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y(3.)24 b Fr(Braidings)18 b(and)g(tensors)50 1758 y Fs(Let)j Fp(C)j Fs(b)q(e)e(a)f(strict)g (monoidal)f(category)h(that)h(is)f(braided.)36 b(Let)21 b Fo(\032)h Fp(2)h Fo(B)1466 1765 y Fm(n)1510 1758 y Fs(b)q(e)f(a)f(braid)g(in)0 1816 y(the)d(braid)h(group)g(with)f (canonical)g(image)p 828 1789 26 2 v 17 w Fo(\032)g Fp(2)g Fo(S)952 1823 y Fm(n)975 1816 y Fs(.)28 b(Let)18 b Fo(\033)h Fs(:=)p 1222 1789 V 17 w Fo(\032)1247 1798 y Fl(\000)p Fk(1)1294 1816 y Fs(.)28 b(Let)18 b Fo(\032)g Fs(:)f Fo(A)1536 1823 y Fk(1)1568 1816 y Fp(\012)12 b Fo(:)c(:)g(:)k Fp(\012)g Fo(A)1777 1823 y Fm(n)0 1874 y Fp(\000)-30 b(!)19 b Fo(A)115 1882 y Fm(\033)q Fk(\(1\))197 1874 y Fp(\012)13 b Fo(:)8 b(:)g(:)13 b Fp(\012)g Fo(A)409 1882 y Fm(\033)q Fk(\()p Fm(n)p Fk(\))500 1874 y Fs(also)21 b(denote)e(the)h(asso)q(ciated)g(braid)g(action)g(on)g(the)g Fo(n)p Fs(-fold)g(tensor)0 1932 y(pro)q(duct.)i(So)17 b Fo(\032)f Fs(is)g(a)h(natural)f(transformation)h(of)f(functors)h(in)f Fo(n)g Fs(v)m(ariables.)50 1990 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y(=)e Fp(h)p Fo(S)s Fs(\()p Fo(a)376 1738 y Fk(\(1\))423 1730 y Fo(b)444 1738 y Fk(\(1\))491 1730 y Fs(\))i Fo(a)552 1738 y Fk(\(2\))615 1730 y Fo(b)636 1738 y Fk(\(2\))699 1730 y Fo(S)s Fs(\()p Fo(b)772 1738 y Fk(\(3\))819 1730 y Fs(\))g Fo(S)s Fs(\()p Fo(a)932 1738 y Fk(\(3\))979 1730 y Fs(\))p Fo(;)8 b(\034)1041 1737 y Fk(23)1078 1730 y Fo(\034)1099 1737 y Fk(56)1137 1730 y Fo(\034)1158 1737 y Fk(45)1196 1730 y Fo(\034)1217 1737 y Fk(34)1254 1730 y Fp(i)325 1788 y Fs(\(c)o(hange)16 b(of)h(braid)f(map\))227 1871 y(=)e Fp(h)p Fo(S)s Fs(\()p Fo(a)376 1878 y Fk(\(1\)\(1\))468 1871 y Fo(b)489 1878 y Fk(\(1\)\(1\))581 1871 y Fs(\))i Fo(a)642 1878 y Fk(\(1\)\(2\))751 1871 y Fo(b)772 1878 y Fk(\(1\)\(2\))880 1871 y Fo(S)s Fs(\()p Fo(b)953 1878 y Fk(\(2\))1000 1871 y Fs(\))g Fo(S)s Fs(\()p Fo(a)1113 1878 y Fk(\(2\))1159 1871 y Fs(\))p Fo(;)8 b(\034)1221 1878 y Fk(23)1259 1871 y Fo(\034)1280 1878 y Fk(56)1317 1871 y Fo(\034)1338 1878 y Fk(45)1376 1871 y Fo(\034)1397 1878 y Fk(34)1434 1871 y Fp(i)325 1929 y Fs(\(b)o(y)16 b(\(36\),)g(the)g(argumen)o(ts)g(are)g Fo(a)957 1936 y Fk(\(1\)\(1\))1049 1929 y Fo(;)8 b(a)1097 1936 y Fk(\(1\)\(2\))1188 1929 y Fo(;)g(a)1236 1936 y Fk(\(2\))1283 1929 y Fo(;)g(b)1326 1936 y Fk(\(1\)\(1\))1418 1929 y Fo(;)g(b)1461 1936 y Fk(\(1\)\(2\))1553 1929 y Fo(;)g(b)1596 1936 y Fk(\(2\))1642 1929 y Fs(\))227 2011 y(=)14 b Fp(h)p Fo(S)s Fs(\(\()p Fo(a)395 2019 y Fk(\(1\))442 2011 y Fo(b)463 2019 y Fk(\(1\))510 2011 y Fs(\))529 2019 y Fk(\(1\))576 2011 y Fs(\))i(\()p Fo(a)656 2019 y Fk(\(1\))703 2011 y Fo(b)724 2019 y Fk(\(1\))771 2011 y Fs(\))790 2019 y Fk(\(2\))853 2011 y Fo(S)s Fs(\()p Fo(b)926 2019 y Fk(\(2\))973 2011 y Fs(\))g Fo(S)s Fs(\()p Fo(a)1086 2019 y Fk(\(2\))1133 2011 y Fs(\))p Fo(;)8 b(\034)1195 2018 y Fk(34)1232 2011 y Fo(\034)1253 2018 y Fk(23)1291 2011 y Fp(i)49 b Fs(\(39\))325 2069 y(\(c)o(hange)16 b(to)h(4)f(argumen)o(ts)g Fo(a)869 2077 y Fk(\(1\))915 2069 y Fo(;)8 b(a)963 2077 y Fk(\(2\))1010 2069 y Fo(;)g(b)1053 2077 y Fk(\(1\))1100 2069 y Fo(;)g(b)1143 2077 y Fk(\(2\))1190 2069 y Fs(,)15 b(apply)i(\(35\))g(t)o(wice\))325 2127 y(\(read)f(from)f(lo)o(w)o(er)h(line)f(to)i(upp)q(er)f(line\))227 2210 y(=)e Fp(h)p Fo(\021)r(")p Fs(\()p Fo(a)392 2218 y Fk(\(1\))439 2210 y Fo(b)460 2218 y Fk(\(1\))507 2210 y Fs(\))i Fo(S)s Fs(\()p Fo(b)615 2218 y Fk(\(2\))662 2210 y Fs(\))g Fo(S)s Fs(\()p Fo(a)775 2218 y Fk(\(2\))822 2210 y Fs(\))p Fo(;)8 b(\034)884 2217 y Fk(34)921 2210 y Fo(\034)942 2217 y Fk(23)980 2210 y Fp(i)325 2268 y Fs(\(b)o(y)16 b(\(38\)\))227 2351 y(=)e Fp(h)p Fo(")p Fs(\()p Fo(a)366 2358 y Fk(\(1\))413 2351 y Fs(\))i Fo(")p Fs(\()p Fo(b)511 2358 y Fk(\(1\))558 2351 y Fs(\))g Fo(S)s Fs(\()p Fo(b)666 2358 y Fk(\(2\))713 2351 y Fs(\))g Fo(S)s Fs(\()p Fo(a)826 2358 y Fk(\(2\))873 2351 y Fs(\))p Fo(;)8 b(\034)935 2358 y Fk(34)972 2351 y Fo(\034)993 2358 y Fk(23)1031 2351 y Fp(i)325 2409 y Fs(\(b)o(y)16 b(\(33\))h(and)f(m)o (ultiplicativ)o(it)o(y)d(of)j Fo(")p Fs(\))227 2491 y(=)e Fp(h)p Fo(")p Fs(\()p Fo(a)366 2499 y Fk(\(1\))413 2491 y Fs(\))i Fo(S)s Fs(\()p Fo(b)p Fs(\))g Fo(S)s Fs(\()p Fo(a)634 2499 y Fk(\(2\))681 2491 y Fs(\))p Fo(;)8 b(\034)743 2498 y Fk(23)780 2491 y Fp(i)325 2549 y Fs(\(b)o(y)16 b(\(38\))h(together)f(with)g(c)o(hange)g(of)h(argumen)o(ts)e(to)i Fo(a)1358 2557 y Fk(\(1\))1405 2549 y Fo(;)8 b(a)1453 2557 y Fk(\(2\))1499 2549 y Fo(;)g(b)p Fs(\))227 2632 y(=)14 b Fp(h)p Fo(S)s Fs(\()p Fo(b)p Fs(\))i Fo(")p Fs(\()p Fo(a)474 2640 y Fk(\(1\))521 2632 y Fs(\))g Fo(S)s Fs(\()p Fo(a)634 2640 y Fk(\(2\))681 2632 y Fs(\))p Fo(;)8 b(\034)743 2639 y Fk(12)780 2632 y Fo(\034)801 2639 y Fk(23)839 2632 y Fp(i)325 2690 y Fs(\(b)o(y)16 b(\(32\)\))p eop %%Page: 11 11 11 10 bop 590 -49 a Fh(ON)17 b(SYMBOLIC)f(COMPUT)m(A)m(TIONS)551 b(11)318 49 y Fs(=)14 b Fp(h)p Fo(S)s Fs(\()p Fo(b)p Fs(\))i Fo(S)s Fs(\()p Fo(a)p Fs(\))p Fo(;)8 b(\034)e Fp(i)416 107 y Fs(\(b)o(y)15 b(\(38\))i(together)g(with)f(c)o(hange)g (of)h(argumen)o(ts)e(to)i Fo(a;)8 b(b)p Fs(\))o Fo(:)1761 173 y Ff(\003)755 272 y Fr(References)0 350 y Fw([JS91])20 b(Jo)o(y)o(al,)10 b(A.)g(and)h(Street,)h(R.:)k Fi(The)c(Ge)n(ometry)g (of)g(T)m(ensor)f(Calculus,)i(I.)d Fw(Adv.)g(Math.)h(88,)f(\(55-112\))g Fb(1991)p Fw(.)0 400 y([P)o(a77])19 b(P)o(areigis,)10 b(B.:)15 b Fi(Non-additive)c(ring)g(and)g(mo)n(dule)g(the)n(ory)g(I.)f (Gener)n(al)h(the)n(ory)g(of)f(monoids.)g Fw(Publicationes)85 450 y(Mathematicae)j(24,)g(Debrecen,)i(\(190-204\))e Fb(1977)p Fw(.)0 500 y([H-Sw62])19 b(Heyneman,)f(R.G.)e(and)h(Sw)o (eedler,)i(M.E.:)25 b Fi(A\016ne)18 b(Hopf)h(algebr)n(as)e(I,)h Fw(J.)f(Algebra)h(13,)f(\(192-241\))85 550 y Fb(1969)0 600 y Fw([P)o(e71])j(P)o(enrose,)14 b(R.:)i Fi(Applic)n(ations)e(of)g (Ne)n(gative)f(Dimensional)i(T)m(ensors.)d Fw(In:)17 b(Com)o(binatorial)10 b(Mathematics)85 649 y(and)k(its)g(Applications.) f(Academic)g(Press.)i(\(221-244\))e Fb(1971)p Fw(.)50 743 y Fv(Ma)m(thema)m(tisches)j(Institut)f(der)h(Universit)886 740 y(\177)885 743 y(at)f(M)992 740 y(\177)991 743 y(unchen,)i(Germany) 50 793 y Fi(E-mail)d(addr)n(ess)s Fw(:)19 b Fa(pareigis@rz.math)o(emat) o(ik.un)o(i-mue)o(nchen)o(.de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF