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b(Do)q(es)24 b(this)g(class)g(ha)o(v)o(e)e(a)i(nice)f(c)o(haracterization?)43 b(Let)24 b(us)g(call)164 1709 y(suc)o(h)15 b(functions)g Fo(r)n(e)n(gularity)h(pr)n(eserving)p Fq(.)21 b(Not)15 b(all)g(functions)g(are)g(regularit)o(y)164 1770 y(preserving:)37 b(for)24 b(example,)g(log)9 b Fn(n)25 b Fq(is)f(not.)46 b(The)24 b(class)h(is)f(closed)g(under)164 1830 y(addition,)17 b(m)o(ultiplication,)d(exp)q(onen)o(tiation,)j(comp)q(osition,)g(and)h (con)o(tains)164 1890 y(arbitrarily)h(fast)h(gro)o(wing)h(functions,)f (including)f(highly)g(noncomputable)164 1950 y(ones.)237 2010 y(In)j(the)f(remainder)f(of)i(this)f(note)h(w)o(e)f(giv)o(e)g(t)o (w)o(o)h(c)o(haracterizations)f(of)164 2071 y(the)e(class)g(of)g (regularit)o(y-preserving)f(functions)h(in)g(terms)e(of)j(the)e (concept)164 2131 y(of)h Fo(ultimate)h(p)n(erio)n(dicity)p Fq(.)27 b(One)18 b(of)h(these)f(c)o(haracterizations)g(in)o(v)o(olv)o (es)e(t)o(w)o(o)164 2191 y(simple)e(indep)q(enden)o(t)h(conditions)i 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y(set)22 b Fn(A)p Fq(,)h(the)f(set)g(lengths\()p Fn(A)p Fq(\))f(is)h(u.p.;)i(and)f(for)f(an)o(y)g(u.p.)g(set)g Fn(U)5 b Fq(,)23 b(the)f(set)164 1553 y Fm(f)p Fn(x)c Fm(j)h(j)p Fn(x)p Fm(j)f(2)h Fn(U)5 b Fm(g)20 b Fq(is)f(regular.)30 b(In)19 b(particular,)g(if)g Fn(A)f Fq(a)i(set)f(of)h(strings)f(o)o(v)o (er)g(a)164 1614 y(single)d(letter)f(alphab)q(et,)h(then)g Fn(A)g Fq(is)g(regular)g(i\013)h(lengths\()p Fn(A)p Fq(\))f(is)g(u.p.) 164 1728 y Fd(De\014nition)i(2)24 b Fq(A)d(function)h Fn(f)28 b Fq(:)23 b Fe(N)g Fm(!)g Fe(N)g Fq(is)e(said)h(to)g Fo(pr)n(eserve)h(ultimate)164 1788 y(p)n(erio)n(dicity)c Fq(if)d Fn(f)478 1770 y Fh(\000)p Fj(1)526 1788 y Fq(\()p Fn(U)5 b Fq(\))16 b(is)g(u.p.)g(whenev)o(er)f Fn(U)21 b Fq(is.)488 b Fc(2)164 1902 y Fd(De\014nition)18 b(3)24 b Fq(A)19 b(function)g Fn(f)25 b Fq(:)19 b Fe(N)h Fm(!)f Fe(N)h Fq(is)f(said)h(to)g(b)q(e)g Fo(ultimately)h(p)n(eri-)164 1962 y(o)n(dic)h(mo)n(dulo)g Fn(m)f Fo(\(u.p.)h Fq(mo)q(d)f Fn(m)p Fo(\))h Fq(if)f(the)g(function)h 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Fq(=)41 b Fm(f)p Fn(a)441 404 y Fl(n)464 425 y Fn(b)14 b Fm(j)f(9)p Fn(y)18 b Fm(j)p Fn(y)r Fm(j)13 b Fq(=)h Fn(f)5 b Fq(\()p Fm(j)p Fn(a)803 404 y Fl(n)826 425 y Fn(b)p Fm(j)p Fq(\))16 b(and)h Fn(a)1017 404 y Fl(n)1040 425 y Fn(by)e Fm(2)f(f)p Fn(a)1198 404 y Fl(n)1221 425 y Fn(ba)1268 404 y Fl(k)1302 425 y Fm(j)g Fn(n)g Fm(\025)g Fq(0)p Fm(gg)311 497 y Fq(=)41 b Fm(f)p Fn(a)441 477 y Fl(n)464 497 y Fn(b)14 b Fm(j)f(9)p Fn(y)18 b Fm(j)p Fn(y)r Fm(j)13 b Fq(=)h Fn(f)5 b Fq(\()p Fn(n)11 b Fq(+)g(1\))17 b(and)g Fn(y)e Fq(=)f Fn(a)1124 477 y Fl(k)1145 497 y Fm(g)311 570 y Fq(=)41 b Fm(f)p Fn(a)441 549 y Fl(n)464 570 y Fn(b)14 b Fm(j)f Fn(k)j Fq(=)e Fn(f)5 b Fq(\()p Fn(n)11 b Fq(+)g(1\))p Fm(g)311 642 y Fq(=)41 b Fm(f)p Fn(a)441 622 y Fl(n)464 642 y Fn(b)14 b Fm(j)f Fn(n)f Fq(+)f(1)j Fm(2)g Fn(f)730 622 y Fh(\000)p Fj(1)777 642 y Fq(\()p Fm(f)p Fn(k)r Fm(g)p Fq(\))p Fm(g)i Fn(;)164 741 y Fq(and)h(b)o(y)f Fd(C1)g Fq(this)g(set)h(is)f(regular,)f (therefore)180 839 y(lengths\()p Fm(f)p 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