%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: slc46tk.dvi %%CreationDate: Tue May 08 17:33:04 2001 %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: D:\PROGRA~1\MIKTEX\TEXMF\MIKTEX\BIN\DVIPS.EXE %+ slc46tk.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.08:1730 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ 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y(the)41 b(space)h(of)e(all)f (linear)g(functionals)g(on)i(the)g(group)g(ring)e(that)i(v)-5 b(anish)40 b(on)h(all)e(elemen)m(ts)166 3144 y(of)33 b Ft(W)14 b Fu(.)47 b(Ev)m(ery)36 b(elemen)m(t)d Ft(x)i Fu(of)e(the)h(orthogonal)e(subspace)k(yields)d(a)h(linear)e(iden)m(tit) m(y)i(for)f(our)166 3261 y(tensor)46 b(co)s(ordinates.)81 b(The)46 b(co)s(e\016cien)m(ts)h(of)e(this)g(iden)m(tit)m(y)g(are)g (the)h(v)-5 b(alues)45 b(of)g Ft(x)g Fu(on)g(the)166 3377 y(p)s(erm)m(utations)32 b(of)g(the)g(symmetric)g(group)g Fs(S)1852 3392 y Fl(r)1891 3377 y Fu(.)43 b(These)34 b(iden)m(tities)e(are)g(the)h(linear)e(iden)m(tities)166 3493 y(for)h(whic)m(h)h(w)m(e)h(searc)m(h)g(to)e(solv)m(e)h(our)f (normal)f(form)g(problem.)266 3609 y(The)37 b(elemen)m(ts)g(of)f(the)h (orthogonal)e(subspace)k(of)d Ft(W)50 b Fu(can)37 b(b)s(e)g(determined) f(as)h(solutions)166 3725 y(of)45 b(a)g(linear)f(equation)h(system.)84 b(T)-8 b(o)45 b(construct)i(this)e(system)h(w)m(e)h(ha)m(v)m(e)g(to)e (\014nd)h(a)f(basis)166 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b(can)h(b)s(e)g(sho)m(wn)h(that)e(this)h (group)f(ring)g(elemen)m(t)g(is)g(equal)h(to)f(a)g(sum)h(of)f(group)p eop %%Page: 5 5 5 4 bop 1113 251 a Fp(NORMAL)33 b(F)n(ORMS)f(OF)h(VECTOR)h(ANAL)-6 b(YSIS)906 b(5)166 500 y Fu(ring)32 b(elemen)m(ts)g Ft(T)822 515 y Fl(b)857 500 y Fu(.)1064 618 y Fk(X)1056 829 y Fl(p)p Fj(2S)1182 837 y Fh(r)1232 712 y Fu(\()p Ft(pT)14 b Fu(\))1487 665 y Fl(j)1516 674 y Fi(1)1589 665 y Fl(j)1618 674 y Fi(2)1691 665 y Fl(:::)38 b(j)1818 677 y Fh(l)1428 738 y Fl(j)1457 747 y Fi(1)1530 738 y Fl(j)1559 747 y Fi(2)1633 738 y Fl(:::)g(j)1760 750 y Fh(l)1822 738 y Fl(i)1846 750 y Fi(2)p Fh(l)p Fi(+1)1978 738 y Fl(:::)o(i)2061 746 y Fh(r)2116 712 y Ft(p)83 b Fu(=)2437 618 y Fk(X)2407 830 y Fl(b)p Fj(2)p Fg(B)2545 842 y Fh(b)2572 857 y Fi(0)2628 712 y Ft(T)2685 727 y Fl(b)2736 712 y Ft(:)166 1024 y Fu(The)32 b(sum)f(runs)h(through)f(a)g(set)g Ff(B)1471 1039 y Fl(b)1501 1048 y Fi(0)1571 1024 y Fu(of)g(v)m(ector)h(tuples)f Ft(b)h 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b(sequence)i(mark)m(es)d(the)g(set)g Ff(B)3531 1504 y Fl(b)3561 1513 y Fi(0)3600 1489 y Fu(.)266 1605 y(Ev)m(ery)g(linear)c (functional)g(that)i(v)-5 b(anishes)32 b(on)f(the)g(sum)2367 1531 y Fk(P)2472 1634 y Fl(b)p Fj(2)p Fg(B)2610 1646 y Fh(b)2637 1661 y Fi(0)2697 1605 y Ft(T)2754 1620 y Fl(b)2820 1605 y Fu(yields)f(a)h(linear)f(iden-)166 1760 y(tit)m(y)39 b(for)g(the)g(expressions)i(\()p Ft(pT)14 b Fu(\))1457 1713 y Fl(j)1486 1722 y Fi(1)1559 1713 y Fl(j)1588 1722 y Fi(2)1661 1713 y Fl(:::)38 b(j)1788 1725 y Fh(l)1398 1786 y Fl(j)1427 1795 y Fi(1)1500 1786 y Fl(j)1529 1795 y Fi(2)1603 1786 y Fl(:::)g(j)1730 1798 y Fh(l)1792 1786 y Fl(i)1816 1798 y Fi(2)p Fh(l)p Fi(+1)1948 1786 y Fl(:::)o(i)2031 1794 y Fh(r)2070 1760 y Fu(.)63 b(Th)m(us)41 b(w)m(e)f(ha)m(v)m(e)h(no)m(w)f(the)f(problem)f(to)166 1888 y(\014nd)d(the)g(smallest)e(subspace)j Ft(W)49 b Fu(whic)m(h)35 b(con)m(tains)f(all)f(sums)2510 1813 y Fk(P)2615 1917 y Fl(b)p Fj(2)p Fg(B)2753 1929 y Fh(b)2780 1944 y Fi(0)2840 1888 y Ft(T)2897 1903 y Fl(b)2966 1888 y Fu(of)h(the)h(tensors)g Ft(T)166 2021 y Fu(b)s(eing)d(considered.) 1717 2331 y(9.)49 b Fx(Slide)266 2506 y Fu(No)m(w)33 b(I)g(turn)f(to)h(the)g(question)g(ho)m(w)g(w)m(e)g(can)g(\014nd)g(the) g(subspaces)i Ft(W)14 b Fu(.)266 2622 y(I)39 b(need)h(t)m(w)m(o)g(op)s (erations.)62 b(A)39 b(group)g(ring)f(elemen)m(t)h(acts)h(as)f(a)g (symmetry)h(op)s(erator)e(on)166 2738 y(tensors.)84 b(This)46 b(is)g(de\014ned)h(in)e(this)g(form)m(ula.)82 b(F)-8 b(urther)45 b(I)h(need)h(the)f(in)m(v)m(olution)f Fs(\003)g Fu(that)166 2854 y(replaces)38 b(ev)m(ery)h(p)s(erm)m(utation)d(of)h(a) g(group)g(ring)f(elemen)m(t)h(b)m(y)i(the)e(corresp)s(onding)g(in)m(v)m (erse)166 2970 y(p)s(erm)m(utation.)266 3087 y(A)i(symmetry)h(class)f (of)g Ft(r)s Fu(-times)f(co)m(v)-5 b(arian)m(t)39 b(tensors)i(is)e (de\014ned)h(in)f(the)h(follo)m(wing)d(w)m(a)m(y)-8 b(.)166 3203 y(Let)32 b Ff(r)f Fu(b)s(e)h(a)f(righ)m(t)f(ideal)g(of)h(the)h (group)f(ring)g Fm(K)17 b Fu([)p Fs(S)1999 3218 y Fl(r)2043 3203 y Fu(])31 b(for)g(whic)m(h)h(an)f(ideal)f(elemen)m(t)i Ft(a)c Fs(2)g Ff(r)j Fu(and)166 3319 y(a)i(tensor)h Ft(T)47 b Fu(exist)33 b(with)g Ft(aT)42 b Fs(6)p Fu(=)29 b(0.)45 b(Then)34 b(the)g(tensor)f(set)h Fs(T)2400 3334 y Fg(r)2460 3319 y Fu(:=)29 b Fs(f)p Ft(aT)42 b Fs(j)29 b Ft(a)f Fs(2)i Ff(r)17 b Ft(;)32 b(T)43 b Fs(2)29 b(T)3388 3334 y Fl(r)3426 3319 y Ft(V)21 b Fs(g)33 b Fu(is)166 3435 y(called)g(the)h(symmetry)g(class)g(of)f(tensors)i(de\014ned)h(b)m(y)f Ff(r)p Fu(.)47 b(If)33 b Ft(e)h Fu(is)g(a)f(generating)h(idemp)s(oten)m (t)166 3552 y(of)d(the)h(righ)m(t)e(ideal)g Ff(r)h Fu(then)h(a)f (tensor)h Ft(T)45 b Fu(b)s(elongs)31 b(to)g(the)h(symmetry)f(class)h Fs(T)3044 3567 y Fg(r)3107 3552 y Fu(if)e(and)h(only)g(if)166 3668 y Ft(eT)42 b Fu(=)27 b Ft(T)14 b Fu(.)266 3784 y(No)m(w)33 b(w)m(e)h(ha)m(v)m(e)g(the)f(follo)m(wing)e(prop)s(osition:)42 b(If)33 b Ft(e)g Fu(is)f(an)h(idemp)s(oten)m(t.)43 b(Then)34 b(a)f(tensor)g Ft(T)166 3900 y Fu(ful\014lls)28 b Ft(eT)42 b Fu(=)27 b Ft(T)44 b Fu(i\013)29 b(all)f(group)i(ring)e(elemen)m(ts)j Ft(T)1974 3915 y Fl(b)2038 3900 y Fu(of)f Ft(T)44 b Fu(lie)28 b(in)h(the)i(left)e(ideal)f Ff(l)i Fu(generated)h(b)m(y)166 4017 y(the)i(idemp)s(oten)m(t)f Ft(e)894 3980 y Fj(\003)934 4017 y Fu(.)266 4133 y(F)-8 b(urther)28 b(w)m(e)h(ha)m(v)m(e)h(a)e (condition)e(whic)m(h)j(guaran)m(tees)g(that)f(the)h(left)e(ideal)g Ff(l)h Fu(is)g(the)g(smallest)166 4249 y(linear)e(subspace)j(whic)m(h)f (con)m(tains)g(all)d Ft(T)1673 4264 y Fl(b)1735 4249 y Fu(of)i(tensors)h(b)s(eing)f(considered.)43 b(Our)27 b(tensors)i(ha)m(v)m(e)166 4365 y Ft(r)36 b Fu(indices)d(and)h(are)f (de\014ned)i(on)e(a)g(v)m(ector)i(space)f Ft(V)22 b Fu(.)46 b(If)33 b(the)h(dimension)e(of)h Ft(V)55 b Fu(is)33 b(greater)g(or)166 4481 y(equal)c(to)g Ft(r)j Fu(then)d(ev)m(ery)i(left)d(ideal)g Ff(l)h Fu(is)g(the)g(span)h(of)e(all)f Ft(T)2318 4496 y Fl(b)2382 4481 y Fu(of)i(tensors)h(whic)m(h)g(b)s(elong)e(to)g(the) 166 4598 y(symmetry)33 b(class)g(of)f(the)h(righ)m(t)e(ideal)g Ff(l)1626 4562 y Fj(\003)1666 4598 y Fu(.)266 4714 y(Th)m(us)j(the)f (subspace)h Ft(W)47 b Fu(is)32 b(a)g(left)g(ideal)f Ff(l)h Fu(in)g(this)g(simple)f(case.)1693 5024 y(10.)48 b Fx(Slide)266 5199 y Fu(The)36 b(structure)h(of)f(the)g(space)h Ft(W)49 b Fu(for)35 b(sums)1982 5124 y Fk(P)2087 5228 y Fl(b)p Fj(2)p Fg(B)2225 5240 y Fh(b)2252 5255 y Fi(0)2312 5199 y Ft(T)2369 5214 y Fl(b)2439 5199 y Fu(is)g(more)g(complicated.)52 b(First)35 b(I)166 5332 y(ha)m(v)m(e)f(to)e(de\014ne)i(some)e (concepts.)266 5448 y(If)i Ft(\025)g Fu(is)g(a)f(partition)g(of)g Ft(r)k Fu(of)d(length)g Ft(k)j Fu(and)d(if)f(\()p Ft(v)2108 5463 y Fn(1)2148 5448 y Ft(;)17 b(:)g(:)g(:)e(;)i(v)2413 5463 y Fl(k)2456 5448 y Fu(\))34 b(a)g Ft(k)s Fu(-tuple)f(of)h(v)m (ectors)i(then)f(I)166 5564 y(denote)c(b)m(y)g(the)f(sym)m(b)s(ol)g Fs(h)p Ft(\025)p Fu(;)17 b Ft(v)1295 5579 y Fn(1)1334 5564 y Ft(;)g(:)g(:)g(:)e(;)i(v)1599 5579 y Fl(k)1642 5564 y Fs(i)30 b Fu(that)f Ft(r)s Fu(-tuple)g(of)h(v)m(ectors)h(that)f (con)m(tains)g Ft(\025)3328 5579 y Fn(1)3398 5564 y Fu(copies)p eop %%Page: 6 6 6 5 bop 166 251 a Fp(6)1482 b(B.)26 b(FIEDLER)166 500 y Fu(of)32 b Ft(v)324 515 y Fn(1)396 500 y Fu(then)h Ft(\025)675 515 y Fn(2)747 500 y Fu(copies)g(of)f Ft(v)1193 515 y Fn(2)1265 500 y Fu(and)h(so)g(on.)588 663 y Fs(h)p Ft(\025)p Fu(;)17 b Ft(v)775 678 y Fn(1)814 663 y Ft(;)g(:)g(:)g(:)e(;) i(v)1079 678 y Fl(k)1122 663 y Fs(i)83 b Fu(:=)g(\()p Ft(v)1515 678 y Fn(1)1554 663 y Ft(;)17 b(:)g(:)g(:)f(;)h(v)1820 678 y Fn(1)1468 719 y Fk(|)p 1513 719 107 12 v 107 w({z)p 1710 719 V 107 w(})1626 807 y Fl(\025)1667 816 y Fi(1)1859 663 y Ft(;)g(v)1950 678 y Fn(2)1990 663 y Ft(;)g(:)g(:)g(:)e(;)i(v)2255 678 y Fn(2)1903 719 y Fk(|)p 1948 719 V 107 w({z)p 2145 719 V 107 w(})2061 807 y Fl(\025)2102 816 y Fi(2)2295 663 y Ft(;)g(:)g(:)g(:)f(;)h(v)2561 678 y Fl(k)2603 663 y Ft(;)g(:)g(:)g(:)f(;)h(v)2869 678 y Fl(k)2513 719 y Fk(|)p 2558 719 110 12 v 110 w({z)p 2758 719 V 110 w(})2673 807 y Fl(\025)2714 819 y Fh(k)2911 663 y Fu(\))28 b Fs(2)g Ft(V)3150 622 y Fl(r)3204 663 y Ft(:)166 958 y Fu(F)-8 b(or)47 b(ev)m(ery)i Ft(b)k Fu(=)g(\()p Ft(v)936 973 y Fn(1)976 958 y Ft(;)17 b(:)g(:)g(:)f(;)h(v)1242 973 y Fl(r)1279 958 y Fu(\))53 b Fs(2)h Ft(V)1568 922 y Fl(r)1606 958 y Fu(,)d(there)d(exists)h(a)e(unique)h(partition)d Ft(\025)53 b Fs(`)g Ft(r)d Fu(and)d(a)166 1074 y(p)s(erm)m(utation)c Ft(q)50 b Fs(2)e(S)1003 1089 y Fl(r)1085 1074 y Fu(suc)m(h)d(that)f Ft(b)g Fu(can)g(b)s(e)g(written)g(as)g Ft(b)j Fu(=)g Ft(q)t Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)2910 1089 y Fn(1)2948 1074 y Ft(;)g(:)g(:)g(:)f(;)h(w)3237 1089 y Fj(j)p Fl(\025)p Fj(j)3321 1074 y Fs(i)44 b Fu(where)166 1190 y Ft(w)236 1205 y Fn(1)275 1190 y Ft(;)17 b(:)g(:)g(:)f(;)h(w)564 1206 y Fj(j)p Fl(\025)p Fj(j)680 1190 y Fu(are)33 b(the)g(pairwise)f (di\013eren)m(t,)h(suitably)f(ren)m(um)m(b)s(ered)h(v)m(ectors)i(from)c Ft(b)p Fu(.)266 1306 y(Here)43 b(y)m(ou)h(see)f(an)g(example.)73 b(This)43 b(is)g(an)f(arbitrary)g(7-tuple)g(of)g(v)m(ectors,)47 b(this)42 b(is)g(the)166 1423 y(sym)m(b)s(ol)29 b Fs(h)p Ft(\025)p Fu(;)17 b Ft(a;)g(b;)g(c)p Fs(i)29 b Fu(and)h(this)f(is)h (the)g(partition)e Ft(\025)h Fu(in)g(this)h(case.)43 b(W)-8 b(e)30 b(ha)m(v)m(e)i(3)d Ft(a)p Fu(,)i(2)e Ft(b)h Fu(and)g(2)g Ft(c)p Fu(.)266 1539 y(W)-8 b(e)33 b(call)d Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)822 1554 y Fn(1)861 1539 y Ft(;)g(:)g(:)g(:)f(;)h(w)1150 1554 y Fj(j)p Fl(\025)p Fj(j)1234 1539 y Fs(i)32 b Fu(a)g Fr(gr)-5 b(ouping)32 b Fu(of)g Ft(b)h Fu(and)g Ft(\025)f Fu(the)h Fr(gr)-5 b(ouping)35 b(p)-5 b(artition)32 b Fu(of)g Ft(b)p Fu(.)1693 1753 y(11.)48 b Fx(Slide)266 1927 y Fu(Let's)37 b(tak)m(e)g(a)f(lo)s (ok)f(at)g(the)i(space)g Ft(W)50 b Fu(for)36 b(sums)2127 1853 y Fk(P)2232 1957 y Fl(b)p Fj(2)p Fg(B)2370 1969 y Fh(b)2397 1984 y Fi(0)2457 1927 y Ft(T)2514 1942 y Fl(b)2548 1927 y Fu(.)55 b(W)-8 b(e)36 b(consider)h Ft(r)s Fu(-times)d(co-)166 2061 y(v)-5 b(arian)m(t)26 b(tensors)i Ft(T)41 b Fu(from)25 b(a)i(certain)f(symmetry)i(class)f(whic)m(h)g(con) m(tain)g Ft(l)i Fu(index)e(con)m(tractions)166 2177 y(and)k(free)g (indices)f(c)m(haracterized)h(b)m(y)h(a)e(v)m(ector)i(tuple)e Ft(b)2240 2192 y Fn(0)2280 2177 y Fu(.)43 b(Then)32 b(the)f(smallest)e (subspace)j Ft(W)166 2293 y Fu(whic)m(h)h(con)m(tains)g(all)d(sums)1208 2218 y Fk(P)1313 2322 y Fl(b)p Fj(2)p Fg(B)1451 2334 y Fh(b)1478 2349 y Fi(0)1538 2293 y Ft(T)1595 2308 y Fl(b)1662 2293 y Fu(of)i(suc)m(h)i(tensors)g(is)e(describ)s(ed)i(b)m(y) f(this)f(form)m(ula.)351 2518 y Ft(W)69 b Fu(=)56 b Fs(L)713 2533 y Fe(K)768 2408 y Fk(n)865 2424 y(X)835 2636 y Fl(b)p Fj(2)p Fg(B)973 2648 y Fh(b)1000 2663 y Fi(0)1055 2518 y Ft(T)1112 2533 y Fl(b)1175 2404 y Fk(\014)1175 2463 y(\014)1175 2523 y(\014)1235 2518 y Ft(T)42 b Fs(2)28 b(T)1482 2533 y Fg(l)1501 2514 y Fd(\003)1542 2408 y Fk(o)1691 2518 y Fu(=)1880 2424 y Fk(X)1850 2636 y Fl(\025)p Fj(2)p Fn(\003)1987 2648 y Fh(b)2014 2663 y Fi(0)2236 2424 y Fk(X)2086 2639 y Fj(h)p Fl(\025)p Fn(;)p Fl(w)2225 2649 y Fh(i)2252 2639 y Fj(i2M)2410 2651 y Fh(b)2437 2666 y Fi(0)2472 2651 y(;)p Fh(\025)2547 2518 y Ft(a)2598 2534 y Fj(h)p Fl(\025)p Fn(;)p Fl(w)2737 2544 y Fh(i)2763 2534 y Fj(i)2817 2518 y Fs(\001)22 b Fu(1)2916 2533 y Fj(H)2976 2541 y Fh(t)3000 2559 y(\025)3071 2518 y Fs(\001)f Fm(K)d Fu([)p Fs(S)3279 2533 y Fl(r)3323 2518 y Fu(])23 b Fs(\001)e Ft(e)389 2800 y Fs(\017)42 b Ft(e)32 b Fu(is)g(an)g(idemp)s (oten)m(t)f(that)h(c)m(haracterizes)i(the)f(symmetry)f(class)g(of)g (the)h(tensors)g Ft(T)14 b Fu(.)389 2916 y Fs(\017)42 b Fu(\003)549 2931 y Fl(b)579 2940 y Fi(0)650 2916 y Fu(is)32 b(the)h(set)g(of)f(all)f(grouping)g(partitions)g(of)h(all)f(v) m(ector)i(tuples)g(from)e Ff(B)3311 2931 y Fl(b)3341 2940 y Fi(0)3380 2916 y Fu(.)389 3033 y Fs(\017)42 b Fu(F)-8 b(or)32 b(ev)m(ery)i Ft(\025)28 b Fs(2)g Fu(\003)1160 3048 y Fl(b)1190 3057 y Fi(0)1262 3033 y Fu(w)m(e)33 b(form)f(the)h(lexicographically)d(smallest)h(Y)-8 b(oung)32 b(tableau)g Ft(t)3607 3048 y Fl(\025)481 3149 y Fu(of)38 b Ft(\025)i Fu(and)f(the)h(group)f Fs(H)1433 3164 y Fl(t)1458 3176 y Fh(\025)1542 3149 y Fu(of)g(the)h(horizon)m(tal)e(p)s(erm)m (utations)g(of)h Ft(t)3056 3164 y Fl(\025)3101 3149 y Fu(.)64 b(1)3241 3164 y Fj(H)3301 3172 y Fh(t)3325 3189 y(\025)3413 3149 y Fu(is)38 b(the)481 3265 y(sum)1562 3428 y(1)1611 3443 y Fj(H)1671 3451 y Fh(t)1695 3468 y(\025)1827 3428 y Fu(=)2018 3333 y Fk(X)1986 3544 y Fl(s)p Fj(2H)2126 3552 y Fh(t)2150 3570 y(\025)2211 3428 y Ft(s)481 3704 y Fu(of)32 b(the)h(elemen)m(ts)g(of)f Fs(H)1354 3719 y Fl(t)1379 3731 y Fh(\025)1424 3704 y Fu(.)389 3820 y Fs(\017)42 b Fu(W)-8 b(e)28 b(can)h(de\014ne)h(a)e (partial)e(ordering)i(for)g(groupings)g(of)g(a)g(\014xed)h(v)m(ector)h (tuple)e Ft(b)p Fu(.)43 b(F)-8 b(or)481 3937 y(instance,)37 b(the)f(tuple)g(\()p Ft(a;)17 b(b;)g(a;)g(a;)g(c;)g(c;)g(b)p Fu(\))35 b(has)h(the)h(t)m(w)m(o)f(groupings)f(\()p Ft(a;)17 b(a;)g(a;)g(b;)g(b;)g(c;)g(c)p Fu(\))481 4053 y(and)31 b(\()p Ft(a;)17 b(a;)g(a;)g(c;)g(c;)g(b;)g(b)p Fu(\).)43 b(W)-8 b(e)32 b(call)d(\()p Ft(a;)17 b(a;)g(a;)g(b;)g(b;)g(c;)g(c)p Fu(\))31 b(smaller)f(than)h(\()p Ft(a;)17 b(a;)g(a;)g(c;)g(c;)g(b;)g(b) p Fu(\))481 4169 y(since)33 b(the)g(tuple)f(\()p Ft(a;)17 b(b;)g(c)p Fu(\))32 b(is)g(lexicographically)e(smaller)g(than)j(\()p Ft(a;)17 b(c;)g(b)p Fu(\).)389 4285 y Fs(\017)42 b(M)601 4300 y Fl(b)631 4309 y Fi(0)665 4300 y Fl(;\025)760 4285 y Fu(is)30 b(the)h(set)h(of)e(suc)m(h)i(v)m(ector)f Ft(r)s Fu(-tuples)f(whic)m(h)h(are)g(the)g(smallest)e(groupings)g(of)481 4402 y(the)k Ft(b)28 b Fs(2)g Ff(B)900 4417 y Fl(b)930 4426 y Fi(0)1001 4402 y Fu(with)33 b(grouping)e(partition)f Ft(\025)p Fu(.)389 4518 y Fs(\017)42 b Fu(F)-8 b(or)31 b(ev)m(ery)i(v)m(ector)g Ft(r)s Fu(-tuple)e Ft(b)h Fu(w)m(e)h(can)f (\014nd)g(a)f(p)s(erm)m(utation)g Ft(p)2806 4533 y Fl(b)2872 4518 y Fu(suc)m(h)i(that)f Ft(b)g Fu(can)g(b)s(e)481 4634 y(written)c(in)f(the)i(form)d Ft(p)1365 4649 y Fl(b)1400 4634 y Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)1610 4649 y Fn(1)1648 4634 y Ft(;)g(:)g(:)g(:)f(;)h(w)1937 4650 y Fj(j)p Fl(\025)p Fj(j)2021 4634 y Fs(i)28 b Fu(where)h Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)2575 4649 y Fn(1)2614 4634 y Ft(;)g(:)g(:)g(:)e(;)i(w) 2902 4650 y Fj(j)p Fl(\025)p Fj(j)2986 4634 y Fs(i)28 b Fu(is)g(the)g(smallest)481 4750 y(grouping)39 b(of)g Ft(b)p Fu(.)66 b(Ob)m(viously)40 b Ft(p)1662 4765 y Fl(b)1737 4750 y Fu(is)f(not)h(unique)h(in)e(general.)65 b(W)-8 b(e)41 b(assume)f(that)g(a)481 4866 y(mapping)29 b Ff(p)h Fu(is)g(giv)m(en)h(whic)m(h)g(assigns)g(exactly)g(one)g(suc)m(h)i(p)s (erm)m(utation)c Ft(p)3246 4881 y Fl(b)3311 4866 y Fu(to)h(ev)m(ery)481 4983 y(v)m(ector)j Ft(r)s Fu(-tuple)f Ft(b)p Fu(.)389 5099 y Fs(\017)42 b Fu(If)d(an)h(ab)s(o)m(v)m(e)g(mapping)e Ff(p)h Fu(has)h(b)s(een)g(de\014ned)h(then)g(w)m(e)f(form)e(suc)m(h)j (a)f(group)f(ring)481 5215 y(elemen)m(t)1271 5378 y Ft(a)1322 5393 y Fj(h)p Fl(\025)p Fn(;)p Fl(w)1461 5403 y Fh(i)1488 5393 y Fj(i)1602 5378 y Fu(:=)1957 5283 y Fk(X)1934 5484 y Fh(b)p Fd(2)p Fc(B)2058 5501 y Fh(b)2085 5516 y Fi(0)1798 5572 y Fc(p)p Fi(\()p Fh(b)p Fi(\))1906 5558 y Fd(\000)p Fi(1)1989 5572 y Fh(b)11 b Fi(=)f Fd(h)p Fh(\025)p Fi(;)p Fh(w)2208 5588 y(i)2234 5572 y Fd(i)2286 5378 y Ff(p)p Fu(\()p Ft(b)p Fu(\))2453 5337 y Fj(\000)p Fn(1)p eop %%Page: 7 7 7 6 bop 1113 251 a Fp(NORMAL)33 b(F)n(ORMS)f(OF)h(VECTOR)h(ANAL)-6 b(YSIS)906 b(7)481 500 y Fu(for)30 b(ev)m(ery)j(smallest)d(grouping)g Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)1873 515 y Fl(i)1900 500 y Fs(i)27 b(2)h(M)2180 515 y Fl(b)2210 524 y Fi(0)2245 515 y Fl(;\025)2310 500 y Fu(.)43 b(The)32 b(sum)f(runs)h(through)f (the)g(set)481 616 y(of)h(all)e(suc)m(h)k(v)m(ector)g Ft(r)s Fu(-tuples)e(that)g(ha)m(v)m(e)i(the)f(smallest)e(grouping)h Fs(h)p Ft(\025)p Fu(;)17 b Ft(w)3203 631 y Fl(i)3230 616 y Fs(i)p Fu(.)1693 854 y(12.)48 b Fx(Slide)266 1028 y Fu(No)m(w)33 b(I)f(will)e(explain)h(ho)m(w)i(the)g(form)m(ula)d(for)i (the)h(space)g Ft(W)46 b Fu(from)31 b(the)i(last)e(transparency)166 1144 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b(a)f(second)h(algorithm)c(of)i(m)m(y)h(Habilita-)166 2657 y(tionssc)m(hrift\))38 b(and)g(solv)m(e)g(the)h(linear)d(equation) i(system)h(for)f(the)g(co)s(e\016cien)m(ts)i Ft(x)3220 2672 y Fl(p)3298 2657 y Fu(of)e(linear)166 2773 y(iden)m(tities.)266 2890 y(In)27 b(the)h(case)g(of)e(large)h(symmetric)f(groups)h Fs(S)1923 2905 y Fl(r)1989 2890 y Fu(computer)g(calculations)e(in)h (the)i(group)f(ring)166 3006 y Fm(K)18 b Fu([)p Fs(S)325 3021 y Fl(r)369 3006 y Fu(])k(lead)g(to)f(hard)i(problems)e(with)h (resp)s(ect)h(to)f(time)f(and)h(memory)-8 b(.)39 b(I)22 b(use)h(discrete)g(F)-8 b(ourier)166 3122 y(transforms,)51 b(whic)m(h)e(transform)d(group)i(ring)f(elemen)m(ts)h(in)m(to)f(blo)s (c)m(k)g(diagonal)f(matrices.)166 3238 y(Since)39 b(all)d(calculations) h(can)h(b)s(e)h(carried)f(out)h(separately)f(for)g(ev)m(ery)j(single)c (blo)s(c)m(k)h(matrix)166 3354 y(discrete)g(F)-8 b(ourier)37 b(transforms)g(reduce)i(the)f(computational)d(problems)i(to)g(a)g(high) g(degree.)166 3471 y(F)-8 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