%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: ln41.dvi %%CreationDate: Mon Jun 26 16:19:47 2000 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: DVIPS16 -a ln41 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.06.26:1522 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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1667 y Fl(:)0 1751 y Fq(Th)o(us)17 b Fl(\014)h Fq(and)f Fl(\014)296 1733 y Fk(\000)p Fi(1)359 1751 y Fq(are)f(m)o(utually)e(in)o(v)o(erse)g (homomorphisms.)p 1765 1751 2 33 v 1767 1719 30 2 v 1767 1751 V 1796 1751 2 33 v 75 1837 a(Since)k Fl(H)249 1819 y Fk(\003)288 1837 y Fq(=)h(Hom)n(\()p Fl(H)q(;)47 b Fq(\))20 b(and)f Fl(S)j Fq(:)d Fl(H)k Fg(\000)-31 b(!)19 b Fl(H)24 b Fq(is)19 b(an)g(algebra)h(an)o(tihomomorphism,)c(the)0 1895 y(dual)g Fl(H)152 1877 y Fk(\003)189 1895 y Fq(is)g(an)g Fl(H)t Fq(-mo)q(dule)g(in)g(four)h(di\013eren)o(t)e(w)o(a)o(ys:)316 1972 y Fg(h)p Fq(\()p Fl(f)20 b(*)13 b(a)p Fq(\))p Fl(;)8 b(g)r Fg(i)14 b Fq(:=)g Fg(h)p Fl(a;)8 b(g)r(f)d Fg(i)p Fl(;)119 b Fg(h)p Fq(\()p Fl(a)14 b(\()g(f)5 b Fq(\))p Fl(;)j(g)r Fg(i)14 b Fq(:=)g Fg(h)p Fl(a;)8 b(f)d(g)r Fg(i)p Fl(;)316 2030 y Fg(h)p Fq(\()p Fl(f)20 b(+)13 b(a)p Fq(\))p Fl(;)8 b(g)r Fg(i)14 b Fq(:=)g Fg(h)p Fl(a;)8 b(S)s Fq(\()p Fl(f)d Fq(\))p Fl(g)r Fg(i)p Fl(;)48 b Fg(h)p Fq(\()p Fl(a)14 b(\))g(f)5 b Fq(\))p 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Fo(Prop)r(osition)17 b(4.1.19.)24 b Fm(L)n(et)17 b Fl(H)22 b Fm(b)n(e)c(a)g(\014nite)h(dimensional)g (Hopf)f(algebr)n(a.)24 b(Then)19 b Fl(H)1685 2522 y Fk(\003)1722 2540 y Fm(is)f(a)0 2598 y(right)f(Hopf)h(mo)n(dule)f(over)h Fl(H)t Fm(.)p eop %%Page: 124 9 124 8 bop 0 118 a Fr(124)534 b(4.)17 b(THE)f(INFINITESIMAL)g(THEOR)m(Y) 75 213 y Ff(Pr)o(oof.)j Fl(H)311 195 y Fk(\003)349 213 y Fq(is)f(a)g(left)f Fl(H)573 195 y Fk(\003)593 213 y Fq(-mo)q(dule)g(b)o(y)g(left)g(m)o(ultiplic)o(ation)e(hence)i(b)o(y)h (2.1.1)g(a)g(righ)o(t)f Fl(H)t Fq(-)0 271 y(como)q(dule)12 b(b)o(y)h Fl(\016)r Fq(\()p Fl(a)p Fq(\))g(=)432 233 y Fj(P)485 285 y Fh(i)507 271 y Fl(b)528 253 y Fk(\003)528 283 y Fh(i)548 271 y Fl(a)5 b Fg(\012)g Fl(b)644 278 y Fh(i)656 271 y Fq(.)21 b(Let)13 b Fl(f)s(;)8 b(g)15 b Fg(2)f Fl(H)k Fq(and)c Fl(a;)8 b(b)13 b Fg(2)h Fl(H)1232 253 y Fk(\003)1252 271 y Fq(.)20 b(The)13 b(\(left\))f(m)o (ultiplication)0 329 y(of)17 b Fl(H)100 311 y Fk(\003)136 329 y Fq(satis\014es)670 402 y 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Fq(\))p Fl(dx)328 1884 y(f)19 b Fq(=)423 1847 y Fj(P)484 1884 y Fl(S)517 1866 y Fk(\000)p Fi(1)564 1884 y Fq(\()p Fl(\016)605 1892 y Fi(\(1\))651 1884 y Fq(\))p Fg(h)699 1871 y Fj(e)689 1884 y Fl(f)6 b(;)i(\016)763 1892 y Fi(\(2\))810 1884 y Fg(i)52 b Fl(a)14 b Fq(=)973 1847 y Fj(P)1025 1884 y Fg(h)1044 1844 y Fj(R)1068 1901 y Fi(\(1\))1115 1884 y Fl(;)7 b Fj(e)-27 b Fl(a)p Fg(i)1190 1844 y Fj(R)1214 1901 y Fi(\(2\))1269 1884 y Fl(:)0 1828 y Fq(\(16\))75 1999 y Ff(Pr)o(oof.)19 b Fq(W)l(e)i(use)f(the)h(isomorphisms)d Fl(H)26 b Fg(\000)-30 b(!)21 b Fl(H)1031 1981 y Fk(\003)1071 1999 y Fq(de\014ned)g(b)o(y)1325 1985 y Fj(b)1316 1999 y Fl(f)26 b Fq(:=)1448 1985 y Fj(e)1439 1999 y Fl(f)h Fq(=)21 b(\()1568 1958 y Fj(R)1622 1999 y Fl(\()g(f)5 b Fq(\))22 b(=)0 2021 y Fj(P)53 2058 y Fg(h)72 2018 y Fj(R)95 2075 y Fi(\(1\))143 2058 y Fl(;)8 b(f)d Fg(i)221 2018 y Fj(R)245 2075 y Fi(\(2\))308 2058 y Fq(and)17 b Fl(H)447 2040 y Fk(\003)481 2058 y Fg(\000)-30 b(!)13 b Fl(H)21 b Fq(de\014ned)16 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867 y Fg(i)14 b Fq(=)g Fg(h)1000 854 y Fj(e)991 867 y Fl(f)j Fg(\003)11 b Fj(e)-28 b Fl(g)r(;)8 b(h)p Fg(i)p Fl(:)0 971 y Fq(F)l(rom)15 b(\(22\))i(w)o(e)f(get)382 960 y Fj(e)384 971 y Fq(1)408 978 y Fh(H)456 971 y Fq(=)508 931 y Fj(R)541 971 y Fq(.)21 b(So)c(w)o(e)f(ha)o(v)o(e)837 958 y Fj(e)828 971 y Fl(f)j Fq(=)925 958 y Fj(f)923 971 y Fq(1)p Fl(f)h Fq(=)1041 960 y Fj(e)1042 971 y Fq(1)12 b Fg(\003)1123 958 y Fj(e)1114 971 y Fl(f)19 b Fq(=)1209 931 y Fj(R)1250 971 y Fg(\003)1284 958 y Fj(e)1275 971 y Fl(f)6 b Fq(.)p 1765 971 2 33 v 1767 940 30 2 v 1767 971 V 1796 971 2 33 v 75 1079 a(If)15 b Fl(G)i Fq(is)f(a)h(\014nite)f (Ab)q(elian)f(group)i(and)g Fl(a;)8 b(b)13 b Fg(2)h Fl(H)980 1061 y Fk(\003)1014 1079 y Fq(=)1113 1053 y Fi(^)1105 1061 y Fh(G)1134 1079 y Fq(.)21 b(Then)503 1177 y(\()p Fl(a)11 b Fg(\003)g Fl(b)p Fq(\)\()p Fl(\026)p Fq(\))j(=)g Fg(j)p Fl(G)p Fg(j)834 1156 y Fk(\000)p Fi(1)955 1130 y Fj(X)889 1244 y Fh(\037;\025)p Fk(2)973 1236 y Fi(^)966 1244 y Fh(G;\037\025)p 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Fq(One)j(of)h(the)f(most)g(imp)q (ortan)o(t)f(form)o(ulas)h(for)h(F)l(ourier)f(transforms)g(is)g(the)g (Planc)o(herel)f(for-)0 1611 y(m)o(ula)g(on)h(the)g(in)o(v)m(ariance)g (of)g(the)g(inner)g(pro)q(duct)h(under)f(F)l(ourier)g(transforms.)21 b(W)l(e)16 b(ha)o(v)o(e)75 1705 y Fo(Theorem)g(4.1.29.)23 b Fm(\(The)18 b(Plancher)n(el)h(formula\))712 1802 y Fg(h)p Fl(a;)8 b(f)d Fg(i)15 b Fq(=)e Fg(h)922 1789 y Fj(e)912 1802 y Fl(f)6 b(;)i(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)p Fl(:)-1088 b Fq(\(25\))75 1899 y Ff(Pr)o(oof.)19 b Fq(First)d(w)o(e)g(ha)o(v)o(e)f(from)g(\(16\))96 1997 y Fg(h)p Fl(a;)8 b(f)d Fg(i)17 b Fq(=)280 1959 y Fj(P)333 1997 y Fg(h)352 1956 y Fj(R)385 2014 y Fi(\(1\))432 1997 y Fl(;)7 b Fj(e)-27 b Fl(a)p Fg(ih)518 1956 y Fj(R)552 2014 y Fi(\(2\))599 1997 y Fl(;)8 b(S)654 1978 y Fk(\000)p Fi(1)701 1997 y Fq(\()p Fl(\016)742 2004 y Fi(\(1\))789 1997 y Fq(\))p Fg(ih)855 1983 y Fj(e)846 1997 y Fl(f)e(;)i(\016)920 2004 y Fi(\(2\))966 1997 y Fg(i)14 b Fq(=)1051 1959 y Fj(P)1104 1997 y Fg(h)1123 1956 y Fj(R)1156 1997 y Fl(;)7 b Fj(e)-27 b Fl(aS)1237 1978 y Fk(\000)p Fi(1)1284 1997 y Fq(\()p Fl(\016)1325 2004 y Fi(\(1\))1372 1997 y Fq(\))p Fg(ih)1438 1983 y Fj(e)1429 1997 y Fl(f)6 b(;)i(\016)1503 2004 y Fi(\(2\))1549 1997 y Fg(i)228 2071 y Fq(=)280 2034 y Fj(P)333 2071 y Fg(h)352 2031 y Fj(R)385 2071 y Fl(;)g(S)440 2053 y Fk(\000)p Fi(1)487 2071 y Fq(\()p Fl(\016)528 2079 y Fi(\(1\))575 2071 y Fq(\))p Fl(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(ih)732 2058 y Fj(e)723 2071 y Fl(f)6 b(;)i(\016)797 2079 y Fi(\(2\))843 2071 y Fg(i)15 b Fq(=)928 2034 y Fj(P)981 2071 y Fg(h)1000 2031 y Fj(R)1033 2071 y Fl(;)8 b(S)1088 2053 y Fk(\000)p Fi(1)1135 2071 y Fq(\()p Fl(S)s Fq(\()p Fl(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\)\))p Fl(\016)1338 2079 y Fi(\(1\))1385 2071 y Fq(\))p Fg(ih)1451 2058 y Fj(e)1442 2071 y Fl(f)6 b(;)i(\016)1516 2079 y Fi(\(2\))1562 2071 y Fg(i)228 2138 y Fq(=)280 2100 y Fj(P)333 2138 y Fg(h)352 2098 y Fj(R)385 2138 y Fl(;)g(S)440 2120 y Fk(\000)p Fi(1)487 2138 y Fq(\()p Fl(\016)528 2146 y Fi(\(1\))575 2138 y Fq(\))p Fg(ih)642 2125 y Fj(e)632 2138 y Fl(f)e(;)i(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fl(\016)797 2146 y Fi(\(2\))843 2138 y Fg(i)15 b Fq(=)928 2100 y Fj(P)981 2138 y Fg(h)1000 2098 y Fj(R)1033 2138 y Fl(;)8 b(S)1088 2120 y Fk(\000)p Fi(1)1135 2138 y Fq(\()p Fl(\016)r Fq(\))1197 2146 y Fi(\(2\))1244 2138 y Fg(ih)1292 2125 y Fj(e)1282 2138 y Fl(f)e(;)i(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fl(S)s Fq(\()p Fl(S)1510 2120 y Fk(\000)p Fi(1)1557 2138 y Fq(\()p Fl(\016)r Fq(\))1619 2146 y Fi(\(1\))1665 2138 y Fq(\))p Fg(i)228 2204 y Fq(=)14 b Fg(h)299 2164 y Fj(R)333 2204 y Fl(;)8 b(S)388 2186 y Fk(\000)p Fi(1)435 2204 y Fq(\()p Fl(\016)r Fq(\))p Fg(ih)544 2191 y Fj(e)535 2204 y Fl(f)d(;)j(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)p Fl(:)0 2298 y Fq(Apply)15 b(this)h(to)h Fg(h)320 2258 y Fj(R)354 2298 y Fl(;)8 b(\016)r Fg(i)p Fq(.)21 b(Then)16 b(w)o(e)g(get)154 2404 y(1)e(=)g Fg(h)263 2364 y Fj(R)296 2404 y Fl(;)8 b(\016)r Fg(i)14 b Fq(=)g Fg(h)446 2364 y Fj(R)479 2404 y Fl(;)8 b(S)534 2383 y Fk(\000)p Fi(1)581 2404 y Fq(\()p Fl(\016)r Fq(\))p Fg(ih)682 2391 y Fj(e)681 2404 y Fl(\016)r(;)g(\027)s Fq(\()776 2385 y Fj(e)773 2364 y(R)806 2404 y Fq(\))p Fg(i)14 b Fq(=)g Fg(h)929 2364 y Fj(R)962 2404 y Fl(;)8 b(S)1017 2383 y Fk(\000)p Fi(1)1064 2404 y Fq(\()p Fl(\016)r Fq(\))p Fg(i)p Fl("\027)1195 2383 y Fk(\000)p Fi(1)1242 2404 y Fl(\027)s Fq(\(1\))15 b(=)f Fg(h)1417 2364 y Fj(R)1450 2404 y Fl(;)8 b(S)1505 2383 y Fk(\000)p Fi(1)1552 2404 y Fq(\()p Fl(\016)r Fq(\))p Fg(i)p Fl(:)0 2504 y Fq(Hence)15 b(w)o(e)h(get)g Fg(h)p Fl(a;)8 b(f)d Fg(i)14 b Fq(=)g Fg(h)508 2491 y Fj(e)498 2504 y Fl(f)6 b(;)i(\027)s Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)p Fl(:)p 1765 2504 V 1767 2473 30 2 v 1767 2504 V 1796 2504 2 33 v 75 2598 a Fo(Corollary)18 b(4.1.30.)23 b Fm(If)17 b Fl(H)22 b Fm(is)17 b(unimo)n(dular)g(then)h Fl(\027)g Fq(=)13 b Fl(S)1163 2580 y Fi(2)1183 2598 y Fm(.)p eop %%Page: 130 15 130 14 bop 0 118 a Fr(130)534 b(4.)17 b(THE)f(INFINITESIMAL)g(THEOR)m (Y)75 213 y Ff(Pr)o(oof.)j Fl(H)i Fq(unimo)q(dular)15 b(means)h(that)g Fl(\016)i Fq(is)e(left)f(and)i(righ)o(t)f(in)o(v)m (arian)o(t.)21 b(Th)o(us)16 b(w)o(e)g(get)195 296 y Fg(h)p Fl(a;)8 b(f)d Fg(i)17 b Fq(=)379 259 y Fj(P)432 296 y Fg(h)451 256 y Fj(R)484 313 y Fi(\(1\))531 296 y Fl(;)7 b Fj(e)-27 b Fl(a)p Fg(ih)617 256 y Fj(R)651 313 y Fi(\(2\))698 296 y Fl(;)8 b(S)753 278 y Fk(\000)p Fi(1)800 296 y Fq(\()p Fl(\016)841 304 y Fi(\(1\))888 296 y Fq(\))p Fg(ih)954 283 y Fj(e)945 296 y Fl(f)e(;)i(\016)1019 304 y Fi(\(2\))1065 296 y Fg(i)327 371 y Fq(=)379 333 y Fj(P)432 371 y Fg(h)451 330 y Fj(R)484 371 y Fl(;)f Fj(e)-27 b Fl(aS)565 353 y Fk(\000)p Fi(1)612 371 y Fq(\()p Fl(\016)653 378 y Fi(\(1\))699 371 y Fq(\))p Fg(ih)766 357 y Fj(e)756 371 y Fl(f)7 b(;)h(\016)831 378 y Fi(\(2\))877 371 y Fg(i)14 b Fq(=)962 333 y Fj(P)1015 371 y Fg(h)1034 330 y Fj(R)1067 371 y Fl(;)8 b(S)1122 353 y Fk(\000)p Fi(1)1169 371 y Fq(\()p Fl(\016)1210 378 y Fi(\(1\))1257 371 y Fl(S)s Fq(\()o Fj(e)-27 b Fl(a)o Fq(\)\))p Fg(ih)1420 357 y Fj(e)1410 371 y Fl(f)6 b(;)i(\016)1484 378 y Fi(\(2\))1531 371 y Fg(i)327 437 y Fq(=)379 400 y Fj(P)432 437 y Fg(h)451 397 y Fj(R)484 437 y Fl(;)g(S)539 419 y Fk(\000)p Fi(1)586 437 y Fq(\()p Fl(\016)627 445 y Fi(\(1\))674 437 y Fq(\))p Fg(ih)741 424 y Fj(e)731 437 y Fl(f)e(;)i(\016)805 445 y Fi(\(2\))852 437 y Fl(S)885 419 y Fi(2)904 437 y Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)49 b Fq(\()16 b(since)g Fl(\016)h Fq(is)f(righ)o(t)g(in)o(v)m(arian)o(t\))327 504 y(=)e Fg(h)398 464 y Fj(R)432 504 y Fl(;)8 b(S)487 486 y Fk(\000)p Fi(1)534 504 y Fq(\()p Fl(\016)r Fq(\))p Fg(ih)643 491 y Fj(e)634 504 y Fl(f)d(;)j(S)718 486 y Fi(2)738 504 y Fq(\()o Fj(e)-27 b Fl(a)o Fq(\))p Fg(i)14 b Fq(=)g Fg(h)914 491 y Fj(e)905 504 y Fl(f)6 b(;)i(S)990 486 y Fi(2)1009 504 y Fq(\()o Fj(e)-27 b Fl(a)p Fq(\))p Fg(i)p Fl(:)0 585 y Fq(Hence)15 b Fl(S)178 567 y Fi(2)211 585 y Fq(=)f Fl(\027)s Fq(.)p 1765 585 2 33 v 1767 554 30 2 v 1767 585 V 1796 585 2 33 v 75 668 a(W)l(e)j(also)h(get)f(a)h(sp) q(ecial)f(represen)o(tation)g(of)h(the)f(inner)g(pro)q(duct)h Fl(H)1359 650 y Fk(\003)1391 668 y Fg(\012)11 b Fl(H)21 b Fg(\000)-31 b(!)72 b Fq(b)o(y)17 b(b)q(oth)0 726 y(in)o(tegrals:)75 813 y Fo(Corollary)h(4.1.31.)404 918 y Fg(h)p Fl(a;)8 b(f)d Fg(i)14 b Fq(=)585 850 y Fj(Z)642 918 y(e)-27 b Fl(a)o Fq(\()p Fl(x)p Fq(\))p Fl(f)5 b Fq(\()p Fl(x)p Fq(\))p Fl(dx)14 b Fq(=)948 850 y Fj(Z)998 863 y Fk(\003)1026 918 y Fl(S)1059 897 y Fk(\000)p Fi(1)1106 918 y Fq(\()p Fl(a)p Fq(\)\()p Fl(x)p Fq(\))1244 905 y Fj(e)1236 918 y Fl(f)t Fq(\()p Fl(x)p Fq(\))p Fl(dx:)-1397 b Fq(\(26\))75 1031 y Ff(Pr)o(oof.)19 b Fq(W)l(e)e(ha)o(v)o(e)f(the)h(rules)g(for)g (the)g(F)l(ourier)g(transform.)23 b(F)l(rom)16 b(\(18\))i(w)o(e)f(get)g Fg(h)p Fl(a;)8 b(f)d Fg(i)16 b Fq(=)0 1097 y Fg(h)19 1057 y Fj(R)53 1097 y Fl(;)7 b Fj(e)-27 b Fl(a)o(f)5 b Fg(i)19 b Fq(=)224 1057 y Fj(R)264 1097 y(e)-27 b Fl(a)p Fq(\()p Fl(x)p Fq(\))p Fl(f)5 b Fq(\()p Fl(x)p Fq(\))p Fl(dx)19 b Fq(and)g(from)f(\(17\))i Fg(h)p Fl(a;)8 b(f)d Fg(i)19 b Fq(=)g Fg(h)p Fl(S)1088 1079 y Fk(\000)p Fi(1)1135 1097 y Fq(\()p Fl(a)p Fq(\))1208 1084 y Fj(e)1199 1097 y Fl(f)5 b(;)j(\016)r Fg(i)18 b Fq(=)1368 1057 y Fj(R)1401 1070 y Fk(\003)1429 1097 y Fl(S)1462 1079 y Fk(\000)p Fi(1)1509 1097 y Fq(\()p Fl(a)p Fq(\)\()p Fl(x)p Fq(\))1647 1084 y Fj(e)1639 1097 y Fl(f)5 b Fq(\()p Fl(x)p Fq(\))p Fl(dx:)p 1765 1155 V 1767 1124 30 2 v 1767 1155 V 1796 1155 2 33 v 75 1238 a Fq(The)16 b(F)l(ourier)g(transform)h(leads)f(to)h (an)g(in)o(teresting)f(in)o(tegral)g(transform)g(on)h Fl(H)k Fq(b)o(y)16 b(double)0 1296 y(application.)75 1387 y Fo(Prop)r(osition)h(4.1.32.)24 b Fm(The)g(double)i(tr)n(ansform) 1066 1374 y Fq(\025)1056 1387 y Fl(f)31 b Fq(:=)26 b(\()p Fl(\016)i(\()f Fq(\()1353 1347 y Fj(R)1412 1387 y Fl(\()f(f)5 b Fq(\)\))25 b Fm(de\014nes)h(an)0 1445 y(automorphism)16 b Fl(H)j Fg(\000)-30 b(!)14 b Fl(H)22 b Fm(with)664 1534 y Fq(\025)654 1547 y Fl(f)5 b Fq(\()p Fl(y)r Fq(\))13 b(=)812 1479 y Fj(Z)870 1547 y Fl(f)5 b Fq(\()p Fl(x)p Fq(\))p Fl(\016)r Fq(\()p Fl(xy)r Fq(\))p Fl(dx:)75 1658 y Ff(Pr)o(oof.)19 b Fq(W)l(e)d(ha)o(v)o(e)290 1730 y Fg(h)p Fl(y)r(;)368 1717 y Fq(\025)357 1730 y Fl(f)6 b Fg(i)16 b Fq(=)e Fg(h)p Fl(y)r(;)8 b Fq(\()p Fl(\016)15 b(\()f Fq(\()679 1690 y Fj(R)726 1730 y Fl(\()f(f)5 b Fq(\)\))p Fg(i)15 b Fq(=)f Fg(h)p Fq(\()979 1690 y Fj(R)1026 1730 y Fl(\()g(f)5 b Fq(\))p Fl(y)r(;)j(\016)r Fg(i)422 1788 y Fq(=)474 1751 y Fj(P)527 1788 y Fg(h)p Fq(\()565 1748 y Fj(R)612 1788 y Fl(\()14 b(f)5 b Fq(\))p Fl(;)j(\016)767 1796 y Fi(\(1\))813 1788 y Fg(ih)p Fl(y)r(;)g(\016)921 1796 y Fi(\(2\))968 1788 y Fg(i)15 b Fq(=)1053 1751 y Fj(P)1106 1788 y Fg(h)1125 1748 y Fj(R)1159 1788 y Fl(;)8 b(f)d(\016)1232 1796 y Fi(\(1\))1279 1788 y Fg(ih)p Fl(y)r(;)j(\016) 1387 1796 y Fi(\(2\))1433 1788 y Fg(i)422 1846 y Fq(=)474 1809 y Fj(P)527 1846 y Fg(h)546 1806 y Fj(R)569 1863 y Fi(\(1\))617 1846 y Fl(;)g(f)d Fg(ih)706 1806 y Fj(R)730 1863 y Fi(\(2\))777 1846 y Fl(;)j(\016)821 1854 y Fi(\(1\))868 1846 y Fg(ih)p Fl(y)r(;)g(\016)976 1854 y Fi(\(2\))1023 1846 y Fg(i)14 b Fq(=)1108 1809 y Fj(P)1160 1846 y Fg(h)1179 1806 y Fj(R)1203 1863 y Fi(\(1\))1250 1846 y Fl(;)8 b(f)d Fg(ih)1339 1806 y Fj(R)1364 1863 y Fi(\(2\))1419 1846 y Fl(y)r(;)j(\016)r Fg(i)422 1912 y Fq(=)474 1875 y Fj(P)527 1912 y Fg(h)546 1872 y Fj(R)569 1930 y Fi(\(1\))617 1912 y Fl(;)g(f)d Fg(ih)706 1872 y Fj(R)730 1930 y Fi(\(2\))777 1912 y Fl(;)j Fq(\()p Fl(y)15 b(*)f(\016)r Fq(\))p Fg(i)g Fq(=)f Fg(h)1066 1872 y Fj(R)1100 1912 y Fl(;)8 b(f)d Fq(\()p Fl(y)16 b(*)d(\016)r Fq(\))p Fg(i)422 1979 y Fq(=)474 1938 y Fj(R)515 1979 y Fl(f)5 b Fq(\()p Fl(x)p Fq(\))p Fl(\016)r Fq(\()p Fl(xy)r Fq(\))p Fl(dx)0 2058 y Fq(since)15 b Fg(h)p Fl(x;)8 b Fq(\()p Fl(y)16 b(*)d(\016)r Fq(\))p Fg(i)h Fq(=)g Fg(h)p Fl(xy)r(;)8 b(\016)r Fg(i)p Fq(.)p 1765 2058 V 1767 2027 30 2 v 1767 2058 V 1796 2058 2 33 v eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF