,clibasmon,climon,clipo lynom\},\n minpoly::polynom,o::\{symbol,string \}) \nlocal ph,d,k,r,q,h,expr,s,var,varx,dclinumber;\noptions `Copyrig ht (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights \+ reserved.`;\ndescription `Last revised: November 1, 2002`;\n########## ###################################\nph:=convert(minpoly,horner);\nvar :=op(remove(type,indets(ph),indexed));\nif not type(eval(clinumber),\{ clibasmon,climon,clipolynom\}) \n then return subs(var=clinumber,ph) \nend if;\nif nops(\{var\})<>1 then varx:=op(select((member,\{var\}, \{x,y,z\}))) else varx:=var end if;\nif nops(\{varx\})<>1 then \n er ror \"expecting only one of x, y, or z as a variable in %1 but found % 2\",minpoly,varx \nend if:\nd:=degree(ph,varx);\nh:=ph:\nfor k from 1 \+ to d do\n r[k]:=rem(h,x,x,'s');\n q[k]:=convert(s,horner);\nh:=q [k];\nend do:\ndclinumber:=displayid(clinumber):\nexpr:=clicollect(r[d ]*Id+q[d]*dclinumber);\nfor k from d-1 to 1 by -1 do\n expr:=r[k]*I d+'cmul'(expr,dclinumber);\nend do:\nif nargs=2 then return simplify(e val(expr))\nelif nargs=3 then \n if args[3]='horner' then return exp r \n else \n error \"third (optional) argument, when used, m ust be 'horner', but received %1 instead\",args[3]\n end if;\nelse e rror \"wrong number of arguments\"\nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 19 "No. 77. Procedure " }{TEXT 396 4 "sexp" }{TEXT -1 427 " finds a power series expansion of a Clifford polynomial p up \+ to and including order n modulo the minimal polynomial of p. It is rec ommended that this procedure be used when n > d, where d is the degree of the minimal polynomial of p. Otherwise, use 'cexp' or 'cexpQ' inst ead. The reason is that 'sexp' is faster than 'cexp' when n > d, but i s is slower when n <= d. This procedure can use an optional argument s uch as K or -K." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 " " {TEXT -1 49 "Typical use: sexp(p,4); sexp(p,4,K);sexp(p,4,-K);" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1525 "s exp:=proc(p::\{numeric,cliscalar,clibasmon,climon,clipolynom\},n::nonn egint) \nlocal k,pp,pol,powrs,co,te,nte,lname,coB,nameB;\noptions `Cop yright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rig hts reserved.`;\ndescription `Last revised: November 1, 2002`;\n###### #######################################\nif nargs=2 then\n coB:=1: \n nameB:=`B`: \n lname:=`B`: \nelif nargs=3 then\n if type(a rgs[3],\{name,symbol,matrix,array\}) then\n coB:=1:\n name B:=args[3];\n lname:=args[3];\n elif type(args[3],`&*`(numeri c,\{name,symbol,matrix,array\})) then\n coB:=op(select(type,\{op (args[3])\},numeric));\n nameB:=op(remove(type,\{op(args[3])\},n umeric));\n lname:=args[3]:\n else \n error \"wrong typ e of third argument in sexp. See ?sexp for more help.\" \n end if;\n else\n error \"two or three arguments expected in sexp. See ?sexp fo r more help.\"\nend if;\n#####################################\nif n=0 then \n if type(p,\{numeric,'cliscalar'\}) then return 1 else retur n Id fi\nend if;\nk:='k':\nif type(p,\{numeric,cliscalar\}) then retur n add(p^k/k!,k=0..n) end if;\nif evalb(vectorpart(p,0)=p) then pp:=sca larpart(p);\n return (add(pp^k/k!,k=0..n)*Id) \nend if;\npol:=climin poly[lname](p,'powers');\npowrs:=pol[2]:\n### readlib(powmod);\nk:='k' :te:='te':\npol:=collect(add(powmod('x',k,pol[1],'x')/k!,k=0..n),'x'); \nco:=[coeffs(pol,'x','te')]:\nte:=[te]:\nnte:=nops(te):\nfor k from 1 to nte do \n te[k]:=powrs[degree(te[k],'x')+1] \nend do;\nreturn c licollect(add(co[k]*te[k],k=1..nte))\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 358 18 "No. 78. Procedure " }{TEXT 397 8 "all_sigs" }{TEXT 398 383 " gives signatures of all real, real simple, real semi-simple, com plex, quaternionic, quaternionic simple, and quaternionic semi_simple \+ Clifford algebras up to and including the dimension specified as the f irst parameter. Second parameter, when used, must be 'real', 'complex' , or 'quat', while the third parameter must be 'simple' or 'semisimple '.\n\nUse: all_sigs(9,'real','simple');\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2806 "all_sigs:=proc(r) \nlocal s1,s2,mi,ma,P,Q,p,q,pq,r_ pq,c_pq,q_pq,x,\nsimple_r_pq,simple_q_pq,semisimple_r_pq,semisimple_q_ pq;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfrie d Fauser. All rights reserved.`;\ndescription `Last revised: November \+ 1, 2002`;\n#############################################\nif nargs=2 t hen \n s1:=args[2]:\nelif nargs=3 then \n s1:=args[2]:\n s2:=arg s[3]:\nend if; \nif not type(r,range) or \n not type(s1, \{string,symbol\}) or\n not type(s2,\{string,symbol\})\nthen\nWARNIN G(`try first argument as range, e.g., 1..9, second argument as 'real', 'complex', or 'quat', and third arguments as 'simple' or 'semisimple' instead of:`);\nreturn 'procname(args)'\nend if;\n################### #####\nmi:=min($r):ma:=max($r):\nP:=\{$0..9\}:Q:=\{$0..9\}:\npq:=[]:\n for p in P do\nfor q in Q do \n if p+q<=ma and p+q>=mi then pq:=[op (pq),[p,q]] end if: \nend do:\nend do:\nr_pq:=[]:c_pq:=[]:q_pq:=[]:\nf or x in pq do\np:=x[1]:q:=x[2]:\nif member((p - q) mod 8,\{0,1,2\}) th en r_pq:=[op(r_pq),x] end if;\nif member((p - q) mod 4,\{3\}) then c_p q:=[op(c_pq),x] end if;\nif member((p - q) mod 8,\{4,5,6\}) then q_pq: =[op(q_pq),x] end if;\nend do:\n##################################\nif nargs=1 then return pq end if;\n##################################\ni f nargs=2 then\n if s1='real' then return r_pq elif\n s1='compl ex' then return c_pq elif\n s1='quat' then return q_pq else\n \+ error \"second input string must be 'real', 'complex' or 'quat' but \+ received %1\",args[2] \n end if:\nend if: \n####################### ###########\nif s1='real' then\n simple_r_pq:=[]:semisimple_r_pq: =[]:\n for x in r_pq do \n if member(x[1]-x[2] mod 8,\{1 \}) then \n semisimple_r_pq:=[op(semisimple_r_pq),x] \n \+ else \n simple_r_pq:=[op(simple_r_pq),x]\n \+ end if;\n end do:\n if s2='simple' then return simple_r_pq \+ elif\n s2='semisimple' then return semisimple_r_pq else\n \+ error \"third argument must be 'simple' or 'semisimple' but receiv ed %1\",args[3]\n fi\nend if;\n################################## \nif s1='complex' then\n if s2='simple' then return c_pq elif\n \+ s2='semisimple' then return [] \n end if:\nend if;\n############### ###################\nif s1='quat' then\n simple_q_pq:=[]:semisimp le_q_pq:=[]:\n for x in q_pq do \n if member(x[1]-x[2] m od 8,\{5\}) then \n semisimple_q_pq:=[op(semisimple_q_pq), x] \n else \n simple_q_pq:=[op(simple_q_pq),x]\n \+ end if;\n end do:\n if s2='simple' then return simp le_q_pq elif\n s2='semisimple' then return semisimple_q_pq els e\n error \"third argument must be 'simple' or 'semisimple' bu t received %1 instead\",args[3]\n end if:\nend if;\nerror \"wrong number of arguments. See ?all_sigs for more help.\"\nend proc:\n" }} {PARA 0 "" 0 "" {TEXT 357 18 "No. 79. Procedure " }{TEXT 399 9 "adfmat rix" }{TEXT 400 116 " accomplishes addition of two matrices of type 'd fmatrix', that is, matrices whose entries belong to a double field\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "adfmatrix:=proc(M1::dfmatrix, M2 ::dfmatrix) local L1, L2;\noptions `Copyright (c) 1995-2003 by Rafal A blamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `L ast revised: November 1, 2002`;\n##################################### ########\n L1:=ddfmatrix(M1);\n L2:=ddfmatrix(M2);\n return c dfmatrix(evalm(L1[1] + L2[1]), evalm(L1[2] + L2[2]))\nend proc:\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 361 22 "No. 80/81: Procedures " } {TEXT 403 9 "beta_plus" }{TEXT 404 5 " and " }{TEXT 401 10 "beta_minus " }{TEXT 402 374 " [originally procedure 'beta' from the package 'doub le'] are now part of \"CLIFFORD\". They give two scalar bilinear forms in the spinor ideal S of Cl(Q).\n\nUsage: beta_plus(psi,phi,f); beta_ plus(psi,phi,f),'s'); beta_minus(psi,phi,f); beta_minus(psi,phi,f),'s' ); where psi and phi are spinors, f is an idempotent, and 's' is an op tional argument that will store 'purescalar'.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2004 "beta_plus:= proc(psi,phi,f) \nlocal locf,locdata,y, m,flag,mons,uu,eq,lambda,sys,sol,Kbas,v,i,vars,flagf;\nglobal B,_prole vel;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfri ed Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#############################################\nif not _pro level then\n if not type(psi,\{cliscalar,clibasmon,climon,clipolynom\} ) then \n error \"first argument must be of type 'cliscalar', 'clibas mon', 'climon', or 'clipolynom'\" \n end if;\n if not type(phi,\{clisc alar,clibasmon,climon,clipolynom\}) then \n error \"second argument m ust be of type 'cliscalar', 'clibasmon', 'climon', or 'clipolynom'\" \+ \n end if;\nend if;\n###Load in pre-computed data and check if idempot ents are the same\nlocdata:=clidata(B):\nlocf:=eval(locdata[4]);\nKbas :=locdata[6];\nif nops(Kbas)>1 then\n flagf:=evalb(f=eval(locf) or f =gradeinv(locf) or \n f=-gradeinv(locf) or f=-eval(locf ));\n if not flagf then\nerror \"when K = C or K = H, primitive idem potent f = plus/minus clidata(B)[4] or its grade involution\"\n end \+ if;\nend if;\n###\n y:=cmul(reversion(expand(psi)),expand(phi));\n \+ if y = 0 then return 0 end if;\n m := 'm';i:='i':\n flag := f alse;\n mons := cbasis(linalg[coldim](B));\n v := array(1 .. nop s(Kbas),[]);\n lambda := add(v[i]*Kbas[i],i=1..nops(Kbas));\n fo r m in mons while not flag do\n uu := m;\n eq := clicoll ect(cmul(m,y) - expand(cmul(lambda,f)));\n sys := \{coeffs(eq, \+ cliterms(eq))\};\n vars := \{seq(v[i], i = 1 .. nops(Kbas))\}; \n sol := solve(sys, vars);\n flag := not evalb(sol = NU LL)\n end do:\n if nargs = 4 then\n if not type(args[4],na me) or type(args[4],protected) then \n error \"fourth optiona l argument, when used, must be of type unprotected name\"\n else assign(args[4],uu) \n end if;\n end if;\n lambda:=subs(so l,lambda):\n if vectorpart(lambda,0)=lambda then return (scalarpart (lambda)) \n else return lambda\n end if;\nend proc:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 2036 "beta_minus:= proc(psi,phi,f) \nlo cal locf,locdata,y,m,flag,mons,uu,eq,lambda,sys,sol,Kbas,v,i,vars,flag f;\nglobal B,_prolevel;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\nif not _prolevel then\n if not type(psi,\{cliscalar,clibasmon, climon,clipolynom\}) then \n error \"first argument must be of type ' cliscalar', 'clibasmon', 'climon', or 'clipolynom'\" \n end if;\n if n ot type(phi,\{cliscalar,clibasmon,climon,clipolynom\}) then \n error \+ \"second argument must be of type 'cliscalar', 'clibasmon', 'climon', \+ or 'clipolynom'\" \n end if;\nend if;\n###Load in pre-computed data an d check if idempotents are the same\nlocdata := clidata(B):\nlocf := e val(locdata[4]);\nKbas := locdata[6];\nif nops(Kbas)>1 then\n flagf: =evalb(f=eval(locf) or f=gradeinv(locf) or \n f=-gradei nv(locf) or f=-eval(locf));\n if not flagf then\n error \"when \+ K = C or K = H, primitive idempotent f = plus/minus clidata(B)[4] or i ts grade involution\"\n end if;\nend if;\n###\n y := cmul(conjuga tion(expand(psi)),expand(phi));\n if y = 0 then return 0 end if;\n \+ m := 'm';i:='i':\n flag := false;\n mons := cbasis(linalg[col dim](B));\n v := array(1 .. nops(Kbas),[]);\n lambda := add(v[i] *Kbas[i],i=1..nops(Kbas));\n for m in mons while not flag do\n \+ uu := m;\n eq := clicollect(cmul(m,y) - expand(cmul(lambda,f )));\n sys := \{coeffs(eq, cliterms(eq))\};\n vars := \{ seq(v[i], i = 1 .. nops(Kbas))\};\n sol := solve(sys, vars);\n \+ flag := not evalb(sol = NULL)\n end do:\n if nargs = 4 th en\n if not type(args[4],name) or type(args[4],protected) then \+ \n error \"fourth optional argument, when used, must be of ty pe unprotected name\"\n else assign(args[4],uu) \n en d if;\n end if;\n lambda:=subs(sol,lambda):\n if vectorpart(l ambda,0)=lambda then \n return scalarpart(lambda) \n else \n \+ return lambda\n end if;\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 362 18 "No. 82. Procedure " }{TEXT 405 9 "cdfmatrix" }{TEXT 406 100 " creates a matrix over double field from a list of two matrices o r from a serquence of to matrices.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 921 "cdfmatrix:=proc() local l1,l2,L,i,j,m,n,m1,m2,MN;\noptions `Cop yright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rig hts reserved.`;\ndescription `Last revised: November 1, 2002`;\n###### #######################################\nif nargs=1 and type(args[1],l ist(\{matrix,array\})) \n then m1,m2:= evalm(args[1][1]) ,evalm(args[1][2]);\nelif nargs=2 and type(args[1],\{matrix,array\}) a nd type(args[2],\{matrix,array\}) \n then m1,m2:= evalm( args[1]),evalm(args[2])\nelse error \"wrong number or types of argumen ts. See ?cdfmatrix for help.\" \nend if;\n l1:=convert(m1,mlist);\n l2:=convert(m2,mlist);\n L:=[];\n for i to nops(l1) do \n \+ L:=[op(L),[l1[i],l2[i]]] \n end do:\n m:=linalg[rowdim](m 1);\n n:=linalg[rowdim](m1);\n MN:=linalg[matrix](m,n,[]);\n \+ for i to m do \n for j to n do MN[i,j]:=L[(i-1)*n+j] \n end \+ do:\n end do:\n return evalm(MN)\nend proc:\n" }}{PARA 0 "" 0 " " {TEXT 363 18 "No. 83. Procedure " }{TEXT 407 9 "ddfmatrix" }{TEXT 408 64 " decomposes a matrix over double field into a pair of matrices .\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "ddfmatrix:=proc(M::dfmatrix ) local m,n,i,L1,L2,L;\noptions `Copyright (c) 1995-2003 by Rafal Abla mowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n######################################## #####\n m:=linalg[rowdim](M);\n n:=linalg[coldim](M);\n L:=co nvert(M,mlist);\n L1:=[seq(L[i][1],i=1..nops(L))];\n L2:=[seq(L[ i][2],i=1..nops(L))];\n return [linalg[matrix](m,n,L1),linalg[matri x](m,n,L2)]\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 364 18 "No. 84. Procedure " }{TEXT 409 11 "diagonalize" }{TEXT 410 42 " tr ies to diagonalize a symmetric matrix.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 784 "diagonalize:=proc(m::symmatrix) local locB,flag,i,j, L,v,S,Bdiag;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: \+ November 1, 2002`;\n#############################################\nif \+ linalg[coldim](m)<>linalg[rowdim](m) then\n error \"expected a squar e matrix as input\" \nend if;\nif type(m,diagmatrix) then \n return \+ evalm(m) \nend if; \nL:=[linalg[eigenvects](m)];\nflag:=true:\nfor i f rom 1 to nops(L) while flag=true do\n if L[i][2]>nops(L[i][3]) then flag:=false end if: \nend do: \nif not flag then \n error \"since m atrix entered does not have a complete set of linearly independent eig envectors, it is not diagonalizable\" \nend if;\nreturn linalg[diag](s eq(seq(L[i][1],j=1..L[i][2]),i=1..nops(L)))\nend proc:\n" }}{PARA 0 " " 0 "" {TEXT 365 6 "No. 85" }{TEXT -1 1 "." }{TEXT 366 11 " Procedure \+ " }{TEXT 411 9 "mdfmatrix" }{TEXT 412 46 " multiplies two matrices ove r a double field.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "mdfmatrix:= proc(M1::dfmatrix,M2::dfmatrix) local L1, L2;\noptions `Copyright (c) \+ 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserve d.`;\ndescription `Last revised: November 1, 2002`;\n################# ############################\n L1:=ddfmatrix(M1);\n L2:=ddfmatri x(M2);\n return cdfmatrix((L1[1]) &cm (L2[1]),(L1[2]) &cm (L2[2])) \nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 370 18 "No. 86. Procedure " } {TEXT 413 7 "cocycle" }{TEXT 414 901 " finds an element x in the given Clifford algebra such that cmul(x,a1) = cmul(a2,x) where a1 and a2 ar e the first two arguments of type 'clibasmon', 'climon', or 'clipolyno m'. \n\nIf only two arguments are passed to the procedure, element x b elongs to the Clifford algebra over the lowest dimension dim = max(max index(a1),maxindex(a2)). \n\nIf three arguments are used with the thi rd argument being a list of elements of type 'clibasmon', 'climon', or 'clipolynom', then x belongs to the set generated by a1, a2, and the \+ elements in the third list a3. \n\nIf the fourth argument a4 is used, \+ then the third argument is expected to be a list of elements of type ' clibasmon', in which case the procedure searches for x from that list. \n\nTypical use:\n\ncocycle(1+2*e1-e1we3,3*e2+e2we4);\ncocycle(1+2*e1- e1we3,3*e2+e2we4, [e1we2+Id,e1we2we3,e4]);\ncocycle(1+2*e1-e1we3,3*e2+ e2we4, [e1we2,e1we2we3,e4],'clibasmon');\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1471 "cocycle:=proc(a1::\{clibasmon,climon,clipolynom\}, \n a2::\{clibasmon,climon,clipolynom\},\n a3 ::list(\{clibasmon,climon,clipolynom\}),\n a4::symbol) \n local g,v,n,llist,i,d,S,x,y,xy,sys,vars,sol,llist1,llist2,llist3;\nopt ions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser . All rights reserved.`;\ndescription `Last revised: November 1, 2002` ;\n#############################################\n#if a1=a2 then retur n [Id] end if;\nif nargs=4 and member(args[4],\{clibasmon,clibasmon\}) then\n llist:=a3:\n S:=[]:\n for i from 1 to nops(llist) do\n x :=cmul(llist[i],a1): y:=cmul(a2,llist[i]):\n if x-y =0 then\n \+ if x <> 0 and y <> 0 then\n if cmul(llist[i],llist[i]) <> 0 the n\n S:=[op(S),llist[i] ]:\n end if: \n end if: \n end if:\n end do:\nreturn S\nend if;\nif nargs=3 then\n llist1:= `union`(op(map(cliterms,remove(member,\{seq(op(\{cmul(a1,g),cmul(g,a1) \}),g=a3)\},\{0\})))):\n llist2:=`union`(op(map(cliterms,remove(membe r,\{seq(op(\{cmul(a2,g),cmul(g,a2)\}),g=a3)\},\{0\})))):\n llist3:=ma p(op@cliterms,convert(a3,set)); \n llist:=convert(`union`(llist1,lli st2,llist3),list):\n llist:=sort([op(llist),op(cliterms(op(a3)))],byg rade):\nelse\n llist:=cbasis(max(maxindex(a1),maxindex(a2))):\nend if ;\nn:=nops(llist):\ng:=add(_X[i]*llist[i],i=1..n);\nvars:=\{seq(_X[i], i=1..n)\}:\nxy:=clicollect(cmul(g,a1)-cmul(a2,g)):\nsys:=\{coeffs(xy,l list)\};\nsys:=map(normal,sys);\nsol:=solve(sys,vars);\nreturn subs(so l,g)\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 374 18 "No. 87. Procedure \+ " }{TEXT 415 8 "clisolve" }{TEXT 416 103 " for solving equations in a \+ Clifford algebra Cl(B). \n\nTypical use:\n\nclisolve(eq,pp);\nclisolve (eq,set);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 592 "clisolve:=proc(eq:: \{clibasmon,climon,clipolynom\},indet::\{list,algebraic\}) \nlocal i,T ,vars,sol,sys;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz a nd Bertfried Fauser. All rights reserved.`;\ndescription `Last revised : November 1, 2002`;\n#############################################\ni f type(indet,list) then\n vars:=convert(indet,set)\nelse\n vars:=sel ect(type,indets(indet),indexed)\nend if;\nT:=cliterms(eq);\nsys:=\{coe ffs(clicollect(simplify(eq)),T)\};\nsol:=[solve(sys,vars)];\nif type(i ndet,list) then\n return sol\nelse\n return [seq(subs(sol[i],indet), i=1..nops(sol))]\nend if;\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 23 "No. 88. This procedure " }{TEXT 372 13 "CLIFFORD_ENV " }{TEXT 417 135 " lists all environnmental variables defined in Clifford, Cliplus, GTP, Octonion, and Bigebra packages, when these packages are loaded. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6578 "CLIFFORD_ENV:=proc() global _warnings_flag:\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revis ed: November 1, 2002`;\n############################################# \nif not assigned(Clifford) then \n lprint(`>>> Package Clifford has not been loaded yet. Type 'with(Clifford)' at the Maple prompt to loa d it first. <<<`)\nelse\n print('``');###Print blank line\n lprint(`>> > Global variables defined in Clifford:-setup are now available and ha ve these values: <<<`);\nlprint(`************* Start *************`); \+ \n########################\nlprint('dim_V'=dim_V);\n #(dimension o f the carrier space for Cl(V,B))\nif not member(dim_V,\{1,2,3,4,5,6,7, 8,9\}) and _warnings_flag then\n lprint(`Warning, value of dim_V is \+ expected to be a positive integer between 1 and 9, inclusive.`);\n p rint('``');###Print blank line\nend if;\n########################\nlpr int('_default_Clifford_product'=_default_Clifford_product);\n #(c ontrols whether cmulRS or cmulNUM is used in Clifford product 'cmul') \n#lprint(`Possible values are: 'cmulRS','cmulNUM','cmulgen','cmul_use r_defined'.`);\nif not member(_default_Clifford_product,\{'cmulRS','cm ulNUM','cmulgen','cmul_user_defined'\}) \n and _warnings_flag then\n lprint(`****** SERIOUS WARNING ******`); \n lprint(`>>> Value of \+ _default_Clifford_product was expected to be 'cmulRS', 'cmulNUM', 'cmu lgen', or 'cmul_user_defined'. <<<`);\n lprint(`******************** *********`);\nend if;\n########################\nlprint('_prolevel'=_p rolevel);\n #(controls whether or not parsing is done)\nif not me mber(_prolevel,\{true,false\}) and _warnings_flag then\n lprint(`War ning, value of _prolevel is expected to be true or false.`);\n print ('``');###Print blank line\nend if;\n########################\nlprint( '_shortcut_in_minimalideal'=_shortcut_in_minimalideal);\n #(contr ols flow in procedure 'minimalideal')\nif not member(_shortcut_in_mini malideal,\{true,false\}) and _warnings_flag then\n lprint(`Warning, \+ value of _shortcut_in_minimalideal is expected to be true or false.`); \n print('``');###Print blank line\nend if;\n####################### #\nlprint('_shortcut_in_Kfield'=_shortcut_in_Kfield);\n #(control s flow in procedure 'Kfield')\nif not member(_shortcut_in_Kfield,\{tru e,false\}) and _warnings_flag then\n lprint(`Warning, value of _shor tcut_in_Kfield is expected to be true or false.`);\n print('``');### Print blank line\nend if;\n########################\nlprint('_shortcut _in_spinorKbasis'=_shortcut_in_spinorKbasis);\n #(controls flow i n procedure 'spinorKbasis')\nif not member(_shortcut_in_spinorKbasis, \{true,false\}) and _warnings_flag then\n lprint(`Warning, value of \+ _shortcut_in_spinorKbasis is expected to be true or false.`);\n prin t('``');###Print blank line\nend if;\n########################\nlprint ('_shortcut_in_spinorKrepr'=_shortcut_in_spinorKrepr);\n #(contro ls flow in procedure 'spinorKrepr')\nif not member(_shortcut_in_spinor Krepr,\{true,false\}) and _warnings_flag then\n lprint(`Warning, val ue of _shortcut_in_spinorKrepr is expected to be true or false.`);\n \+ print('``');###Print blank line\nend if;\n########################\nl print('_warnings_flag'=_warnings_flag);\n #(controls whether some procedures, e.g., 'wedge', give warnings)\nif not member(_warnings_fl ag,\{true,false\}) then\n lprint(`Warning, value of _warnings_flag i s expected to be true or false.`);\n print('``');###Print blank line \nend if;\n########################\nlprint('_scalartypes'=_scalartype s);\n #(defines types considered to be 'scalars' by 'clibilinear' and 'clilinear')\n########################\nlprint('_quatbasis'=_quat basis);\n #(defines default quaternionic basis')\nlprint(`******* ****** End *************`);\nprint('``');###Print blank line \nend if; \n########################\nif assigned(Cliplus) then\n print('``');## #Print blank line\n lprint(`>>> Global variables defined in Cliplus:-s etup are now available and have these values: <<<`);\n lprint(`****** ******* Start *************`);\n lprint('macro(cmul=climul)');\n \+ #('cmul' is now extended by 'climul') \n lprint('macro(cmulQ=climul)') ;\n #('cmulQ' is now extended by 'climul')\n lprint('macro(`&c`=c limul)');\n #('&c' is now extended by 'climul')\n lprint('macro(` &cQ`=climul)');\n #('&cQ' is now extended by 'climul')\n lprint(' macro(reversion=clirev)');\n #('reversion' is now extended by 'cl irev')\n lprint('macro(LC=LCbig)');\n #('LC' is now extended by ' LCbig')\n lprint('macro(RC=RCbig)');\n #('RC' is now extended by \+ 'RCbig')\n if _warnings_flag then \n lprint(`Warning, new definiti ons for type/climon and type/clipolynom now include &C`);\n end if;\n \+ lprint(`************* End *************`);\n print('``');###Print blan k line \nend if;\n\n################################################## ##\n### Executable Bigebra file for Maple 6 is Bigebra6\n############# #######################################\nif assigned(Bigebra6) then\n \+ print('``');###Print blank line\n lprint(`>>> Global variables defined in Bigebra:-init are now available and have these values: <<<`);\n l print(`************* Start *************`);\n lprint('_CLIENV[_SILENT] '=_CLIENV[_SILENT]); #controls messaging upon starting 'Bigebra'\n lp rint('_CLIENV[_QDEF_PREFACTOR]'=_CLIENV[_QDEF_PREFACTOR]); #prefactor in 'switch'\n lprint(`************* End *************`);\n print('``' );###Print blank line\nend if;\n###################################### ####\nif assigned(GTP) then\n print('``');###Print blank line\n lprint (`************* Start *************`);\n lprint(`>>> There are no new \+ global variables or macros in GTP yet. <<<`);\n lprint(`************* \+ End *************`);\n print('``');###Print blank line \nend if;\n#### ######################################\nif assigned(Octonion) then\n p rint('``');###Print blank line\n lprint(`>>> Global variables defined \+ in Octonion:-setup are now available and have these values: <<<`);\n \+ print('``');###Print blank line\n lprint(`************* Start ******** *****`); \n lprint('_octbasis'=_octbasis); #standard octonio n basis as Maple global variable\n lprint('_pureoctbasis'=_pureoctbasi s); #pure octonion basis as Maple global variable\n lprint('_default_ Fano_triples'=_default_Fano_triples); #default list of Fano triples\n \+ lprint('_default_squares'=_default_squares); #default squares of e1,e2 ,e3,e4,e5,e6,e7\n lprint('_default_Clifford_product'=_default_Clifford _product); #selects cmulNUM for numeric B\n lprint(`************* End \+ *************`);\n print('``');###Print blank line \nend if;\n######## ##################################\n\nreturn NULL\nend proc:\n" }} {PARA 0 "" 0 "" {TEXT 373 18 "No. 89. Procedure " }{TEXT 418 13 "makec libasmon" }{TEXT 419 402 " that takes a list and makes Grassmann basis monomials. It is expected, that the list contains positive integers b etween 1 and 9 inclusive, or symbolic indices consisting of one-charac ter strings. If the list is empty, then Id is returned. If any two ele ments in the list are peated, then 0 is returned. This procedure has a remember table.\n\nTypical use: makeclibasmon([]); makeclibasmon([1,7 ,i,j,3]);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 473 "makeclibasmon:=proc (x::list) \nlocal result,N,i;\noptions `Copyright (c) 1995-2003 by Raf al Ablamowicz and Bertfried Fauser. All rights reserved.`,remember;\nd escription `Last revised: November 1, 2002`;\n######################## #####################\n N:=nops(x);\n if N = 0 then return Id end if ;\n if N > nops(convert(x,set)) then return 0 end if;\n result:=ca t(e,x[1]);\n for i from 2 to N do\n result:=cat(result,cat(we, x[i]));\n end do:\nreturn result\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT -1 18 "No. 90. Procedure " }{TEXT 474 12 "rd_clibasmon" }{TEXT -1 405 " generates a random Grassmann basis monomial. It can be used w ithout any arguments in which case default values are used internally, or with 1 or 2 arguments as follows:\n\nNT1 = maximum allowed index v alue (default 9)\nNT2 = maximum allowed grade (default 4)\n\nrd_clibas mon(); then NT1 = 9, NT2 = 4 \nrd_clibasmon(a1); the n NT1 = a1, NT2 = 4\nrd_clibasmon(a1,a2); then NT1 = a1, NT2 = a2\n \n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1403 "rd_clibasmon:=proc() local ind,NT1,NT2,nt1d,nt2d,L;\noptions `Copyright (c) 1995-2003 by Rafal A blamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `L ast revised: November 1, 2002`;\n##################################### ########\n### NT1 = maximum allowed index value (default 9)\n### NT2 = maximum allowed grade (default 4) (must be less than or equal to NT1) \nnt1d,nt2d:=9,4:\n#############################################\nif n args=0 then\n NT1,NT2:=nt1d,rand(0..nt2d)(): #defaults\n L:=[[]]: \nelif nargs=1 then\n if not type(args[1],nonnegint) or not evalb(ar gs[1]<=9 and args[1]>= 0) then\n error \"argument must be non neg ative integer between 0 and 9 giving the maximum monomial index\"\n \+ end if;\n NT1,NT2:=args[1],rand(0..args[1])():\n L:=[[]]: \n eli f nargs>=2 then\n if evalb(not type([args],list(nonnegint)) or \n not evalb(args[1]<=9 and args[1]>=0) or\n not evalb(args[2] <=args[1] and args[2]>=0)) then\nerror \"first argument must be non ne gative integer between 0 and 9 giving maximum monomial index. Second a rgument must be non negative integer between 0 and first argument givi ng maximum possible grade. Other arguments, if present, are ignored.\" \n end if;\n NT1,NT2:=args[1],min(args[1],args[2]):\n L:=[]:\n \+ end if:\n##############\nL:=[op(L),op(combinat[choose](NT1,NT2))];\n ind:=sort(L[rand(1..nops(L))()]);\nreturn Clifford:-makeclibasmon(ind) \nend proc:\n" }}{PARA 0 "" 0 "" {TEXT -1 18 "No. 91. Procedure " } {TEXT 475 9 "rd_climon" }{TEXT -1 560 " generates a random Grassmann m onomial. It can be used without any arguments in which case default va lues are used internally, or with 1, 2, or 3 arguments as follows:\n\n NT1 = maximum allowed index value (default 9)\nNT2 = maximum allowed g rade (default 4)\nNT3 = maximum absolute value of coefficients allowed (default 12)\n\nrd_climon(); then NT1 = 9, NT2 = 4, N T3 = 12 \nrd_climon(a1); then NT1 = a1, NT2 = 4, NT3 = 12 \nrd_climon(a1,a2); then NT1 = a1, NT2 = a2, NT3 = 12\nrd_climo n(a1,a2,a3); then NT1 = a1, NT2 = a2, NT3 = a3\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 1992 "rd_climon:=proc() local rcf,NT1,NT2,NT3,nt1d,nt2d ,nt3d;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertf ried Fauser. All rights reserved.`;\ndescription `Last revised: Novemb er 1, 2002`;\n#############################################\n### NT1 = maximum allowed index value (default 9)\n### NT2 = maximum allowed gr ade (default 4)\n### NT3 = maximum absolute value of coefficient allow ed (default 12)\nnt1d,nt2d,nt3d:=9,4,12:\n############################ #################\nif nargs=0 then\n NT1,NT2,NT3:=nt1d,rand(0..nt2d) (),rand(1..nt3d)(): #defaults\nelif nargs=1 then\n if not type(args[ 1],nonnegint) or not evalb(args[1]<=9 and args[1]>= 0) then\n err or \"argument must be non negative integer between 0 and 9 giving the \+ maximum monomial index\"\n end if;\n NT1,NT2,NT3:=args[1],rand(0.. args[1])(),rand(1..nt3d)(); \nelif nargs=2 then\n if evalb(not type( [args],list(nonnegint)) or \n not evalb(args[1]<=9 and args [1]>=0) or\n not evalb(args[2]<=args[1] and args[2]>=0)) th en\nerror \"first argument must be non negative integer between 0 and \+ 9 giving maximum monomial index. Second argument must be non negative \+ integer between 0 and first argument giving maximum possible grade.\" \n end if;\n NT1,NT2,NT3:=args[1],min(args[1],args[2]),rand(1..nt3 d)():\nelif nargs>=3 then\n if evalb(not type([args],list(nonnegint) ) or \n not evalb(args[1]<=9 and args[1]>=0) or\n \+ not evalb(args[2]<=args[1] and args[2]>=0)) then\nerror \"first argu ment must be non negative integer between 0 and 9 giving maximum monom ial index. Second argument must be non negative integer between 0 and \+ first argument giving maximum possible grade. Third argument must be a positive integer giving max value of coefficient. Other arguments, if present, are ignored.\"\n end if;\n NT1,NT2,NT3:=args[1],min(args [1],args[2]),args[3]:\nend if:\n#############\nrcf:=[rand(-NT3..-1)(), rand(1..NT3)()]:\nrcf:=rcf[rand(1..nops(rcf))()];\nreturn rcf*rd_cliba smon(NT1,NT2)\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT -1 18 "No. 92. Pro cedure " }{TEXT 476 13 "rd_clipolynom" }{TEXT -1 761 " generates a ran dom Grassmann polynomial. It can be used without any arguments in whic h case default values are used internally, or with 1, 2, 3, or 4 argum ents as follows:\n\nNT1 = maximum allowed index value (default 9)\nNT2 = maximum allowed grade (default 4)\nNT3 = maximum absolute value of \+ coefficients allowed (default 12)\nNT4 = maximum number of terms allo wed (default 4)\n\nrd_clipolynom(); then NT1 = 9, NT2 = 4, NT3 = 12, NT4 = 4 \nrd_clipolynom(a1); then NT1 = a1, NT2 = 4, NT3 = 12, NT4 = 4\nrd_clipolynom(a1,a2); \+ then NT1 = a1, NT2 = a2, NT3 = 12, NT4 = 4\nrd_clipolynoma1,a2,a3); then NT1 = a1, NT2 = a2, NT3 = a3, NT4 = 4\nrd_clipolynom(a1,a 2,a3,a4); then NT1 = a1, NT2 = a2, NT3 = a3, NT4 = a4\n" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 3535 "rd_clipolynom:=proc() \nlocal rnt,rcf,NT1, nt1d,NT2,nt2d,NT3,nt3d,NT4,nt4d,L,newL,i,inde,x,m;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights re served.`;\ndescription `Last revised: November 1, 2002`;\n############ #################################\n### NT1 = maximum allowed index val ue (default 9)\n### NT2 = maximum allowed grade (default 4) (must be l eq. than NT1)\n### NT3 = maximum absolute value of coefficient allowed (default 12)\n### NT4 = maximum number of terms allowed (default 5)\n nt1d,nt2d,nt3d,nt4d:=9,4,12,5:\n###################################### ###############\nif nargs=0 then\n NT1,NT2,NT3,NT4:=\n nt1d,rand(0 ..nt2d)(),rand(1..nt3d)(),rand(1..nt4d)(): #defaults\nelif nargs=1 the n\n if not type(args[1],nonnegint) or not evalb(args[1]<=9 and args[ 1]>= 0) then\n error \"argument must be non negative integer betw een 0 and 9 giving the maximum monomial index\"\n end if;\n NT1,NT 2,NT3,NT4:=args[1],rand(0..args[1])(),\n rand(1..nt 3d)(),rand(1..nt4d)():\nelif nargs=2 then\nif evalb(not type([args],li st(nonnegint)) or \n not evalb(args[1]<=9 and args[1]>=0) o r\n not evalb(args[2]<=args[1] and args[2]>=0)) then\nerror \"first argument must be non negative integer between 0 and 9 giving \+ maximum monomial index. Second argument must be non negative integer b etween 0 and first argument giving maximum possible grade.\"\n end i f;\n NT1,NT2,NT3,NT4:=args[1],rand(0..min(args[1],args[2]))(),\n \+ rand(1..nt3d)(),rand(1..nt4d)(): \nelif nargs=3 then \n if evalb(not type([args],list(nonnegint)) or \n not ev alb(args[1]<=9 and args[1]>=0) or\n not evalb(args[2]<=args [1] and args[2]>=0)) then\nerror \"first argument must be non negative integer between 0 and 9 giving maximum monomial index. Second argumen t must be non negative integer between 0 and first argument giving max imum possible grade. Third argument must be a positive integer giving \+ max value of coefficient.\";\n end if;\n NT1,NT2,NT3,NT4:=args[1], rand(0..min(args[1],args[2]))(),\n args[3],rand(1.. nt4d)():\nelif nargs>=4 then\n if evalb(not type([args],list(nonnegi nt)) or \n not evalb(args[1]<=9 and args[1]>=0) or\n \+ not evalb(args[2]<=args[1] and args[2]>=0)) then\nerror \"first a rgument NT1 must be non negative integer between 0 and 9 giving maximu m monomial index. Second argument NT2 must be non negative integer bet ween 0 and NT1 (inclusive) giving maximum possible grade. Third argume nt NT3 must be a positive integer giving max value of coefficient. Fou rth argument NT4 must be a positive integer giving maximum number of t erms (it is expected to be no larger that number of combinations NT1 c hoose NT2. Other arguments, if present, are ignored.\"\n end if:\n \+ NT1,NT2,NT3,NT4:=args[1],min(args[1],args[2]),args[3],args[4]:\nend i f:\n#############\n### NT1 = maximum allowed index value (default 9)\n ### NT2 = maximum allowed grade (default 5)\n### NT3 = maximum absolut e value of coefficient allowed (default 12)\n### NT4 = maximum number \+ of terms allowed (default 4)\n#############\nL:=\{\}:\nfor i from 0 to NT2 do\n L:=\{op(L),op(combinat[choose](NT1,i))\};\nend do:\nm:=mi n(nops(L),NT4):\nL:=convert(L,list):\nnewL:=[[],[[]]]:\nnewL:=newL[ran d(1..2)()]:\nfor i from 1 to m do\n inde:=rand(1..nops(L))();\n \+ x:=L[inde];\n newL:=[op(newL),x];\n L:=subsop(inde=NULL,L);\nend do;\nL:=map(makeclibasmon,newL);\nrcf:=[rand(-NT3..-1)(),rand(1..NT3) ()]:\nreturn add(rcf[rand(1..nops(rcf))()]*L[i],i=1..nops(L))\nend pro c:\n" }}{PARA 258 "" 0 "" {TEXT -1 33 "No. 93. Initialization procedur e " }{TEXT 420 5 "setup" }{TEXT -1 26 " for the Clifford package." }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 94 "This \+ package is loaded automatically into Maple session when command with(C lifford); is given." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1632 "setup:=proc() \nlocal x,y,i,j;\nglobal libname,B ,\n_quatbasis,qi,qj,qk,\n_prolevel,\n_shortcut_in_minimalideal,\n_shor tcut_in_Kfield,\n_shortcut_in_spinorKbasis,\n_shortcut_in_spinorKrepr, \ndim_V,\n_warnings_flag,\n_scalartypes,\n_CLIENV,\n_default_Clifford_ product,\npause,\n###################################\n`convert/dfmatr ix`,`convert/mlist`,`convert/str_to_int`,`type/clibasmon`,\n`type/anti symmatrix`,`type/climatrix`,`type/climon`,`type/clipolynom`,\n`type/cl iprod`,`type/cliscalar`,`type/dfmatrix`,`type/diagmatrix`, `type/evene lement`,`type/fieldelement`,`type/gencomplex`,`type/genquatbasis`,\n`t ype/genquaternion`,`type/idempotent`,`type/nilpotent`,`type/oddelement `,\n`type/primitiveidemp`,`type/purequatbasis`,`type/quaternion`,\n`ty pe/symmatrix`,`type/tensorprod`,\n`&c`,`&cQ`,`&cQm`,`&cm`,`&om`,`&q`,` &qm`,`&rm`,`&w`,`&wm`;\n###################################\noptions ` Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All \+ rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n### ######################################################\n_prolevel:=fal se: #assigning default value\n_shortcut_in_minimalideal :=true: #assigning default value\n_shortcut_in_Kfield:=true: #as signing default value\n_shortcut_in_spinorKbasis:=true: #assigning def ault value\n_shortcut_in_spinorKrepr:=true: #assigning default value \n_warnings_flag:=true: #assigning default value\ndim_V:=9: #default value\n_scalartypes:=\{RootOf,mathfun c,function,numeric,rational,constant,indexed,complex,`^`\}:\n_CLIENV[_ QDEF_PREFACTOR]:=-1:\n_default_Clifford_product:=cmulRS: #default Clif ford product\n" }}{PARA 0 "" 0 "" {TEXT 371 98 "(1) Global variable _s calartypes contains all types declared by the user to be of type 'scal ar'. \n" }}{PARA 258 "" 0 "" {TEXT -1 303 "(2) Standard quaternion bas is as Maple global variable as in P. Lounesto \"Clifford Algebras and \+ Spinors\", page 49. To avoid conflicts with i, j, k, etc. traditional ly used in summations, loops, user could define qi, qj, and qk in plac e of \{i, j, k\} used to denote pure quaternion part of a quaternion. \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "_quatbasis:=[[Id,e3we2,e1we3, e2we1],\{`Maple has assigned qi:=-e2we3, qj:=e1we3, qk:=-e1we2`\}];\n " }}{PARA 0 "" 0 "" {TEXT 367 48 "(3) Defining abbreviations for quate rnion basis:" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "un protect(qi,qj,qk);\nqi:=-e2we3:\nqj:=e1we3:\nqk:=-e1we2:\n" }}{PARA 0 "" 0 "" {TEXT 368 31 "(4) Defining useful functions:\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 83 "pause:=proc(s::nonnegint) local s1:\ns1:=time( ):\nwhile time()-s1 < s do od end proc:" }}{PARA 0 "" 0 "" {TEXT 369 37 "\n(5) Protecting all procedure names:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "protect(Clifford,e,'qi','qj','qk',Id,w);\n" }}{PARA 258 "" 0 "" {TEXT 473 22 "Types and conversions:" }{TEXT -1 32 "\n\nNo . 1. Definition of the type " }{TEXT 436 9 "clibasmon" }{TEXT -1 87 ", i.e., a basis monomial. \n\nTypical use: type(e2we1,clibasmon); type( e1we2,clibasmon);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 907 "`type/cliba smon`:=proc(a)\nlocal a1,i,str,lst,e_set,w_set,ind_lst,N;\noptions `Co pyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All ri ghts reserved.`;\ndescription `Last revised: November 1, 2002`;\n##### ########################################\na1:=simplify(a):\n if a1 = Id then return true end if:\n if type(a1,\{string,name,symbol\}) th en\n str:=convert(a1,string);\n lst:=[seq(str[i],i=1..length(a 1))];\n N:=(nops(lst)+1)/3;\n if N=1 then \n e_set:=\{ls t[1]\};\n w_set:=\{\"w\"\};\n ind_lst:=[lst[2]];\n els e\n e_set:=\{seq(lst[3*i-2],i=1..N)\};\n w_set:=\{seq(lst[ 3*i],i=1..N-1)\};\n ind_lst:=[seq(lst[3*i-1],i=1..N)];\n end if:\n# print(e_set,w_set,ind_lst,N,lst);\n if (e_set=\{\"e\"\}) a nd (w_set=\{\"w\"\}) and (N=nops(\{op(ind_lst)\})) then\n return true\n else\n return false \n end if:\n else\n re turn false \n end if: \nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 30 "No. 2. Definition of the type " }{TEXT 437 9 "cliscalar" }{TEXT -1 255 ", i.e., Clifford scalar. A Clifford scalar is essentially any \+ number, function, constant, or an algebraic expression not containing \+ any basis monomials (this means that 2*Id is not of type 'cliscalar'). \n\nTypical use: type(e1+e2we3+2*Pi*B[1,2],cliscalar);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "`type/cliscalar`:=proc(a::anything) local a1 ,locscalartypes;\nglobal `&C`,_scalartypes; \noptions `Copyright (c) 1 995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved .`;\ndescription `Last revised: November 1, 2002`;\n################## ###########################\na1:=simplify(a):\nlocscalartypes:=remove( member,_scalartypes,\{`^`\}):\nif type(a1,\{matrix,list\}) or hastype( a1,clibasmon) or \n hastype(a1,tensorprod) or has(a1,`&C`) then retu rn false \nend if: \nif type(a1,locscalartypes) or evalb(op(map(type, \{op(a1)\},locscalartypes))=true)\n then return true \nend if:\nif \+ type(a1,`^`) then\n if select(hastype,\{a1\},clibasmon)=\{\} then\n \+ return true else error \"illegal expression in %1\",a1 \n end i f:\nend if:\nreturn cliparse(a1)\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 31 "No. 3. Definition of the type " }{TEXT 438 6 "climon" } {TEXT -1 197 ", i.e., Clifford monomial. A Clifford monomial is essent ially any basis monomial (of type 'clibasmon') multiplied by a Cliffor d scalar (of type 'cliscalar').\n\nTypical use: type(e1we2+2*e2,climon );\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 755 "`type/climon`:=proc(x1) lo cal x,S,xx,flag6plus:\noptions `Copyright (c) 1995-2003 by Rafal Ablam owicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last \+ revised: November 1, 2002`;\n######################################### ####\nx:=simplify(x1):\nflag6plus:=assigned(Cliplus):\nif hastype(x,cl iprod) and not flag6plus and _warnings_flag then \n WARNING(`argumen t to 'type/climon' contains type 'cliprod'. Load 'Cliplus' to extend \+ functionality of CLIFFORD. Type ?cliprod for help.`);\nend if:\n###### ############\nif not flag6plus then S:=\{'clibasmon'\} else S:=\{'clib asmon','cliprod'\} end if:\nxx:=simplify(x):\nif type(xx,cliscalar) th en false\nelif evalb(type(xx,`*`) and nops(select(type,\{op(xx)\},S))= 1) then\n true \nelse \n false\nend if:\nend proc:\n" }}{PARA 258 " " 0 "" {TEXT -1 30 "No. 4. Definition of the type " }{TEXT 439 10 "cli polynom" }{TEXT -1 265 ", i.e., Clifford polynomial. A Clifford polyn omial is a multivariate polynomial in the unknowns of type 'climon' or 'cliprod', i.e., Clifford monomial, with coefficients of the type 'cl iscalar', i.e., Clifford scalar.\n\nTypical use: type(e1+2*Pi*e2we3,cl ipolynom);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 941 "`type/clipolynom`: =proc(x1) local x,flag6plus:\noptions `Copyright (c) 1995-2003 by Rafa l Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n################################## ###########\nx:=simplify(x1):\nif type(eval(x),\{matrix,list,set,clisc alar\}) or \n (not type(eval(x),algebraic)) or \n \+ hastype(eval(x),tensorprod) then \nreturn false \nend if:\n flag6plus:=assigned(Cliplus):\nif hastype(x,cliprod) and not flag6plus and _warnings_flag then \n WARNING(`argument to 'type/clipolynom' c ontains type 'cliprod'. Load 'Cliplus' to extend functionality of CLI FFORD. Type ?cliprod for help.`);\nend if:\nif evalb(not flag6plus and type(expand(x),`+`) and hastype(x,clibasmon) and not hastype(x,clipro d)) \n then return true \nend if:\nif evalb(flag6plus and type(expa nd(x),`+`) and hastype(x,\{clibasmon,cliprod\})) then \n return true \nend if: \nreturn false \nend proc:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT 432 24 "No. 5. Conv erts strings " }{TEXT 440 10 "str_to_int" }{TEXT 441 98 " : `1`, `2`, \+ ..., `0` to appropriate digit.\n\nTypical use: map(convert,extract(e1w e2),str_to_int);\n" }{MPLTEXT 0 21 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 629 "`convert/str_to_int`:=proc(a1::symbol)\noptions `Copyright (c ) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reser ved.`,remember;\ndescription `Last revised: November 1, 2002`;\n###### #######################################\nif args[1] = `0` then return \+ 0 elif\n args[1] = `1` then return 1 elif\n args[1] = `2` then ret urn 2 elif\n args[1] = `3` then return 3 elif\n args[1] = `4` then return 4 elif\n args[1] = `5` then return 5 elif\n args[1] = `6` \+ then return 6 elif\n args[1] = `7` then return 7 elif\n args[1] = \+ `8` then return 8 elif\n args[1] = `9` then return 9 else\n return a1\nend if:\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 26 "No. 6. Def inition of type " }{TEXT 442 9 "nilpotent" }{TEXT -1 914 ". The follo wing procedure verifies whether or not its non-zero argument is a nilp otent element in the given Clifford algebra Cl(B). It is expected tha t a matrix of the bilinear form B has been specified. If the element happens to be an idempotent, or if some power of that element equals \+ the element itself, or if the element is of type 'cliscalar' then the \+ procedure returns 'false'. Otherwise, the procedure checks if any po wer of its argument up to and including order of 2^N, where N is the m aximum index found in the input, is zero.\n\nThis procedurecan also te st for nilpotency w.r.t. to a name/symbol/matrix/array which may be pa ssed on as a second element of list why the first element in the list \+ is the element to be checked for nilpotency. \n\nTypical use: type((1 /2)*(e1 +e1we3),nilpotent); #this is a nilpotent element in Cl(3,0) \+ \ntype(p,nilpotent);\ntype([p,K],nilpotent);\ntype([p,-K],nilpotent); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2103 "`type/nilpotent`:=proc(a11) \nlocal a1,i,x,y,xx,k,flagB,S,lname,flagindexed;global B;\noptions `C opyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All r ights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#### #########################################\n########################### ###############\n##This code allows for passing name of the matrix K a s a second element in a list:\n##To test element p for nilpotency w.r. t. matrix K enter [p,K];\n##To test element p for nilpotency w.r.t. B \+ enter p, or, [p,B].\n##########################################\nif ty pe(a11,\{cliscalar,clibasmon,climon,clipolynom\}) then\n a1:=a11:\n \+ lname:=`B`:\n flagindexed:=false:\n if not type(B,matrix) the n error \"matrix must be assigned to B\" \n else flagB:=typ e(B,diagmatrix) \n end if:\nelif type(a11,list) then\n if nops( a11)<>2 then error \"list must have exactly two elements\"\n elif not type(a11[1],\{cliscalar,clibasmon,climon,clipolynom\}) or\n \+ not type(a11[2],\{name,symbol,matrix,array,`&*`(numeric,\{name,sy mbol,matrix,array\})\})\n then error \"list must contain clipolyn om and name\"\n else\n a1:=a11[1]:\n lname:=a11[2]:\n flaginde xed:=true:\n if not type(evalm(lname),matrix) then error \"matrix must be assigned to %1\",lname \n else flagB:=type(evalm(l name),diagmatrix) \n end if: \n end if:\nelse\n error \"unexp ected argument type\"\nend if:\n###################################\nx :=displayid(a1):\nif a1=0 then return true \n elif type(a1,cliscala r) then \n return false \n elif (type(x,clibasmon) and \+ flagB and linalg[det](evalm(lname))<>0) then \n return fal se \nend if:\n####################################\nxx:=cmul[lname](x ,x):\nif evalb(xx=0) then return true end if:\nif evalb(simplify(xx-x) =0) or not evalb(solve(xx=k*x,k)=NULL) then return false end if:\ny:=x x:\nfor i from 1 to 2^maxindex(a1) do\n if y=vectorpart(y,0) or \+ y=x then return false end if: \n y:=cmul(x,y);\n if y=0 t hen return true end if:\n end do:\nerror \"Sorry, but I am unabl e to determine nilpotency of %1\",a1\nend proc:\n" }}{PARA 258 "" 0 " " {TEXT -1 26 "No. 7. Definition of type " }{TEXT 443 10 "idempotent" }{TEXT -1 311 ". The following procedure verifies whether or not its \+ argument is an idempotent in the given Clifford algebra Cl(B). It is \+ expected that a matrix of the bilinear form B has been specified. It c an also check element p for being idempotent in Cl(K) if K is entered \+ as a second argument in a list such as [p,K].\n" }}{PARA 0 "" 0 "" {TEXT 431 124 "Typical use: type((1/2)*(1 + e1),idempotent); #this is an idempotent in Cl(3,0)\ntype(p,idempotent);\ntype([p,K],idempotent) ;" }}{PARA 0 "" 0 "" {TEXT 435 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1577 "`type/idempotent`:=proc(a11) \nlocal f,ff,lname,a1,flagindexed,f lagB; global B;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz \+ and Bertfried Fauser. All rights reserved.`;\ndescription `Last revise d: November 1, 2002`;\n#############################################\n ##########################################\n##This code allows for pas sing name of the matrix K as a second element in a list:\n##To test el ement p for being idempotent w.r.t. matrix K enter [p,K];\n##To test e lement p for being idempotent w.r.t. B enter p, or, [p,B].\n########## ################################\nif type(a11,\{cliscalar,clibasmon,cl imon,clipolynom\}) then\n a1:=a11:\n lname:=`B`:\n flagindexed:= false:\n if not type(B,matrix) then error \"matrix must be assign ed to B\" \n else flagB:=type(B,diagmatrix) \n end if: \nelif type(a11,list) then\n if nops(a11)<>2 then error \"list must \+ have exactly two elements\"\n elif not type(a11[1],\{cliscalar,cl ibasmon,climon,clipolynom\}) or\n not type(a11[2],\{name,sym bol,matrix,array,`&*`(numeric,\{name,symbol,matrix,array\})\})\n \+ then error \"list must contain clipolynom and name\"\n else\n a1:= a11[1]:\n lname:=a11[2]:\n flagindexed:=true:\n if not type(e valm(lname),matrix) then error \"matrix must be assigned to %1\",lname \n else flagB:=type(evalm(lname),diagmatrix) \n end i f: \n end if:\nelse\n error \"unexpected argument type\"\nend if: \n########################################\nf:=displayid(a1):\nff:=cmu l[lname](f,f):\nif evalb(ff=0) then return false end if:\nreturn evalb (simplify(ff-f)=0)\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. \+ 8. A new type " }{TEXT 444 9 "climatrix" }{TEXT -1 424 " is a matrix w ith at least one entry of type 'clipolynom'. Note that anything in Map le that has been defined via the procedure linalg[matrix] is of the st andard Maple type 'matrix' including matrices with entries in a Cliffo rd algebra. Since a matrix with numerical entries is not of the type ' climatrix', this procedure allows one to distinguish such matrix from \+ those that do have at least one entry in a Clifford algebra." }}{PARA 258 "" 0 "" {TEXT -1 208 "\nMatrices of the type 'matrix' but not 'cli matrix' may be multiplied using standard Maple matrix multiplication o perator `&*`.\n\nMatrices of the type 'climatrix' must be multiplied u sing the procedure 'rmulm'." }}{PARA 0 "" 0 "" {TEXT 430 104 "\nTypica l use: M:=linalg[matrix](2,2,[e1,e3we4+e3,e4,Id-e1]);\n \+ type(M,climatrix);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "` type/climatrix`:=proc(x)\noptions `Copyright (c) 1995-2003 by Rafal Ab lamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `La st revised: November 1, 2002`;\n###################################### #######\nif type(x,array) then\n return evalb(select(type,convert(x,s et),\{clipolynom,climon,clibasmon\})<>\{\})\nelse \n return false\nen d if:\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 429 37 "No. 9. Useful conv ersion function to " }{TEXT 445 5 "mlist" }{TEXT 446 20 " needed by 'r mulm'.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 368 "`convert/mlist`:=proc( a1::matrix) local i,longlist;\noptions `Copyright (c) 1995-2003 by Raf al Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescriptio n `Last revised: November 1, 2002`;\n################################# ############\nlonglist:=[]:\nfor i from 1 to linalg[rowdim](a1) do\nlo nglist:=[op(longlist),op(convert(linalg[row](a1,i),list))] od\nend pro c:\n" }}{PARA 0 "" 0 "" {TEXT 428 19 "No. 10. A new type " }{TEXT 447 12 "fieldelement" }{TEXT 448 2 ":\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 547 "`type/fieldelement`:=proc(a1::algebraic) global f; \noptions `Cop yright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rig hts reserved.`;\ndescription `Last revised: November 1, 2002`;\n###### #######################################\nif not assigned(f) then \n \+ error \"primitive idempotent f has not been assigned yet\" \nend if:\n if not type(f,primitiveidemp) then \n error \"although f has been as signed, it is not of type/primitiveidemp\"\nend if:\nif member(squarem odf(args[1],f),\{-1,1\}) then return true else return false end if \ne nd proc:\n" }}{PARA 0 "" 0 "" {TEXT 427 20 "No. 11. A new type: " } {TEXT 449 9 "symmatrix" }{TEXT 450 25 " - a symmetric matrix:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 469 "`type/symmatrix`:=proc(a1::\{name, symbol,matrix,`&*`(algebraic,matrix)\}) \noptions `Copyright (c) 1995- 2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`; \ndescription `Last revised: November 1, 2002`;\n##################### ########################\nif evalb(evalm(a1)=a1) then return false end if:\nif linalg[coldim](a1)<>linalg[rowdim](a1) then\n error \"B mus t be assigned square matrix\" \nend if:\nreturn linalg[equal](a1,linal g[transpose](a1))\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 426 20 "No. 12 . A new type: " }{TEXT 451 13 "antisymmatrix" }{TEXT 452 31 " - an a nti-symmetric matrix:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "`type/a ntisymmatrix`:=proc(a1::\{name,symbol,matrix,`&*`(algebraic,matrix)\}) \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried \+ Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#############################################\nif evalb(evalm (a1)=a1) then return false end if:\nif linalg[coldim](a1)<>linalg[rowd im](a1) then\n error \"B must be assigned square matrix\" \nend if: \nreturn linalg[equal](a1,-linalg[transpose](a1))\nend proc:\n" }} {PARA 0 "" 0 "" {TEXT 425 20 "No. 13. A new type: " }{TEXT 453 10 "dia gmatrix" }{TEXT 454 25 " - a diagonal matrix.\n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 482 "`type/diagmatrix`:=proc(a1::anything) local N,i,DD ;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried \+ Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n#############################################\nif not type(a1 ,\{matrix,`&*`(algebraic,matrix)\}) then return false end if:\nif not \+ type(a1,symmatrix) then return false end if:\n N:=linalg[coldim](a1): \n DD:=linalg[diag](seq(a1[i,i],i=1..N)):\n return linalg[iszero](ev alm(a1-DD))\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 18 "No. 14. New type: " }{TEXT 455 14 "primitiveidemp" }{TEXT -1 1109 " - primitive i dempotent. This procedure determines the number of factors in the giv en idempotent of the type (1/2)*(Id+e[i]), i=1..n, where \{e[i],i=1..n \} is a set of commuting basis monomials with square equal to 1 mod Id . \nIt returns 'true' if n = q - RHnumber(q-p), where 'RHnumber' is t he Radon-Hurwitz function and [p,q] is signature of the current quadra tic form which is assumed to have been defined, i.e., the bilinear for m B has been defined as a diagonal matrix, and 'false' if n < q - RHnu mber(q-p).\n\nIf the argument is the identity element 'Id' of the alge bra Cl(Q), the procedure checks if Cl(Q) is simple or semi-simple, and it returns 'true' or 'false' respectively. It is known that when Cl( Q) is semi-simple, 'Id' can be written as a sum of mutually annihilati ng idempotents (1/2)*(Id+p) and (1/2)*(Id-p), where p is the unit pseu do-scalar element (volume element) in Cl(Q).\n\nThe procedure expects \+ that the bilinear form B has been defined as a diagonal matrix.\n\nTyp ical use: type(cmul((1/2)*(Id+e1),(1/2)*(Id+e2we3we4we5),primitiveidem p);\n type(Id,primitiveidemp);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 508 "`type/primitiveidemp`:=proc(f::idempotent) l ocal p,q,numfact;global B;\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription ` Last revised: November 1, 2002`;\n#################################### #########\nif not type(B,matrix) then \n error \"B must be assigned \+ square matrix\" \nelse\n p:=Bsignature(B)[1]:q:=Bsignature(B)[2]\nen d if:\nnumfact:=q-RHnumber(q-p):\nif scalarpart(f)=1/2^numfact then \n return true \nelse \n return false \nend if:\nend proc:\n" }} {PARA 258 "" 0 "" {TEXT -1 13 "No. 15. Type " }{TEXT 456 13 "purequatb asis" }{TEXT -1 109 " is a procedure which checks if the given list of three basis monomials can be a basis for pure quaternions.\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 920 "`type/purequatbasis`:=proc(l1::lis t(\{clibasmon,climon,clipolynom\})) \nlocal p,q,r;global B;\noptions ` Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All \+ rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n### ##########################################\nif nops(l1) <> 3 then \n \+ error \"list must have exactly 3 elements of type 'clibasmon', 'climo n', or 'clipolynom' but received a list with %1 elements\",nops(l1)\ne nd if:\nif not type(B,matrix) then \n error \"square matrix must be \+ assigned to B\"\nend if: \np:=l1[1]:q:=l1[2]:r:=l1[3]:\nif cmul(p,p)<> -Id then return false elif\n cmul(q,q)<>-Id then return false elif\n cmul(r,r)<>-Id then return false elif\n not member(cmul(p,q),\{r, -r\}) then return false elif\n cmul(p,q)+cmul(q,p)<>0 then return fa lse elif\n cmul(p,r)+cmul(r,p)<>0 then return false elif\n cmul(q, r)+cmul(r,q)<>0 then return false else\n return true\nend if:\nend p roc:\n" }}{PARA 258 "" 0 "" {TEXT -1 20 "No. 16. A new type: " }{TEXT 457 10 "gencomplex" }{TEXT -1 413 " - a generalized complex element of Cl(B). A Clifford polynomial p in Cl(B) is of this type if it belong s to a subalegbra A of Cl(B) isomorphic to complex numbers C. Knowing \+ that the given polynomial p is of that type allows for finding the inv erse of p in A < Cl(B) a more efficient way by the procedure 'cinv'.\n \nNote that elements of grade 0 (eg., 2*Id) are not of this type.\n\nT ypical use: type(p,gencomplex);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 886 "`type/gencomplex`:=proc(a1::\{cliscalar,clibasmon,climon,clipolyn om\}) local L;global B;\noptions `Copyright (c) 1995-2003 by Rafal Abl amowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Las t revised: November 1, 2002`;\n####################################### ######\nif not type(B,matrix) then \n error \"can't check type since B is not assigned a matrix\" \nend if:\nif type(a1,cliscalar) then re turn false end if:\nL:=[op(cliterms(reorder(a1)))];\nif nops(L)>2 then return false end if:\nif nops(L)=1 and L=[Id] then return false end i f:\nif nops(L)=2 and not member(Id,L) then return false end if:\nL:=re move(member,L,[Id]);\nif maxindex(L)>linalg[coldim](B) then \n error \"can't check type since the largest index in %1 is greater than size %2 of current form B\", a1,linalg[coldim](B)\nend if:\nif cmul(L[1],L [1])=-Id then \n return true \nelse \n return false \nend if:\nen d proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 20 "No. 17. A new type: " } {TEXT 458 13 "genquaternion" }{TEXT -1 513 " - a generalized quaternio nic element of Cl(B). A Clifford polynomial p in Cl(B) is of this typ e if it belongs to a subalegbra A of Cl(B) isomorphic to a division ri ng H of quaternions. Knowing that the given polynomial p is of that t ype allows for finding the inverse of p in A < Cl(B) a more efficient \+ way by the procedure 'cinv'.\n\nNote that elements of grade 0 (eg., 2* Id) and elements of type 'gencomplex' - a generalized complex element \+ of Cl(B), are not of this type.\n\nTypical use: type(p,genquaternion); \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "`type/genquaternion`:=proc(a 1::\{cliscalar,clibasmon,climon,clipolynom\}) local L;global B;\noptio ns `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. \+ All rights reserved.`;\ndescription `Last revised: November 1, 2002`; \n#############################################\nif not type(B,matrix) then \n error \"square matrix must be assigned to B\" \nend if:\nif type(a1,cliscalar) then return false end if:\nL:=[op(cliterms(reorder (a1)))];\nif nops(L)>4 or type(a1,gencomplex) then return false end if :\nL:=remove(member,L,[Id]);\nif nops(L)=1 then return false end if:\n if nops(L)=2 then L:=[op(L),cmul(L[1],L[2])] end if:\nreturn type(L,pu requatbasis)\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 26 "No. 18/19. Two new types: " }{TEXT 460 11 "evenelement" }{TEXT -1 5 " and " } {TEXT 459 10 "oddelement" }{TEXT -1 242 " in Cl(B). These two type-ch ecking procedures determine whether their inputs are even elements, od d elements, or neither in Cl(B).\n\nTypical use: type(p,evenelement); \n type(p,oddelement);\n\nwhere p is a Clifford pol ynomial.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "`type/evenelement`:= proc(a1::\{cliscalar,clibasmon,climon,clipolynom\})\noptions `Copyrigh t (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights r eserved.`;\ndescription `Last revised: November 1, 2002`;\n########### ##################################\nif type(a1,cliscalar) then return \+ true end if:\nreturn evalb(reorder(displayid(a1-gradeinv(a1)))=0)\nend proc:\n\n`type/oddelement`:=proc(a1::\{cliscalar,clibasmon,climon,cli polynom\})\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and B ertfried Fauser. All rights reserved.`;\ndescription `Last revised: No vember 1, 2002`;\n#############################################\nif ty pe(a1,cliscalar) then return false end if:\nreturn evalb(reorder(displ ayid(a1+gradeinv(a1)))=0)\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 424 18 "No. 20. New type: " }{TEXT 461 10 "quaternion" }{TEXT 462 1 "\n" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 896 "`type/quaternion`:=proc(q::algebr aic) local aa1,aa2,S;global B,qi,qj,qk;\noptions `Copyright (c) 1995-2 003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\n description `Last revised: November 1, 2002`;\n####################### ######################\nif not assigned(B) or not type(B,matrix) then \+ \n error \"bilinear form B has not been assigned yet. It must be def ined as the identity 3 x 3 matrix.\"\nend if:\nif not linalg[equal](B, linalg[diag](1$3)) then \n error \"identity 3 x 3 matrix must be ass igned to B\" \nend if:\nif not type(eval(q),\{'clibasmon','climon','cl ipolynom'\}) then \n error \"wrong input type: input must be of type 'clibasmon','climon', or 'clipolynom'\" \nend if:\naa1:=\{op(cliterms (reorder(expand(eval(q)))))\};\naa2:=\{Id,e1we2,e1we3,e2we3\};#standar d basis to be compared to\nS:=aa1 minus aa2;\nif op(S) = NULL then \n \+ return true else return false \nend if:\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 423 17 "No. 21. New type " }{TEXT 463 10 "tensorprod" } {TEXT 464 183 " is needed to include new types from the package 'GTP' \+ for 'Graded Tensor Product'. This is an experimental package for comp utations with graded tensor products of Clifford algebras." }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "`type/tensorprod`:=proc(a 1::anything)\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: \+ November 1, 2002`;\n#############################################\nif \+ type(a1,function) and op(0,a1)=`&t` then return true else return false end if:\nreturn false\nend proc:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT 422 18 "No. 22. New type: " }{TEXT 465 12 "genquatbasis" }{TEXT 466 187 ". This procedure checks i f the given list or set of four elements is a basis for generalized qu aternionic ring.\n\nUse: type([p1,p2,p3,p4], genquatbasis);type(\{p1,p 2,p3,p4\}, genquatbasis);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1475 "`t ype/genquatbasis`:=proc(L::\{list(\{cliscalar,clibasmon,climon,clipoly nom\}),\n set(\{cliscalar,clibasmon,climo n,clipolynom\})\}) \nlocal f,p,q,k,loc,i;global B;\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights re served.`;\n#############################################\ndescription \+ `Last revised: November 1, 2002`;\nif nops(L) <> 4 or nops(L)<>nops(co nvert(L,set)) then \n error \"list or set must have exactly 4 differ ent elements\" \nend if:\nif not type(B,matrix) then \n error \"squa re matrix must be assigned to B first\" \nend if: \nf:=op(select(type, L,idempotent)): #select idempotent in L\nif f=NULL then \n error \"o ne element in the list must be an idempotent\" \nend if:\nloc:=remove( member,L,\{f\}); #assign remaining elements of L to loc \np,q, k:=seq(loc[i],i=1..3): #assign elements of loc to p,q,k\n###### ############################\nif cmul(p,p)<>cmul(-Id,f) then return fa lse elif\n cmul(q,q)<>cmul(-Id,f) then return false elif\n cmul(k, k)<>cmul(-Id,f) then return false \nend if:\n######################### ######### \nif (cmul(p,q)=cmul(k,f) and cmul(q,p)=-cmul(k,f) and \n \+ cmul(q,k)=cmul(p,f) and cmul(k,q)=-cmul(p,f) and \n cmul(k,p)=cm ul(q,f) and cmul(p,k)=-cmul(q,f)) \nor\n (cmul(p,q)=-cmul(k,f) and c mul(q,p)=cmul(k,f) and \n cmul(q,k)=-cmul(p,f) and cmul(k,q)=cmul(p ,f) and \n cmul(k,p)=-cmul(q,f) and cmul(p,k)=cmul(q,f))\nthen retu rn true \nelse\n return false\nend if:\nend proc:\n" }}{PARA 258 "" 0 "" {TEXT -1 16 "No. 23. New type" }{TEXT 421 2 ": " }{TEXT 467 7 "cl iprod" }{TEXT 468 117 "\n\nUse: type(e1we2 &C e3, cliprod); type(`&C`( e1,e2),cliprod); type(`&C`[K](e1,e2),cliprod); type(&C(e1,e2),cliprod) ;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "`type/cliprod`:=proc(f::\{f unction,anything\}) local p;\noptions `Copyright (c) 1995-2003 by Rafa l Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n################################## ###########\nevalb(member(op(0,f),\{`&C`\}) or member(op(0,op(0,f)),\{ `&C`\}))\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 433 18 "No. 24. Procedu re " }{TEXT 469 16 "convert/dfmatrix" }{TEXT 470 84 " converts a list \+ of matrices or a pair of matrices inot a matrix over double field.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "`convert/dfmatrix`:=proc() local l1,l2,L,i,j,m,n,m1,m2,MN;\noptions `Copyright (c) 1995-2003 by Rafal \+ Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription ` Last revised: November 1, 2002`;\n#################################### #########\nif nargs=1 and type(args[1],dfmatrix) \n then return args[1]\nelif nargs=1 and type(args[1],list(\{matrix,array\})) \n then m1,m2:=evalm(args[1][1]),evalm(args[1][2]);\nel if nargs=2 and type(args[1],\{matrix,array\}) and type(args[2],\{matri x,array\}) \n then m1,m2:=evalm(args[1]),evalm(args[2]) \nelse error \"wrong number or types of arguments\" \nend if:\n l1 \+ := convert(m1,mlist);\n l2 := convert(m2,mlist);\n L := [];\n \+ for i to nops(l1) do L := [op(L), [l1[i], l2[i]]] end do:\n m := l inalg[rowdim](m1);\n n := linalg[rowdim](m1);\n MN := linalg[mat rix](m, n, []);\n for i to m do for j to n do MN[i, j] := L[(i - 1) *n + j] od\n end do:\n return evalm(MN)\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 434 18 "No. 25. Procedure " }{TEXT 471 13 "type/dfmatrix" }{TEXT 472 73 " checks if a matrix is of type 'dfma trix', that is, over a double field.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "`type/dfmatrix`:=proc(m::anything) local mm;\noptions `Copyri ght (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n######### ####################################\nif not type(m,matrix) and not ty pe(m,list(matrix)) then return false end if:\nif type(m,matrix) then \+ \n return type(convert(m,mlist),\n list(list(\{cliscalar,cl ibasmon,climon,clipolynom,numeric,symbol,algebraic\})))\nelse\n retu rn false\nend if:\nend proc:\n" }}{PARA 0 "" 0 "" {TEXT 477 79 "In thi s version we define all ampersand operators as global in Clifford:-set up:" }{TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2304 "`&c`:=pro c() local NP,ARGS,coB,nameB,lname,decindex,flagdec;\noptions `Copyrigh t (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights r eserved.`;\ndescription `Last revised: November 1, 2002`;\n########### ##################################\n################################## #####\n### Works when &c[''K''] or &c[''-K''] is entered and K is a ma trix\n#######################################\nflagdec:=true:\nif type (op(procname),procedure) then\n if type([args],listlist) then\n \+ if type(op(args),array) then\n WARNING(\"enclose index in dou ble quotes as in &c[''B''] or &c[''-B''] when B has been assigned a ma trix to avoid the following:\");\n return 'procname(args)';\n \+ end if;\n else coB:=1:\n nameB:=`B`:\n lname:=`B`:\n \+ ARGS:=[args]:\n flagdec:=false:\n end if;\nelse lname:=o p(procname);\n ARGS:=[args];\n if type(lname,`&*`(numeric,name )) then\n coB:=op(select(type,\{op(lname)\},numeric));\n \+ nameB:=op(select(type,\{op(lname)\},name));\n else\n co B:=1:\n nameB:=lname:\n end if;\n flagdec:=false:\n e nd if;\n#######################################\ndecindex:=proc() loca l ARGS,coB,nameB;global B;\nif type([args],listlist) then\n if type( op(args),function) then\n ARGS:=op(op(args));\n coB:=1:\n \+ nameB:=eval(op(0,op(args)));\n if type(nameB,`&*`(numeric,name )) then\n coB:=op(select(type,\{op(nameB)\},numeric));\n \+ nameB:=op(select(type,\{op(nameB)\},name));\n end if;\n eli f type(op(args),`&*`(numeric,function)) then\n nameB:=\{op(op(arg s))\}:\n coB:=op(select(type,nameB,numeric));\n nameB:=op(se lect(type,nameB,function));\n ARGS:=op(nameB);\n nameB:=op(0 ,nameB);\n else\n error \"unable to determine index or wrong in dex, use name in double quotes as in &c[''B''] or &c[''-B'']\"\n en d if;\nelif\n type([args],list) then\n ARGS:=args;\n coB:=1:\n \+ nameB:=`B`; #default name \nelse\n error \"cannot determine argumen ts and/or index from arguments\"\n end if;\nreturn coB,nameB,[ARGS];\n end proc:\n#####################################\nif flagdec then \n \+ coB,nameB,ARGS:=decindex(args);\n lname:=coB*nameB;\nend if;\nNP:=n ops(ARGS);\nif member(0,ARGS) then return 0 end if;\nif NP <=1 then re turn op(ARGS) end if;\nreturn cmul[eval(lname)](op(ARGS)); \nend proc: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2380 "`&cQ`:=proc() local NP,ARGS ,coB,nameB,lname,decindex,flagdec;\noptions `Copyright (c) 1995-2003 b y Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescr iption `Last revised: November 1, 2002`;\n############################ #################\n#######################################\n### Works \+ when &cQ[''K''] or &cQ[''-K''] is entered and K is a matrix\n######### ##############################\nflagdec:=true:\nif type(op(procname),p rocedure) then\n if type([args],listlist) then\n if type(op(arg s),array) then\n WARNING(\"enclose index in double quotes as i n &cQ[''B''] or &cQ[''-B''] when B has been assigned a matrix to avoid the following:\");\n return 'procname(args)';\n end if;\n else coB:=1:\n nameB:=`B`:\n lname:=`B`:\n ARGS:=[ args]:\n flagdec:=false:\n end if;\nelse lname:=op(procname); \n ARGS:=[args];\n if type(lname,`&*`(numeric,name)) then\n \+ coB:=op(select(type,\{op(lname)\},numeric));\n nameB:=op (select(type,\{op(lname)\},name));\n else\n coB:=1:\n \+ nameB:=lname:\n end if;\n flagdec:=false:\n end if;\n#### ###################################\ndecindex:=proc() local ARGS,coB,n ameB;global B;\nif type([args],listlist) then\n if type(op(args),fun ction) then\n ARGS:=op(op(args));\n coB:=1:\n nameB:=ev al(op(0,op(args)));\n if type(nameB,`&*`(numeric,name)) then\n \+ coB:=op(select(type,\{op(nameB)\},numeric));\n nameB:=op (select(type,\{op(nameB)\},name));\n end if;\n elif type(op(ar gs),`&*`(numeric,function)) then\n nameB:=\{op(op(args))\}:\n \+ coB:=op(select(type,nameB,numeric));\n nameB:=op(select(type,na meB,function));\n ARGS:=op(nameB);\n nameB:=op(0,nameB);\n \+ else\n error \"unable to determine index from or wrong index, us e name in double quotes as in &cQ[''B''] or &cQ[''-B'']\"\n end if; \nelif\n type([args],list) then\n ARGS:=args;\n coB:=1:\n name B:=`B`; #default name \nelse\n error \"cannot determine arguments an d/or index from arguments\"\nend if;\nreturn coB,nameB,[ARGS];\nend pr oc:\n#####################################\nif flagdec then \n coB,n ameB,ARGS:=decindex(args);\n lname:=coB*nameB;\nend if;\nNP:=nops(AR GS);\nif member(0,ARGS) then return 0 end if;\nif NP <=1 then return o p(ARGS) end if;\nreturn cmul[eval(lname)](op(ARGS));\n#return cmulQ[ev al(lname)](op(ARGS)); ###Causes an error in `&cQ` \nend proc:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1855 "`&cQm`:=proc() local ARGS,lname,N P,coB,nameB,decindex;\noptions `Copyright (c) 1995-2003 by Rafal Ablam owicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last \+ revised: November 1, 2002`;\n######################################### ####\n#######################################\nif type([args],listlist ) then\n if type(op(args),array) then\n WARNING(\"enclose index in double quotes as in &cQm[''B''] or &cQm[''-B''] when B has been as signed a matrix to avoid the following:\");\n return ('procname(a rgs)');\n end if;\nend if;\n####################################### \ndecindex:=proc() local ARGS,coB,nameB;global B;\nif type([args],list list) then\n if type(op(args),function) then\n ARGS:=op(op(args ));\n coB:=1:\n nameB:=eval(op(0,op(args)));\n if type( nameB,`&*`(numeric,name)) then\n coB:=op(select(type,\{op(name B)\},numeric));\n nameB:=op(select(type,\{op(nameB)\},name)); \n end if;\n elif type(op(args),`&*`(numeric,function)) then\n nameB:=\{op(op(args))\}:\n coB:=op(select(type,nameB,numeri c));\n nameB:=op(select(type,nameB,function));\n ARGS:=op(na meB);\n nameB:=op(0,nameB);\n else\n error \"unable to det ermine index or wrong index type for &cQm, try enclosing name of the i ndex in double quotes as in &cQm[''B''] or &cQm[''-B'']\"\n end if; \nelif\n type([args],list) then\n ARGS:=args;\n coB:=1:\n name B:=`B`; #default name \nelse\n error \"cannot determine arguments an d/or index\"\nend if;\nreturn coB,nameB,[ARGS];\nend proc:\n########## ###########################\ncoB,nameB,ARGS:=decindex(args);\nlname:=c oB*nameB:\n NP:=nops(ARGS);\n if member(0,ARGS) then return 0 end if ;\n if NP <=1 then \n return op(ARGS)\n elif NP = 2 then \n \+ return rmulm(eval(ARGS[1]),eval(ARGS[2]),cmulQ,lname) \n else\n e rror \"only two arguments and index are allowed\"\n end if;\nend proc :\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2447 "`&cm`:=proc() local NP,ARG S,coB,nameB,lname,decindex,flagdec;\noptions `Copyright (c) 1995-2003 \+ by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndesc ription `Last revised: November 1, 2002`;\n########################### ##################\n#######################################\n### Works when &cm[''K''] or &cm[''-K''] is entered and K is a matrix\n######## ###############################\nflagdec:=true:\nif type(op(procname), procedure) then\n if type([args],listlist) then\n if type(op(ar gs),array) then\n WARNING(\"enclose index in double quotes as \+ in &cm[''B''] or &cm[''-B''] when B has been assigned a matrix to avoi d the following:\");\n return 'procname(args)';\n end if; \n else coB:=1:\n nameB:=`B`:\n lname:=`B`:\n A RGS:=[args]:\n flagdec:=false:\n end if;\nelse lname:=op(proc name);\n ARGS:=[args];\n if type(lname,`&*`(numeric,name)) the n\n coB:=op(select(type,\{op(lname)\},numeric));\n nam eB:=op(select(type,\{op(lname)\},name));\n else\n coB:=1: \n nameB:=lname:\n end if;\n flagdec:=false:\nend if; \n#######################################\ndecindex:=proc() local ARGS ,coB,nameB;global B;\nif type([args],listlist) then\n if type(op(arg s),function) then\n ARGS:=op(op(args));\n coB:=1:\n nam eB:=eval(op(0,op(args)));\n if type(nameB,`&*`(numeric,name)) the n\n coB:=op(select(type,\{op(nameB)\},numeric));\n nam eB:=op(select(type,\{op(nameB)\},name));\n end if;\n elif type (op(args),`&*`(numeric,function)) then\n nameB:=\{op(op(args))\}: \n coB:=op(select(type,nameB,numeric));\n nameB:=op(select(t ype,nameB,function));\n ARGS:=op(nameB);\n nameB:=op(0,nameB );\n else\n error \"unable to determine index or wrong index: u se name in double quotes as in &cm[''B''] or &cm[''-B'']\"\n end if; \nelif\n type([args],list) then\n ARGS:=args;\n coB:=1:\n name B:=`B`; #default name \nelse\n error \"cannot determine arguments an d/or index\"\nend if;\nreturn coB,nameB,[ARGS];\nend proc:\n########## ###########################\nif flagdec then \n coB,nameB,ARGS:=deci ndex(args);\n lname:=coB*nameB;\n end if;\n#return (coB,nameB,lname, ARGS);\nNP:=nops(ARGS);\n if member(0,ARGS) then return 0 end if;\n \+ if NP <=1 then \n return op(ARGS)\n elif NP = 2 then \n retur n rmulm(eval(ARGS[1]),eval(ARGS[2]),cmul,lname) \n else\n error \+ \"only two arguments and index are allowed\"\n end if;\nend proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "`&q`:=proc()\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights re served.`;\ndescription `Last revised: November 1, 2002`;\n############ #################################\nreturn qmul(args) \nend proc:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "`&qm`:=proc() local NP: \noptions \+ `Copyright (c) 1995-2003 by Rafal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescription `Last revised: November 1, 2002`;\n## ###########################################\n NP:=nops([args]);\n if member(0,[args]) then return 0 end if;\n if NP <=1 then \n retur n args\n elif NP = 2 then \n return rmulm(eval(args[1]),eval(args [2]),qmul) \n else\n error \"only two arguments are allowed in &q m\"\n end if;\nend proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 349 "`&o m`:=proc()\noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and B ertfried Fauser. All rights reserved.`;\ndescription `Last revised: No vember 1, 2002`;\n#############################################\nif no t assigned(Octonion) then\n error \"package 'Octonion' must be loade d first\"\nend if;\nreturn subs(Id=1,rmulm(args,Octonion:-omul))\nend \+ proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1847 "`&rm`:=proc() local AR GS,lname,NP,coB,nameB,decindex;\noptions `Copyright (c) 1995-2003 by R afal Ablamowicz and Bertfried Fauser. All rights reserved.`;\ndescript ion `Last revised: November 1, 2002`;\n############################### ##############\n#######################################\nif type([args ],listlist) then\n if type(op(args),array) then\n WARNING(\"enc lose index in double quotes as in &rm[''B''] or &rm[''-B''] when B has been assigned a matrix to avoid the following:\");\n return 'pro cname(args)';\n end if;\nend if;\n################################## #####\ndecindex:=proc() local ARGS,coB,nameB;global B;\nif type([args] ,listlist) then\n if type(op(args),function) then\n ARGS:=op(op (args));\n coB:=1:\n nameB:=eval(op(0,op(args)));\n if \+ type(nameB,`&*`(numeric,name)) then\n coB:=op(select(type,\{op (nameB)\},numeric));\n nameB:=op(select(type,\{op(nameB)\},nam e));\n end if;\n elif type(op(args),`&*`(numeric,function)) th en\n nameB:=\{op(op(args))\}:\n coB:=op(select(type,nameB,nu meric));\n nameB:=op(select(type,nameB,function));\n ARGS:=o p(nameB);\n nameB:=op(0,nameB);\n else\n error \"unable to determine index or wrong index type for &rm, try enclosing name of th e index in double quotes as in &rm[''B''] or &rm[''-B'']\"\n end if; \nelif\n type([args],list) then\n ARGS:=args;\n coB:=1:\n name B:=`B`; #default name \nelse\n error \"cannot determine arguments an d/or index\"\nend if;\nreturn coB,nameB,[ARGS];\nend proc:\n########## ###########################\ncoB,nameB,ARGS:=decindex(args);\nlname:=c oB*nameB:\n NP:=nops(ARGS);\n if member(0,ARGS) then return 0 end if ;\n if NP <=1 then \n return op(ARGS)\n elif NP = 2 then \n \+ return rmulm(eval(ARGS[1]),eval(ARGS[2]),`&r`,lname) \n else\n er ror \"only two arguments and index are allowed\"\n end if;\n end proc :\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "`&w`:=proc() return wedge(ar gs) end proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 466 "`&wm`:=proc() l ocal NP: \noptions `Copyright (c) 1995-2003 by Rafal Ablamowicz and Be rtfried Fauser. All rights reserved.`;\ndescription `Last revised: Nov ember 1, 2002`;\n#############################################\n NP:= nops([args]);\n if member(0,[args]) then return 0 end if;\n if NP <= 1 then \n return args\n elif NP = 2 then \n return rmulm(eval (args[1]),eval(args[2]),wedge) \n else\n error \"only two argumen ts are allowed in &wm\"\n end if;\nend proc:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "#################################################### \nend proc: ###<< " 0 "" {MPLTEXT 1 0 8 "libname;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q6C:\\Maple6/Cliffordlib6\"Q.C:\\Maple6/libF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Let's add library files to the main library in libname[1] :\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 449 "march('add',libname [1],`C:\\\\Maple6/Clifforddata/matrealL.m`,`matrealL.m`);\nmarch('add' ,libname[1],`C:\\\\Maple6/Clifforddata/matrealR.m`,`matrealR.m`);\nmar ch('add',libname[1],`C:\\\\Maple6/Clifforddata/matcompL.m`,`matcompL.m `);\nmarch('add',libname[1],`C:\\\\Maple6/Clifforddata/matcompR.m`,`ma tcompR.m`);\nmarch('add',libname[1],`C:\\\\Maple6/Clifforddata/matquat L.m`,`matquatL.m`);\nmarch('add',libname[1],`C:\\\\Maple6/Clifforddata /matquatR.m`,`matquatR.m`);" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, m ember \"matrealL.m\" already in archive, skipping\n" }}{PARA 7 "" 1 " " {TEXT -1 58 "Warning, member \"matrealR.m\" already in archive, skip ping\n" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, member \"matcompL.m\" \+ already in archive, skipping\n" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning , member \"matcompR.m\" already in archive, skipping\n" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, member \"matquatL.m\" already in archive, s kipping\n" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, member \"matquatR.m \" already in archive, skipping\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and verify that indeed addition has taken place:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matrealL);" }}{PARA 12 "" 1 "" {XPPMATH 20 "657#7$\"\"\"F%7#7$\"\"$F%7#7$\"\"!\"\")7#7$\"\"%F/7#7$F%F ,7#7$F/\"\"#7#7$F4F47#7$F%\"\"(7#7$F(F47#7$\"\"&F/7#7$F(F(7#7$F,F+7#7$ F4F%7#7$\"\"*F+7#7$F4F+7#7$F+\"\"'7#7$F>F(7#7$F+F97#7$F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matrealR);" }}{PARA 12 "" 1 "" {XPPMATH 20 "657#7$\"\"\"F%7#7$\"\"$F%7#7$\"\"!\"\")7#7$\"\"%F/7# 7$F%F,7#7$F/\"\"#7#7$F4F47#7$F%\"\"(7#7$F(F47#7$\"\"&F/7#7$F(F(7#7$F,F +7#7$F4F%7#7$\"\"*F+7#7$F4F+7#7$F+\"\"'7#7$F>F(7#7$F+F97#7$F/F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matcompL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "607#7$\"\"$\"\"!7#7$\"\"&\"\"#7#7$\"\"%\"\"\"7 #7$F*F%7#7$\"\")F.7#7$F*\"\"(7#7$F&\"\"*7#7$F%F-7#7$F.F*7#7$F&F)7#7$\" \"'F%7#7$F6F&7#7$F.FB7#7$F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matcompR);" }}{PARA 11 "" 1 "" {XPPMATH 20 "607#7$\"\"$ \"\"!7#7$\"\"&\"\"#7#7$\"\"%\"\"\"7#7$F*F%7#7$\"\")F.7#7$F*\"\"(7#7$F& \"\"*7#7$F%F-7#7$F.F*7#7$F&F)7#7$\"\"'F%7#7$F6F&7#7$F.FB7#7$F-F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matquatL);" }}{PARA 12 "" 1 "" {XPPMATH 20 "657#7$\"\"\"\"\"$7#7$\"\"(\"\"#7#7$F&\"\"&7#7$ F%F-7#7$F-F%7#7$F-\"\"!7#7$F)F%7#7$\"\"'F%7#7$F*F97#7$F4\"\"%7#7$F&F97 #7$F%F>7#7$F>F47#7$F9F*7#7$F*F>7#7$F9F47#7$F*F-7#7$F4F*7#7$F4F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "indices(matquatR);" }}{PARA 12 "" 1 "" {XPPMATH 20 "657#7$\"\"\"\"\"$7#7$\"\"(\"\"#7#7$F&\"\"&7#7$ F%F-7#7$F-F%7#7$F-\"\"!7#7$F)F%7#7$\"\"'F%7#7$F*F97#7$F4\"\"%7#7$F&F97 #7$F%F>7#7$F>F47#7$F9F*7#7$F*F>7#7$F9F47#7$F*F-7#7$F4F*7#7$F4F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 " " {TEXT -1 24 "Last revised: 10-15-2002" }}{PARA 0 "" 0 "" {TEXT -1 952 "NOTES:\n\n1. The table name, e.g., Clifford, and the file name, e .g., Clifford.m must be the same.\n2. March commands useful in creatin g and viewing library file (issue in DOS window):\n\nC:\\Maple6>bin.wn t\\march -c Cliffordlib 20 - creates library in a existing empty di rectory \\Cliffordlib\nC:\\Maple6>bin.wnt\\march -l Cliffordlib - lis t all entries in the library Cliffordlib\nC:\\Maple6>bin.wnt\\march -l Cliffordlib > list.txt - list all entries in the library Cliffordlib and write them into file list.txt\nC:\\Maple6>bin.wnt\\march -d Cliff ordlib Clifford.m - delete Clifford.m from the library Cliffordlib\n \n3. Global variable savelibname is empty, but savelib() automatically assigns libname[1] to savelibname for the purpose of saving package t here with the command savelib().\n4. Maple initialization file maple.i ni contains libname augmented by the path and the directory name \\Cli ffordlib where the Clifford library with Clifford.m will be located. \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 198 "################################################## #####\n###end module:\n###march('create',Cliffordlib,500);\n###savelib (Clifford,`Clifford.m`):\n############################################ ############" }}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }