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y(If)j(there)f(is)h(a)h(lo)q(op)g Fr(p)d Ft(:)g Fr(a)f Fq(\000)-30 b(!)14 b Fr(a)i Ft(then)g(the)g(lo)q (op)h(n)o(um)o(b)q(er)d(of)j Fr(a)f Ft(is)g Fq(j)p Fr(pp)p Fq(j)11 b(\000)g(j)p Fr(p)p Fq(j)j Ft(=)g(1.)0 2004 y(Let)i Fr(a;)8 b(b)13 b Fq(2)h(X)7 b Ft(.)22 b(W)l(e)15 b(sho)o(w)i(that)g (the)f(lo)q(op)h(n)o(um)o(b)q(ers)e(of)h Fr(a)g Ft(and)h(of)g Fr(b)f Ft(are)g(equal.)21 b(W)l(e)16 b(get)0 2074 y Fw(Lemma)11 b(4.2.)17 b Fm(The)d(lo)n(op)g(numb)n(er)g(is)g(c)n(onstant)h(on)f(any) g(str)n(ongly)g(c)n(onne)n(cte)n(d)h Fq(X)7 b Fm(,)15 b(so)f(the)g(lo)n(op)g(numb)n(er)0 2133 y(of)j(a)h(str)n(ongly)f(c)n (onne)n(cte)n(d)h Fq(X)25 b Fm(is)17 b(a)g(wel)r(l)j(de\014ne)n(d)e (numb)n(er)g Fq(L)p Ft(\()p Fq(X)7 b Ft(\))p Fm(.)0 2220 y(Pr)n(o)n(of.)19 b Ft(Let)f Fr(p)262 2202 y Fh(0)291 2220 y Ft(:)e Fr(a)g Fq(\000)-30 b(!)16 b Fr(b)i Ft(and)g Fr(q)597 2202 y Fh(0)625 2220 y Ft(:)e Fr(b)g Fq(\000)-30 b(!)17 b Fr(a)g Ft(b)q(e)h(w)o(alks.)26 b(Then)18 b Fr(p)1189 2202 y Fh(0)1201 2220 y Fr(pq)1249 2202 y Fh(0)1261 2220 y Fr(;)8 b(p)1307 2202 y Fh(0)1319 2220 y Fr(q)r(q)1367 2202 y Fh(0)1394 2220 y Ft(:)16 b Fr(b)g Fq(\000)-30 b(!)16 b Fr(b)i Ft(are)g(closed)f(w)o(alks)0 2278 y(with)g Fq(j)p Fr(p)150 2260 y Fh(0)162 2278 y Fr(pq)210 2260 y Fh(0)221 2278 y Fq(j)12 b(\000)f(j)p Fr(p)335 2260 y Fh(0)347 2278 y Fr(q)r(q)395 2260 y Fh(0)405 2278 y Fq(j)k Ft(=)g Fr(t)p Ft(,)i(so)g(the)g(lo)q(op)h(n)o(um)o(b)q(er)d(of)i Fr(b)g Ft(is)g(less)g(than)g(or)g(equal)g(to)g(the)g(lo)q(op)h(n)o(um)o (b)q(er)0 2336 y(of)f Fr(a)p Ft(.)j(By)c(symmetry)d(the)j(lo)q(op)h(n)o (um)o(b)q(er)d(is)i(constan)o(t)h(on)g Fq(X)7 b Ft(.)721 b Ff(\003)0 2423 y Fw(Lemma)17 b(4.3.)k Fm(L)n(et)e(the)g(lo)n(op)f (numb)n(er)i(of)e Fq(X)26 b Fm(b)n(e)19 b Fr(t)p Fm(.)26 b(L)n(et)19 b Fr(p)1106 2405 y Fh(0)1134 2423 y Ft(:)d Fr(a)1190 2430 y Fp(i)1220 2423 y Fq(\000)-29 b(!)17 b Fr(a)1323 2430 y Fp(j)1359 2423 y Fm(and)i Fr(q)1479 2405 y Fh(0)1507 2423 y Ft(:)d Fr(a)1563 2430 y Fp(i)1593 2423 y Fq(\000)-29 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2688 y Fh(0)505 2709 y Fr(q)r Fq(j)i Ft(=)g Fq(j)p Fr(p)645 2688 y Fh(0)657 2709 y Fq(j)e Ft(+)g Fq(j)p Fr(p)p Fq(j)h(\000)e(j)p Fr(q)882 2688 y Fh(0)893 2709 y Fq(j)h(\000)g(j)p Fr(q)r Fq(j)i Ft(=)h Fr(r)q(t)d Ft(+)g Fr(s)g Fq(\000)g Fr(t)i Ft(=)h(\()p Fr(r)e Fq(\000)f Ft(1\))p Fr(t)g Ft(+)g Fr(s:)p eop %%Page: 6 6 6 5 bop 0 -47 a Fk(6)224 b(OMAR)13 b(COL)465 -57 y(\023)460 -47 y(ON-REYES,)g(REINHARD)g(LA)o(UBENBA)o(CHER,)e(AND)i(BODO)g(P)m (AREIGIS)0 50 y Ft(Hence)22 b(there)g(are)h(w)o(alks)g Fr(p)535 32 y Fh(00)557 50 y Fr(;)8 b(q)603 32 y Fh(00)649 50 y Ft(:)24 b Fr(a)713 57 y Fp(i)752 50 y Fq(\000)-30 b(!)25 b Fr(a)862 57 y Fp(j)903 50 y Ft(with)e Fq(j)p Fr(p)1059 32 y Fh(00)1081 50 y Fq(j)15 b(\000)h(j)p Fr(q)1203 32 y Fh(00)1223 50 y Fq(j)25 b Ft(=)h Fr(s)p Ft(.)41 b(Let)23 b Fr(p)1522 32 y Fh(\003)1568 50 y Ft(:)i Fr(a)1633 57 y Fp(j)1676 50 y Fq(\000)-30 b(!)25 b Fr(a)1786 57 y Fp(i)1822 50 y Ft(b)q(e)f(a)0 108 y(w)o(alk.)30 b(Then)20 b Fq(j)p Fr(p)311 90 y Fh(\003)331 108 y Fr(p)355 90 y Fh(00)376 108 y Fq(j)13 b(\000)g(j)p Fr(p)493 90 y Fh(\003)513 108 y Fr(q)537 90 y Fh(00)558 108 y Fq(j)19 b Ft(=)g Fr(s)g Ft(=)g(0)h(b)q(ecause)f(of)h(the)f(minimali)o(t)o(y)d (of)k(the)f(lo)q(op)h(n)o(um)o(b)q(er)d Fr(t)p Ft(.)31 b(So)0 166 y Fq(j)p Fr(p)38 148 y Fh(0)50 166 y Fq(j)11 b(\000)g(j)p Fr(q)163 148 y Fh(0)174 166 y Fq(j)i(2)h Ft(\()p Fr(t)p Ft(\).)1563 b Ff(\003)0 264 y Fw(Corollary)23 b(4.4.)f Fm(L)n(et)f(the)h(lo)n(op)e(numb)n(er)i(of)f Fq(X)28 b Fm(b)n(e)21 b Fr(t)p Fm(.)33 b(L)n(et)21 b Fr(p)g Ft(:)f Fr(a)h Fq(\000)-29 b(!)20 b Fr(a)h Fm(b)n(e)g(a)g(close)n (d)h(walk.)34 b(Then)0 322 y Fq(j)p Fr(p)p Fq(j)14 b(2)g Ft(\()p Fr(t)p Ft(\))p Fm(.)0 420 y(Pr)n(o)n(of.)19 b Ft(Im)c(Lemma)f(4.3)i(tak)o(e)g Fr(p)h Ft(and)g Fr(pp)p Ft(.)1101 b Ff(\003)0 517 y Fw(Prop)r(osition)20 b(4.5.)h Fm(L)n(et)e(the)h(lo)n(op)f(numb)n(er)h(of)f Fq(X)27 b Fm(b)n(e)20 b Fr(t)d Fq(\025)g Ft(1)p Fm(.)29 b(F)l(or)19 b(e)n(ach)g Fr(a;)8 b(b)17 b Fq(2)h(X)26 b Fm(ther)n(e)19 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y Fq(\031)e Fr(a)658 1828 y Fp(i)671 1821 y Ft(.)1196 b Ff(\003)0 1919 y Fw(Lemma)16 b(4.7.)j Fm(Ther)n(e)f(ar)n(e)e(exactly) j Fr(t)e Fm(lo)n(op)g(e)n(quivalenc)n(e)j(classes)f(in)e Fq(X)7 b Fm(.)0 2016 y(Pr)n(o)n(of.)19 b Ft(Since)14 b(there)g(are)h(w)o(alks)f(starting)h(in)g Fr(a)871 2023 y Fp(i)899 2016 y Ft(for)g(all)f(lengths)h Fq(\025)f Ft(0,)g(tak)o(e)h(a)g(w)o(alk)f Fr(a)1592 2023 y Fp(i)1619 2016 y Fq(\000)-30 b(!)14 b Fr(a)1718 2023 y Fp(i)p Fn(+1)1790 2016 y Fq(\000)-30 b(!)14 b Fr(:)8 b(:)g(:)0 2075 y Fq(\000)-30 b(!)13 b Fr(a)98 2082 y Fp(i)p Fn(+)p Fp(t)169 2075 y Ft(of)j(length)g Fr(t)p Ft(.)21 b(The)16 b Fr(a)552 2082 y Fp(i)p Fn(+)p Fp(j)610 2075 y Ft(,)f Fr(j)i Ft(=)d(0)p Fr(;)8 b(:)g(:)g(:)g(;)g(t)i Fq(\000)h Ft(1)17 b(are)f(in)g(di\013eren) o(t)f(equiv)m(alence)g(classes)740 2158 y(\026)740 2165 y(\026)-24 b Fr(a)766 2172 y Fp(i)779 2165 y Fr(;)802 2158 y Ft(\026)802 2165 y(\026)f Fr(a)827 2172 y Fp(i)p Fn(+1)886 2165 y Fr(;)8 b(:)g(:)g(:)f(;)996 2158 y Ft(\026)996 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Fp(i)463 1868 y Fj(1)482 1856 y Fm(,)18 b Fr(y)539 1863 y Fn(2)572 1856 y Ft(=)c Fr(x)652 1863 y Fp(i)664 1868 y Fj(1)681 1863 y Fn(+1)739 1856 y Fq(\001)d Fr(:)d(:)g(:)i Fq(\001)h Fr(x)885 1863 y Fp(i)897 1868 y Fj(2)916 1856 y Fm(,)18 b Fr(:)8 b(:)g(:)o Fm(,)17 b Fr(y)1062 1863 y Fp(t)1091 1856 y Ft(=)d Fr(x)1171 1863 y Fp(i)1183 1868 y Fi(t)p Fe(\000)p Fj(1)1236 1863 y Fn(+1)1294 1856 y Fq(\001)d Fr(:)d(:)g(:)i Fq(\001)h Fr(x)1440 1863 y Fp(i)1452 1867 y Fi(t)1468 1856 y Fm(.)0 1914 y(F)l(urthermor)n(e)481 1975 y Fr(f)510 1954 y Fp(mt)p Fn(+1)616 1975 y Ft(=)i(\()p Fr(y)710 1982 y Fn(2)730 1975 y Fr(;)8 b(:)g(:)g(:)f(;)h(y)863 1982 y Fn(2)686 2003 y Fd(|)p 708 2003 54 6 v 54 w({z)p 806 2003 V 54 w(})717 2045 y Fp(s)733 2050 y Fj(1)769 2045 y Fc(times)883 1975 y Fr(;)g(y)929 1982 y Fn(3)948 1975 y Fr(;)g(:)g(:)g(:)f(;)h(y) 1081 1982 y Fn(3)904 2003 y Fd(|)p 926 2003 V 54 w({z)p 1024 2003 V 54 w(})944 2045 y Fp(s)960 2050 y Fj(2)978 2045 y Fc(times)1101 1975 y Fr(;)g(:)g(:)g(:)f(;)h(y)1234 1982 y Fn(1)1253 1975 y Fr(;)g(:)g(:)g(:)g(;)g(y)1387 1982 y Fn(1)1210 2003 y Fd(|)p 1232 2003 V 54 w({z)p 1330 2003 V 54 w(})1243 2045 y Fp(s)1259 2049 y Fi(t)1290 2045 y Fc(times)1406 1975 y Ft(\))p Fr(:)953 2103 y Fm(.)953 2120 y(.)953 2137 y(.)398 2209 y Fr(f)427 2189 y Fp(mt)p Fn(+)p Fp(j)531 2209 y Ft(=)14 b(\()p Fr(y)626 2216 y Fp(j)r Fn(+1)689 2209 y Fr(;)8 b(:)g(:)g(:)f(;)h(y)822 2216 y Fp(j)r Fn(+1)602 2241 y Fd(|)p 624 2241 97 6 v 97 w({z)p 765 2241 V 97 w(})676 2284 y Fp(s)692 2289 y Fj(1)727 2284 y Fc(times)885 2209 y Fr(;)g(y)931 2216 y Fp(j)r Fn(+2)994 2209 y Fr(;)g(:)g(:)g(:)f(;)h(y)1127 2216 y Fp(j)r Fn(+2)907 2241 y Fd(|)p 929 2241 V 97 w({z)p 1070 2241 V 97 w(})990 2284 y Fp(s)1006 2289 y Fj(2)1024 2284 y Fc(times)1190 2209 y Fr(;)g(:)g(:)g(:)g(;)g(y)1324 2216 y Fp(j)1342 2209 y Fr(;)g(:)g(:)g(:)f(;)h(y)1475 2216 y Fp(t)1300 2241 y Fd(|)p 1322 2241 51 6 v 51 w({z)p 1417 2241 V 51 w(})1329 2284 y Fp(s)1345 2288 y Fi(t)1377 2284 y 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Fm(has)f(at)h(le)n(ast)g(one)h(non-)g(zer)n(o)0 2498 y(diagonal)e(entry.)0 2592 y(Pr)n(o)n(of.)h Ft(W)l(e)i(\014rst)g (pro)o(v)o(e)f(that)i(the)e(lo)q(op)i(n)o(um)o(b)q(er)e(of)h Fq(X)28 b Ft(is)20 b(the)h(greatest)h(common)c(divisor)j(of)g(the)0 2651 y(lengths)d(of)g(all)f(circuits)f(of)i Fq(X)7 b Ft(.)25 b(The)18 b(problem)e(in)h(pro)o(ving)h(this)f(is,)h(that)g(the) f(lo)q(op)i(n)o(um)o(b)q(er)c(cannot)0 2709 y(b)q(e)h(represen)o(ted)f (in)h(general)g(as)h(the)f(di\013erence)f(of)i(the)f(lengths)g(of)h(t)o (w)o(o)f(circuits.)p eop %%Page: 9 9 9 8 bop 512 -49 a Fk(BOOLEAN)16 b(MONOMIAL)g(D)o(YNAMICAL)g(SYSTEMS)494 b(9)0 50 y Ft(Observ)o(e)12 b(that)i(a)f(closed)g(w)o(alk)g Fr(p)h Ft(can)f(b)q(e)g(decomp)q(osed)g(in)o(to)g(a)g(n)o(um)o(b)q(er)f (of)h(circuits)f Fr(p)1585 57 y Fn(1)1605 50 y Fr(;)c(:)g(:)g(:)g(;)g (p)1739 57 y Fp(l)1765 50 y Ft(sharing)0 108 y(v)o(ertices.)0 166 y(Let)20 b Fr(d)h Ft(b)q(e)f(the)g(greatest)h(common)d(divisor)i 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y(greatest)16 b(common)d(divisor)i(of)h(the)f(n)o (um)o(b)q(ers)f Fr(i)p Ft(,)h(suc)o(h)h(that)g Fr(A)1168 1285 y Fp(i)1197 1303 y Ft(has)g(at)g(least)f(one)h(non-zero)g (diagonal)0 1361 y(en)o(try)f(is)h(the)g(greatest)h(common)d(divisor)i (of)g(the)g(lengths)h(of)f(all)g(circuits)f(of)h Fq(X)7 b Ft(.)368 b Ff(\003)827 1467 y Ft(5.)24 b Fs(Glueing)0 1554 y Fw(De\014nition)18 b(5.1.)j Ft(Let)c Fq(X)23 b Ft(and)18 b Fq(Y)j Ft(b)q(e)c(dep)q(endency)g(graphs)h(of)f(functions)g Fr(f)j Ft(:)15 b Fo(F)1527 1536 y Fp(r)1527 1566 y Fn(2)1565 1554 y Fq(\000)-30 b(!)15 b Fo(F)1666 1536 y Fp(r)1666 1566 y Fn(2)1706 1554 y Ft(and)i Fr(g)g Ft(:)e Fo(F)1897 1536 y Fp(s)1897 1566 y Fn(2)0 1612 y Fq(\000)-30 b(!)14 b Fo(F)101 1594 y Fp(s)101 1625 y Fn(2)123 1612 y Ft(,)j(resp)q(ectiv)o (ely)l(.)j(A)c Fm(glueing)k Fq(X)7 b Ft(#)p Fq(Y)22 b Fm(of)17 b Fq(Y)22 b Fm(to)c Fq(X)24 b Ft(consists)17 b(of)g(a)g(digraph)g(with)g(v)o(ertices)e Fr(V)1798 1619 y Fh(X)1840 1606 y Ft(_)1830 1612 y Fq([)p Fr(V)1891 1619 y Fh(Y)0 1670 y Ft(and)k(edges)g Fr(E)266 1677 y Fh(X)308 1664 y Ft(_)298 1670 y Fq([)p Fr(E)367 1677 y Fh(Y)414 1670 y Ft(\(disjoin)o(t)f(union\),)h(together)f(with)h(a)g (set)f(of)h(additional)f(directed)g(edges)g(from)0 1729 y(v)o(ertices)c(in)i Fq(Y)21 b Ft(to)c(v)o(ertices)d(in)i Fq(X)7 b Ft(.)0 1787 y(The)16 b(function)g(of)h Fq(X)7 b Ft(#)p Fq(Y)20 b Ft(is)c(denoted)g(b)o(y)g Fr(f)5 b Ft(#)p Fr(g)16 b Ft(:)d Fo(F)949 1766 y Fp(r)q Fn(+)p Fp(s)949 1799 y Fn(2)1028 1787 y Fq(\000)-30 b(!)14 b Fo(F)1128 1766 y Fp(r)q Fn(+)q Fp(s)1128 1799 y Fn(2)1194 1787 y Ft(.)0 1856 y(Observ)o(e)21 b(that)h Fq(X)28 b Ft(is)22 b(a)g(subgraph)h(of)e Fq(X)7 b Ft(#)p Fq(Y)26 b Ft(and)c(that)g Fr(f)28 b Ft(:)23 b Fo(F)1216 1838 y Fp(r)1216 1868 y Fn(2)1262 1856 y Fq(\000)-30 b(!)23 b Fo(F)1371 1838 y Fp(r)1371 1868 y Fn(2)1415 1856 y Ft(is)f(a)g(quotien)o(t)f(system)f(of)0 1914 y Fr(f)5 b Ft(#)p Fr(g)16 b Ft(:)d Fo(F)164 1894 y Fp(r)q Fn(+)p Fp(s)164 1926 y Fn(2)243 1914 y Fq(\000)-30 b(!)14 b 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Fq(2)0 2638 y Fo(F)27 2617 y Fp(r)q Fn(+)q Fp(s)27 2650 y Fn(2)93 2638 y Ft(.)28 b(Let)19 b Fr(m)e Fq(2)h Fo(N)h Ft(b)q(e)g(suc)o(h)f(that)h(\()p Fr(f)5 b Ft(#)p Fr(g)r Ft(\))812 2620 y Fp(m)845 2638 y Ft(\()p Fr(\013;)j(\014)s Ft(\))17 b(=)h(\()p Fr(f)5 b Ft(#)p Fr(g)r Ft(\))1173 2620 y Fp(m)p Fn(+1)1251 2638 y Ft(\()p Fr(\013;)j(\014)s Ft(\).)27 b(Then)19 b(w)o(e)f(ha)o(v)o(e)g Fr(f)1762 2620 y Fp(m)1795 2638 y Ft(\()p Fr(\013)p Ft(\))g(=)0 2696 y Fr(f)29 2678 y Fp(m)p Fn(+1)108 2696 y Ft(\()p Fr(\013)p Ft(\))e(so)h(that)g Fr(f)k Ft(is)16 b(a)h(\014xed)f(p)q(oin)o (t)g(system.)981 b Ff(\003)p eop %%Page: 10 10 10 9 bop 0 -47 a Fk(10)205 b(OMAR)13 b(COL)465 -57 y(\023)460 -47 y(ON-REYES,)g(REINHARD)g(LA)o(UBENBA)o(CHER,)e(AND)i(BODO)g(P)m (AREIGIS)0 50 y Fw(Theorem)h(5.3.)k Fm(L)n(et)e Fq(X)22 b Fm(b)n(e)16 b(a)f(\014xe)n(d)i(p)n(oint)e(system)g(and)h(let)h Fq(Y)j Fm(b)n(e)c(str)n(ongly)f(c)n(onne)n(cte)n(d)i(of)e(lo)n(op)g (length)0 108 y Fq(\025)f Ft(1)p Fm(.)22 b(L)n(et)17 b Fq(X)7 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y(times)i(and)j(get)f(that)h Fr(x)440 290 y Fp(i)454 283 y Fr(f)i Ft(=)14 b(\()p Fr(x)596 290 y Fp(i)610 283 y Fr(f)634 290 y Fn(1)653 283 y Fr(;)8 b(:)g(:)g(:)g(;)g(x)791 290 y Fp(i)805 283 y Fr(f)829 290 y Fp(n)852 283 y Ft(\))16 b(is)g(a)h(\014xed)f(p)q(oin)o(t)g (system.)498 b Ff(\003)815 381 y Fs(References)0 460 y Fv([A)o(O])41 b(Alb)q(ert,)14 b(R.,)e(and)h(Othmer,)g(H.)g(G.,)f(The) i(top)q(ology)f(of)g(the)h(regulatory)f(in)o(teractions)h(predicts)h (the)f(expression)127 510 y(pattern)h(of)e(the)h(segmen)o(t)g(p)q (olarit)o(y)f(genes)i(in)e(Drosophila)g(melanogaster,)f Fl(J.)i(The)n(or.)g(Biol)g Fb(223)p Fv(,)f(2003,)f(1{18.)0 560 y([BR])43 b(Barrett,C.L.,)10 b(and)h(Reidys,)f(C.M.,)g(Elemen)o(ts) g(of)g(a)h(Theory)g(of)e(Sim)o(ulation,)f(I:)i(Sequen)o(tial)g(CA)h(Ov) o(er)g(Random)127 610 y(Graphs,)i Fl(Appl.)i(Math.)g(and)g(Comput.)f Fb(98)p Fv(,)f(1999,)g(241{259.)0 660 y([BMR1])20 b(Barrett,)e(C.L.,)d (Mortv)o(eit,)i(H.S.,)f(and)g(Reidys,C.M.,)f(Elemen)o(ts)h(of)g(a)g 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Fb(CT-6)p Fv(,)f(1959,)g (45{60.)0 1108 y([G])70 b(Germ)o(undsson,)13 b(R.,)h(Gunnarsson,)g(J.,) g(and)g(Plan)o(tin,)g(J.,)g(Sym)o(b)q(olic)e(algebraic)i(discrete)i (systems{applied)e(to)127 1158 y(the)j(JAS39)f(\014gh)o(ter)h (aircraft,)g(tec)o(hnical)f(rep)q(ort,)i(Link\177)-21 b(oping)15 b(Univ)o(ersit)o(y)m(,)h(Link\177)-21 b(oping,)16 b(Sw)o(eden,)h(Decem)o(b)q(er,)127 1207 y(1994.)0 1257 y([H])72 b(Hern\023)-21 b(andez)15 b(T)m(oledo,)e(Ren)o(\023)-20 b(e)14 b(A,)g(Linear)f(Finite)h(Dynamical)d(Systems,)j(preprin)o(t,)g (2003.)0 1307 y([K])71 b(Kau\013man,)11 b(S.A.,)h(Metab)q(olic)h (stabilit)o(y)f(and)h(epigenesis)h(in)f(randomly)e(constructed)k (genetic)f(nets,)g Fl(J.)g(The)n(or.)127 1357 y(Biol.)f Fb(22)p Fv(,)g(1969,)g(437{467.)0 1407 y([KN])40 b(Khrennik)o(o)o(v,)13 b(A.,)g(and)g(Nilsson,)g(M.,)g(On)g(the)i(n)o(um)o(b)q(er)d(of)h (cycles)i(of)e Fa(p)p Fv(-adic)g(dynamical)e(systems,)i Fl(J.)h(Numb)n(er)127 1457 y(The)n(ory)f Fb(90)p Fv(,)h(2001,)e (255-264.)0 1506 y([L])77 b(Laub)q(en)o(bac)o(her,)14 b(R.,)f(A)h(computer)f(algebra)h(approac)o(h)f(to)h(biological)e (systems,)h(Pro)q(c.)h(In)o(tl.)f(Symp.)f(Sym)o(b)q(olic)127 1556 y(and)i(Alg.)e(Comp.,)g(Philadelphia,)g(Asso)q(c.)i(Comp.)e(Mac)o (h.,)h(2003.)0 1606 y([LP1])28 b(Laub)q(en)o(bac)o(her,)13 b(R.,)d(and)h(P)o(areigis,)h(B.,)f(Equiv)n(alence)h(relations)f(on)g (\014nite)h(dynamical)d(systems,)j Fl(A)n(dv.)g(Applie)n(d)127 1656 y(Math.)i Fb(26)p Fv(,)f(2001,237{251.)0 1706 y([LP2])28 b(Laub)q(en)o(bac)o(her,)12 b(R.,)f(and)g(P)o(areigis,)g(B.,)h(Decomp)q (osition)d(and)i(sim)o(ulation)e(of)i(sequen)o(tial)g(dynamical)e (systems,)127 1755 y Fl(A)n(dv.)14 b(Applie)n(d)h(Math.)f Fb(30)p Fv(,)f(2003,)g(655{678.)0 1805 y([LS])54 b(Laub)q(en)o(bac)o (her,)13 b(R.,)f(and)h(Stigler,)f(B.,)g(A)h(computational)d(algebra)i (approac)o(h)h(to)g(the)g(rev)o(erse-)i(engineering)e(of)127 1855 y(gene)h(regulatory)g(net)o(w)o(orks,)g(preprin)o(t,)g(2003.)0 1905 y([LN])46 b(Lidl,)16 b(R.,)g(and)h(Niederreiter,)i(H.,)d Fl(Finite)i(Fields)p Fv(,)e(Encyclop)q(edia)i(of)e(Math)g(and)h(its)g (Appl.)f Fb(20)p Fv(,)h(Cam)o(bridge)127 1955 y(Univ)o(ersit)o(y)d (Press,)h(London,)e(1997.)0 2004 y([M1])44 b(Marc)o(hand,)17 b(H.,)g(and)f(LeBorgne,)i(M.,)f(On)g(the)g(optimal)e(con)o(trol)h(of)g (p)q(olynomial)e(dynamical)g(systems)j(o)o(v)o(er)127 2054 y Fb(Z)p Fa(=p)p Fv(,)c Fl(F)m(ourth)i(workshop)g(on)g(Discr)n (ete)f(Event)i(Systems)p Fv(,)d(Cagliari,)f(Italy)m(,)g(IEEE,)i(1998.)0 2104 y([M2])44 b(Marc)o(hand,)22 b(H.,)f(and)f(LeBorgne,)i(M.,)g(P)o (artial)d(order)i(discrete)h(ev)o(en)o(t)f(systems)g(mo)q(deled)e(as)h (p)q(olynomial)127 2154 y(dynamical)11 b(systems,)j(IEEE)g(In)o(tl.)f (Conf.)g(on)h(Con)o(trol)f(Applications,)g(T)m(rieste,)h(Italy)m(,)e (1998.)0 2204 y([MW])22 b(Milligan,)11 b(D.K.,)h(and)i(Wilson,)e (M.J.D.,)g(The)i(b)q(eha)o(viour)g(of)f(a\016ne)h(sequen)o(tial)g(Bo)q (olean)g(net)o(w)o(orks,)f Fl(Conne)n(c-)127 2254 y(tion)i(Scienc)n(e)f Fb(5)p Fv(,)g(1993,)e(153{167.)0 2303 y([W])60 b(W)m(olfram,)10 b(S.,)j Fl(Cel)r(lular)h(A)o(utomata)h(and)g(Complexity)p Fv(,)e(W)m(estview)h(Press,)h(1994.)0 2397 y Fu(Vir)o(ginia)g(Bioinf)o (orma)m(tics)g(Institute,)g(Vir)o(ginia)g(Tech,)h(Bla)o(cksbur)o(g,)h (V)-5 b(A)15 b(24061-0477,)g(USA)0 2481 y(Ma)m(thema)m(tisches)i (Institut)d(der)i(Universit)836 2478 y(\177)835 2481 y(at)f(M)942 2478 y(\177)941 2481 y(unchen,)j(Germany)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF