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[(of)-342.7(le)4.1(ngth)]TJ /F12 1 Tf 4.1454 0 TD 0 Tc (l)Tj /F6 1 Tf 0.6625 0 TD 0.0006 Tc [(are)-347.1(i)1.6(n)32.7(t)0(e)4.2(r)0(c)34.3(h)2.6(ange)4.2(able)]TJ -27.844 -1.1643 TD -0.0008 Tc [(if)-323.9(and)-330.1(o)-3.1(nly)-327.2(i)0.2(f)-334(t)-1.4(he)-328.5(equation)]TJ /F12 1 Tf 3.5834 -1.9272 TD 0 Tc (x)Tj /F14 1 Tf 7.9701 0 0 7.9701 137.52 511.38 Tm (1)Tj /F12 1 Tf 11.9552 0 0 11.9552 142.32 513.1801 Tm (C)Tj /F14 1 Tf 7.9701 0 0 7.9701 150.72 511.38 Tm (1)Tj /F4 1 Tf 11.9552 0 0 11.9552 158.04 513.1801 Tm 0.1739 Tc [()-52.1()-30.1()]TJ /F12 1 Tf 3.3926 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 205.2 511.38 Tm (M)Tj /F12 1 Tf 11.9552 0 0 11.9552 214.68 513.1801 Tm (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 223.08 511.38 Tm (M)Tj /F4 1 Tf 11.9552 0 0 11.9552 235.2 513.1801 Tm 0.1739 Tc [()-52.1()-30.1()]TJ /F12 1 Tf 3.3926 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 282.36 511.38 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 285.84 510.3 Tm (1)Tj /F12 1 Tf 11.9552 0 0 11.9552 290.52 513.1801 Tm (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 298.92 511.38 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 302.4 510.3 Tm (1)Tj /F4 1 Tf 11.9552 0 0 11.9552 309.6 513.1801 Tm 0.1739 Tc [()-52.1()-30.1()]TJ /F12 1 Tf 3.3926 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 356.76 511.38 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 360.24 510.3 Tm (2)Tj /F12 1 Tf 11.9552 0 0 11.9552 364.8 513.1801 Tm (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 373.2 511.38 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 376.68 510.3 Tm (2)Tj /F4 1 Tf 11.9552 0 0 11.9552 383.88 513.1801 Tm 0.1739 Tc [()-52.1()-30.1()]TJ /F12 1 Tf 3.3926 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 431.04 511.38 Tm (N)Tj /F12 1 Tf 11.9552 0 0 11.9552 439.2 513.1801 Tm (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 447.6 511.38 Tm (N)Tj /F6 1 Tf 11.9552 0 0 11.9552 458.88 513.1801 Tm 0.2823 Tc [(=0)-2048.7(\()281.7(1)280(\))]TJ -31.0159 -1.9372 TD 0.0003 Tc [(has)-326.1(a)-323.2(s)5.1(o)-2(lution)-329(w)4.9(i)1.3(th)]TJ /F12 1 Tf 8.4916 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 196.2 488.22 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 199.68 487.14 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 207.72 490.02 Tm (=)Tj /F12 1 Tf 1.0338 0 TD (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 226.68 488.22 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 230.16 487.14 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 238.2 490.02 Tm -0.0008 Tc [(=)-273.1(1)-334.3(and)-320(w)3.8(ith)-340.1(t)-1.4(he)-328.5(follo)27.1(wing)-324.3(prop)-28.9(erties:)]TJ -11.5532 -1.1643 TD 0.0015 Tc (\(1\))Tj /F4 1 Tf 1.7164 0 TD 0 Tc ({)Tj /F12 1 Tf 0.5019 0 TD (C)Tj /F14 1 Tf 7.9701 0 0 7.9701 135 474.3 Tm (1)Tj /F12 1 Tf 11.9552 0 0 11.9552 139.68 476.1 Tm 0.1697 Tc [(,...)10.1(C)]TJ /F5 1 Tf 7.9701 0 0 7.9701 169.08 474.3 Tm 0 Tc (M)Tj /F4 1 Tf 11.9552 0 0 11.9552 178.44 476.1 Tm (})Tj /F6 1 Tf 0.9736 0 TD -0.0017 Tc [(is)-468.7(the)-469.9(i)-0.7(n)30.4(t)-2.3(ersectio)-4(n)-481.5(o)-4(f)]TJ /F4 1 Tf 9.5957 0 TD 0 Tc (R)Tj /F5 1 Tf 7.9701 0 0 7.9701 314.88 474.3 Tm (<)Tj /F6 1 Tf 11.9552 0 0 11.9552 327.6 476.1 Tm -0.0008 Tc [(with)-480.6(an)-460.6(arbitrary)-467.7(but)-463.2(xed)-480.6(minimal)]TJ -20.0349 -1.1643 TD -0.0029 Tc [(cycle)-350.6(b)-0.9(a)-5.2(s)1.9(is,)-343.2(a)-5.2(nd)]TJ /F4 1 Tf 7.0161 0 TD 0 Tc ({)Tj /F12 1 Tf 0.5019 0 TD (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 186.36 460.38 Tm (M)Tj /F14 1 Tf 1.1293 0 TD 0.0017 Tc (+1)Tj /F12 1 Tf 11.9552 0 0 11.9552 206.64 462.18 Tm 0.1697 Tc [(,...)10.1(,C)]TJ /F5 1 Tf 7.9701 0 0 7.9701 241.32 460.38 Tm 0 Tc (N)Tj /F4 1 Tf 11.9552 0 0 11.9552 249.24 462.18 Tm (})Tj /F6 1 Tf 0.793 0 TD (=)Tj /F4 1 Tf 1.0439 0 TD (R)Tj /F14 1 Tf 7.9701 0 0 7.9701 281.28 460.38 Tm (=)Tj /F6 1 Tf 11.9552 0 0 11.9552 288.48 462.18 Tm -0.0023 Tc [(.)-453(T)2.3(he)-340(f)-4.2(a)-4.6(ct)-324.1(tha)-4.6(t)-334.2(instea)-4.6(d)-341.6(o)-4.6(f)]TJ /F4 1 Tf 11.3021 0 TD 0 Tc (R)Tj /F5 1 Tf 7.9701 0 0 7.9701 433.68 460.38 Tm (<)Tj /F6 1 Tf 11.9552 0 0 11.9552 444.72 462.18 Tm -0.0025 Tc [(w)32.2(e)-350.2(ca)-4.8(n)-331.8(r)-3.1(estrict)]TJ -29.8315 -1.1643 TD 0.0003 Tc [(ours)5.1(e)3.9(l)1.3(v)35.3(e)3.9(s)-346.2(t)-0.3(o)-323.2(a)-323.2(s)5.1(ubs)5.1(e)3.9(t)-351.7(of)-322.8(a)-323.2(m)3.2(inim)3.2(al)-330(c)3.9(y)5.2(c)3.9(l)1.3(e)-347.4(b)2.3(as)5.1(is)-326.1(follo)28.2(w)4.9(s)-326.1(from)-328(the)]TJ /F11 1 Tf 26.9402 0 TD 0.001 Tc [(mat)4.8(r)52.2(oid)-342.1(p)-0.9(r)52.2(o)-0.9(p)49.3(e)-0.7(rt)4.8(y)]TJ /F6 1 Tf 7.1064 0 TD 0 Tc (.)Tj -33.0429 -1.1744 TD -0.0011 Tc [(\(2\))-242.6(T)3.5(he)-258.5(solution)-260.1(is)-257.3(minimal)-261.1(i)-0.1(n)-250.1(t)-1.7(he)-248.5(follo)26.8(wing)-264.3(sense:)-411.7(if)-254(w)33.6(e)-268.5(tak)33.9(e)-258.5(an)31(y)-247.2(s)3.7(trict)-262.7(subset)-272.8(of)-243.9(the)]TJ -1.0037 -1.1644 TD -0.0021 Tc [(co)-34.5(ecien)30(ts)-318.5(with)]TJ /F12 1 Tf 7.1265 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 179.88 418.5 Tm (k)Tj /F6 1 Tf 11.9552 0 0 11.9552 188.4 420.3 Tm -0.0019 Tc [(=)-284.2(1)-295.2(then)-311.1(there)-299.5(i)-0.9(s)-308.3(n)0.1(o)-295.2(s)2.9(o)-4.2(l)-0.9(utio)-4.2(n)-311.1(w)2.7(ith)-301.1(e)1.7(xa)-4.2(ctly)-318.2(these)-309.5(c)1.7(o)-34.3(e)1.7(cien)30.2(t)-2.5(s)-318.3(b)-30.1(eing)]TJ -8.3914 -1.1643 TD -0.0013 Tc [(nonzero.)-492.2(This)-357.8(is)-347.8(equiv)53.8(a)-3.6(len)30.8(t)-363.3(to)-344.8(the)-349(f)-3.2(act)-343.2(t)-1.9(hat)-353.2(w)33.4(e)-349(h)0.7(a)26.6(v)33.7(e)-359.1(a)-344.8(minimally)-357.8(linearly)-347.7(d)0.7(ep)-29.5(enden)30.8(t)]TJ T* -0.0039 Tc [(set)-335.8(o)-6.2(f)-327(c)-0.3(ycles.)]TJ 1.0037 -1.1744 TD -0.0016 Tc (Let)Tj /F12 1 Tf 1.7866 0 TD 0 Tc (A)Tj /F6 1 Tf 1.074 0 TD 0.3425 Tc (=\()Tj /F12 1 Tf 1.4855 0 TD 0 Tc (C)Tj /F14 1 Tf 7.9701 0 0 7.9701 160.44 376.62 Tm (1)Tj /F12 1 Tf 11.9552 0 0 11.9552 165.12 378.42 Tm 0.1697 Tc [(,...)10.1(,C)]TJ /F5 1 Tf 7.9701 0 0 7.9701 199.8 376.62 Tm 0 Tc (M)Tj /F12 1 Tf 11.9552 0 0 11.9552 209.16 378.42 Tm 0.1697 Tc (,C)Tj /F5 1 Tf 7.9701 0 0 7.9701 222.84 376.62 Tm 0 Tc (M)Tj /F14 1 Tf 1.1293 0 TD 0.0017 Tc (+1)Tj /F12 1 Tf 11.9552 0 0 11.9552 243.12 378.42 Tm 0.1697 Tc [(,...)20.1(,C)]TJ /F5 1 Tf 7.9701 0 0 7.9701 277.68 376.62 Tm 0 Tc (N)Tj /F6 1 Tf 11.9552 0 0 11.9552 285.72 378.42 Tm 0.362 Tc [(\)b)333.8(e)-5.8(t)361.4(h)364(e)4.2(\()]TJ /F4 1 Tf 4.2157 0 TD 0 Tc (|)Tj /F12 1 Tf 0.281 0 TD (E)Tj /F4 1 Tf 0.7829 0 TD 0.2442 Tc (|)Tj /F12 1 Tf 1.5458 0 TD 0 Tc (N)Tj /F6 1 Tf 0.8833 0 TD -0.0017 Tc [(\)-)-6.5(ma)-4(trix)-368.2(w)2.9(ith)-371.1(t)-2.3(he)-359.5(cycles)]TJ /F12 1 Tf 10.76 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 514.92 376.62 Tm (k)Tj /F6 1 Tf 11.9552 0 0 11.9552 88.08 364.5 Tm -0.0026 Tc [(represen)29.5(ted)-392.1(a)-4.9(s)-379.2(c)1(o)-4.9(l)-1.6(umn)-372(v)32.4(ecto)-4.9(r)-3.2(s.)]TJ /F12 1 Tf 13.8013 0 TD 0 Tc (A)Tj /F6 1 Tf 1.1041 0 TD -0.0009 Tc [(can)-370.3(b)-29.1(e)-368.7(trans)3.9(formed)-380.4(in)31.2(to)-374.5(the)-368.7(r)-1.5(educed)-390.4(ro)27(w)-367.7(e)2.7(c)32.8(h)1.1(elon)]TJ -14.9055 -1.1945 TD 0.0014 Tc (form)Tj 2.5796 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F6 1 Tf 1.064 0 TD -0.0013 Tc [(b)30.8(y)-337.7(G)-5.9(a)-3.6(u-)-6.1(J)-0.2(ordan)-320.5(e)2.3(limination.)-472.1(Then)-340.6(exactly)-337.7(t)-1.9(he)-339(rst)]TJ /F12 1 Tf 22.4435 0 TD 0 Tc (R)Tj /F6 1 Tf 1.0439 0 TD 0.2823 Tc [(=r)281.7(a)280(n)284.3(k)116.5(\()]TJ /F12 1 Tf 3.5231 0 TD 0 Tc (A)Tj /F6 1 Tf 0.7327 0 TD 0.5025 Tc [(\)r)501.9(o)530.4(w)507.1(s)166(o)500.2(f)]TJ 4.5374 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F6 1 Tf -35.4023 -1.2446 TD -0.0018 Tc [(a)-4.1(r)-2.4(e)-389.7(n)0.2(o)-4.1(n)0.2(zero)-4.1(.)-623.1(No)-4.1(tice)-389.7(tha)-4.1(t)-383.9(the)-389.7(upp)-30(er-)-6.6(l)-0.8(ef)-3.7(t)]TJ /F12 1 Tf 17.5352 0 TD 0 Tc (M)Tj /F4 1 Tf 1.3149 0 TD ()Tj /F12 1 Tf 1.0439 0 TD (M)Tj /F6 1 Tf 1.0439 0 TD 0.0007 Tc [(-matrix)-385.9(o)-1.6(f)]TJ 5.009 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F6 1 Tf 1.1242 0 TD -0.0007 Tc [(is)-387.4(the)-388.6(i)0.3(den)31.4(t)-1.3(it)28.8(y)-397.3(m)2.2(atrix)]TJ -26.8102 -1.1643 TD -0.0029 Tc (since)Tj /F4 1 Tf 2.3989 0 TD 0 Tc ({)Tj /F12 1 Tf 0.5019 0 TD (C)Tj /F14 1 Tf 7.9701 0 0 7.9701 131.16 319.62 Tm (1)Tj /F12 1 Tf 11.9552 0 0 11.9552 135.84 321.42 Tm 0.1697 Tc [(,...)10.1(,C)]TJ /F5 1 Tf 7.9701 0 0 7.9701 170.52 319.62 Tm 0 Tc (M)Tj /F4 1 Tf 11.9552 0 0 11.9552 179.88 321.42 Tm (})Tj /F6 1 Tf 0.8331 0 TD -0.0014 Tc [(is)-327.8(a)-324.9(s)3.4(ubset)-343.3(of)-324.5(a)-334.9(c)2.2(ycle)-339.1(basis)-327.8(b)30.7(y)-337.8(c)2.2(onstruction,)-341.7(see)-329.1(F)-4.8(ig.)-321.6(1)-3.7(.)]TJ -7.508 -1.2547 TD -0.0006 Tc [(W)82(e)-328.3(in)31.5(tro)-33(duce)-328.3(a)-324.1(coloring)-334.1(of)-323.7(the)-328.3(c)3(olumns)]TJ /F12 1 Tf 17.1639 0 TD 0 Tc (M)Tj /F6 1 Tf 1.2747 0 TD 0.2221 Tc (+1)Tj /F12 1 Tf 1.4755 0 TD 0.1697 Tc [(,...)20.1(,N)]TJ /F6 1 Tf 3.4027 0 TD 0.0023 Tc (of)Tj 1.3755 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F6 1 Tf 0.7327 0 TD (:)Tj -26.1678 -1.1643 TD -0.0008 Tc [(\(1\))-382.9(T)3.8(w)33.9(o)-394.5(columns)]TJ /F12 1 Tf 7.7689 0 TD 0 Tc (j)Tj /F8 1 Tf 7.9701 0 0 7.9701 186.48 296.82 Tm ()Tj /F6 1 Tf 11.9552 0 0 11.9552 193.92 292.5 Tm 0.0002 Tc (and)Tj /F12 1 Tf 1.9573 0 TD 0 Tc (j)Tj /F8 1 Tf 7.9701 0 0 7.9701 222.84 296.82 Tm -0.0021 Tc ()Tj /F6 1 Tf 11.9552 0 0 11.9552 232.56 292.5 Tm 0 Tc (\()Tj /F12 1 Tf 0.3814 0 TD 0.3727 Tc (>M)Tj /F6 1 Tf 2.1781 0 TD -0.0015 Tc [(\))-393.6(h)0.5(a)26.4(v)33.5(e)-399.4(t)-2.1(he)-389.4(same)-389.4(color)-383.6(i)-0.5(f)-394.9(t)-2.1(here)-389.4(exists)-398.2(a)-385.2(r)-2.1(o)26.4(w)]TJ /F12 1 Tf 18.8903 0 TD 0 Tc (i)Tj /F6 1 Tf 0.7127 0 TD -0.0034 Tc [(suc)30.3(h)]TJ -34.2476 -1.2547 TD 0.0008 Tc (that)Tj 2.3789 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 122.16 275.7 Tm -0.0004 Tc (ij)Tj /F17 1 Tf 5.9776 0 0 5.9776 129 277.98 Tm 0 Tc ()Tj /F6 1 Tf 11.9552 0 0 11.9552 135.48 277.5 Tm (=)Tj 1.2948 0.2509 TD ()Tj /F12 1 Tf -0.2509 -0.2509 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 156.72 275.7 Tm -0.0004 Tc (ij)Tj /F17 1 Tf 5.9776 0 0 5.9776 163.56 277.98 Tm -0.0043 Tc ()Tj /F6 1 Tf 11.9552 0 0 11.9552 172.2 277.5 Tm 0.2723 Tc [(=1)270(.)]TJ -7.0363 -1.1644 TD -0.003 Tc [(\()-3.6(2)-5.3(\))-324.8(U)-1.7(se)-340.7(a)-5.3(s)-319.4(ma)-5.3(n)29.1(y)-339.4(co)-5.3(lo)-5.3(r)-3.6(s)-329.4(a)-5.3(s)-329.4(p)-31.2(o)-5.3(ssible.)]TJ /F1 1 Tf 0 -1.5458 TD -0.0005 Tc [(Denition)-459.5(2.)]TJ /F11 1 Tf 6.6548 0 TD 0.0014 Tc [(Two)-402(r)52.6(elevant)-396.3(cycles)]TJ /F12 1 Tf 8.8931 0 TD 0 Tc (C)Tj /F8 1 Tf 7.9701 0 0 7.9701 283.2 249.42 Tm ()Tj /F12 1 Tf 11.9552 0 0 11.9552 285.96 245.1 Tm 0.1697 Tc (,C)Tj /F8 1 Tf 7.9701 0 0 7.9701 300.48 249.42 Tm -0.0021 Tc ()Tj /F4 1 Tf 11.9552 0 0 11.9552 310.2 245.1 Tm 0.3973 Tc (R)Tj /F11 1 Tf 2.3186 0 TD 0.0019 Tc [(ar)53.1(e)]TJ /F6 1 Tf 1.7967 0 TD 0.0001 Tc (color-related)Tj /F11 1 Tf 5.3298 0 TD 0.4327 Tc [(,i)431.6(f)30.1(\()431.2(i)431.6(\))]TJ /F4 1 Tf 3.2521 0 TD 0 Tc (|)Tj 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[(wit)-3.4(h)-302(t)26.7(w)31.9(o)-306.2(c)0.8(o)-35.2(e)0.8(cien)29.3(t)-3.4(s)]TJ 9.1944 0.2509 TD 0 Tc ()Tj /F12 1 Tf -0.2509 -0.2509 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 291.48 386.94 Tm 0.0325 Tc (rk)Tj /F16 1 Tf 5.9776 0 0 5.9776 300 385.86 Tm 0 Tc (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 307.92 388.74 Tm (=)Tj 1.2948 0.2509 TD ()Tj /F12 1 Tf -0.261 -0.2509 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 329.04 386.94 Tm 0.0325 Tc (rk)Tj /F16 1 Tf 5.9776 0 0 5.9776 337.56 385.86 Tm 0 Tc (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 345.48 388.74 Tm 0.2823 Tc [(=1)-11(s)287.1(u)284.3(c)316(h)-26.9(t)281.7(h)284.3(a)280(t)]TJ /F12 1 Tf 6.0927 0 TD 0 Tc (k)Tj /F14 1 Tf 7.9701 0 0 7.9701 424.44 386.94 Tm (1)Tj /F4 1 Tf 11.9552 0 0 11.9552 432.36 388.74 Tm 0.2869 Tc (J)Tj /F14 1 Tf 7.9701 0 0 7.9701 451.92 386.94 Tm 0 Tc (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 460.08 388.74 Tm 0.0002 Tc (and)Tj /F12 1 Tf 1.8669 0 TD 0 Tc (k)Tj /F14 1 Tf 7.9701 0 0 7.9701 488.52 386.94 Tm (2)Tj /F4 1 Tf 11.9552 0 0 11.9552 496.56 388.74 Tm 0.2768 Tc (J)Tj /F14 1 Tf 7.9701 0 0 7.9701 516 386.94 Tm 0 Tc (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 520.68 388.74 Tm (,)Tj -35.864 -1.1643 TD -0.0026 Tc [(see)-340.3(F)-6(ig)-4.9(.)-322.8(1)-4.9(.)]TJ 1.0038 -1.1644 TD -0.0016 Tc [(No)26.3(w)-258(s)3.2(upp)-29.8(ose)]TJ /F12 1 Tf 5.7614 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 181.2 359.1 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 184.68 358.02 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 192.36 360.9 Tm 0.0002 Tc (and)Tj /F12 1 Tf 1.8268 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 222.6 359.1 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 226.08 358.02 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 233.64 360.9 Tm [(are)-257.4(i)1(n)32.1(t)-0.6(e)3.6(r)-0.6(c)33.7(h)2(ange)3.6(able)3.6(.)-430.6(T)4.6(he)3.6(n)-259(t)-0.6(he)3.6(re)-257.4(e)3.6(x)4.9(is)4.8(ts)-266.2(a)-253.2(m)2.9(inim)2.9(al)-260(s)4.8(o)-2.3(lution)]TJ -11.8543 -1.1643 TD 0.0004 Tc [(of)-292.6(e)4(q)5.3(u.)-299.7(\(1\))-281.3(w)5(ith)]TJ /F12 1 Tf 6.8756 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 180.72 345.18 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 184.2 344.1 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 192.12 346.98 Tm (=)Tj /F12 1 Tf 1.0439 0 TD (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 211.2 345.18 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 214.68 344.1 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 222.6 346.98 Tm 0.2823 Tc [(=1)280(.)-138.3(S)284.3(e)285.9(t)]TJ /F12 1 Tf 3.8744 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 275.52 345.18 Tm (k)Tj /F6 1 Tf 11.9552 0 0 11.9552 284.04 346.98 Tm 0.2723 Tc [(=0)-21(f)270.4(o)270(r)-19.4(a)270(l)273.3(l)]TJ /F12 1 Tf 4.5971 0 TD 0 Tc (k)Tj /F4 1 Tf 0.8231 0 TD 0.2768 Tc (J)Tj /F14 1 Tf 7.9701 0 0 7.9701 368.28 345.18 Tm 0 Tc (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 376.44 346.98 Tm -0.001 Tc [(in)-290.1(this)-297.3(solution)-300.2(\(this)-287.3(includes)]TJ /F12 1 Tf -23.7989 -1.1744 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 98.52 331.14 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 102 330.06 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 109.92 332.94 Tm -0.0011 Tc [(=)-283.4(0)-3.4(\).)-421.7(If)-284.1(the)-288.6(r)-1.7(es)3.7(ulting)-294.4(v)33.9(ector)-302.9(\()]TJ /F12 1 Tf 12.4563 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 265.44 337.26 Tm ()Tj /F5 1 Tf 0 -0.9034 TD (i)Tj /F6 1 Tf 11.9552 0 0 11.9552 268.8 332.94 Tm 0.0001 Tc [(\))-281.6(i)1.1(s)-286.2(a)-283.2(s)4.9(o)-2.2(lution)-289(of)-282.9(eq)5(u.)-290(\(1\),)-290(the)-287.4(o)-2.2(riginal)-280(s)4.9(olution)-289(w)34.8(as)]TJ -14.7953 -1.1644 TD 0.0002 Tc [(not)-321.6(m)3.1(inim)3.1(al,)-340.1(c)3.8(on)32.3(tradic)3.8(t)-0.4(ing)-333.3(t)-0.4(he)-327.5(as)5(s)5(u)2.2(m)3.1(p)2.2(tion)-339.1(that)]TJ /F12 1 Tf 20.7673 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 348.6 317.22 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 352.08 316.14 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 360.48 319.02 Tm 0.0002 Tc (and)Tj /F12 1 Tf 1.9071 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 391.68 317.22 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 395.16 316.14 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 403.68 319.02 Tm -0.0008 Tc [(w)33.9(e)2.8(re)-328.5(in)31.3(terc)32.9(hangeable.)]TJ -25.0737 -1.1643 TD -0.0012 Tc [(Hence)-389.1(w)33.5(e)-389.1(a)-3.5(s)3.6(s)3.6(u)0.8(me)-389.1(that)-383.3(the)-379.1(r)-1.8(es)3.6(ulting)-394.9(v)33.8(ector)-393.3(\()]TJ /F12 1 Tf 19.2617 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 340.8 309.42 Tm ()Tj /F5 1 Tf 0 -0.9184 TD (i)Tj /F6 1 Tf 11.9552 0 0 11.9552 344.16 305.1 Tm 0.0008 Tc [(\))-381.3(m)3.7(a)28.7(y)-385.8(not)-371.2(b)-27.4(e)-377(a)-382.9(s)5.6(o)-1.5(lution)-378.7(an)32.9(y)-385.8(m)3.7(ore)4.4(.)]TJ -21.0988 -1.2547 TD -0.0017 Tc [(This)-308.1(happ)-29.9(ens)-308.1(w)2.9(hen)-300.9(t)-2.3(here)-309.3(is)-298(a)-295(r)-2.3(o)26.2(w)]TJ /F12 1 Tf 14.6043 0 TD 0 Tc (r)Tj /F6 1 Tf 0.7729 0 TD 0.031 Tc [(w)35.6(i)32(t)30.4(h)-268.2(a)28.7(n)-258.1(od)33(d)-268.2(n)63.1(u)33(m)64.1(be)34.6(r)-280.8(o)28.7(f)-262(c)34.6(oe)34.6()33.9(c)34.6(i)32(e)34.6(n)63.1(t)30.4(s)]TJ 15.2372 0.261 TD 0 Tc ()Tj /F12 1 Tf -0.261 -0.261 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 463.56 288.3 Tm 0.0325 Tc (rn)Tj /F6 1 Tf 11.9552 0 0 11.9552 476.76 290.1 Tm 0.0003 Tc [(for)-291.4(w)4.9(hic)34(h)]TJ /F12 1 Tf -32.1903 -1.1945 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 98.52 280.26 Tm ()Tj /F5 1 Tf 0 -0.9184 TD (n)Tj /F6 1 Tf 11.9552 0 0 11.9552 109.32 278.94 Tm ()Tj /F12 1 Tf -0.261 -0.261 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 114.96 274.02 Tm 0.0325 Tc (rn)Tj /F6 1 Tf 11.9552 0 0 11.9552 128.76 275.82 Tm -0.0021 Tc [(=)-354.7(1)-4.4(.)-553.2(I)-0.2(n)-371.5(t)-2.7(his)-368.7(c)1.5(ase,)-392.6(ho)25.8(w)32.6(e)1.5(v)32.9(e)1.5(r,)-402.6(w)32.6(e)-369.9(m)31(u)-0.1(st)-384.2(ha)25.8(v)32.9(e)]TJ /F12 1 Tf 18.3583 0 TD 0 Tc (r)Tj /F4 1 Tf 0.813 0 TD ()Tj /F12 1 Tf 1.1242 0 TD (M)Tj /F6 1 Tf 1.4153 0 TD 0.0002 Tc (and)Tj /F12 1 Tf 1.9472 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 418.2 280.26 Tm ()Tj /F5 1 Tf 0 -0.9184 TD (r)Tj /F6 1 Tf 11.9552 0 0 11.9552 427.92 278.94 Tm ()Tj /F12 1 Tf -0.261 -0.261 TD (A)Tj /F5 1 Tf 7.9701 0 0 7.9701 433.56 274.02 Tm 0.0325 Tc (rr)Tj /F6 1 Tf 11.9552 0 0 11.9552 446.28 275.82 Tm -0.0032 Tc [(=)-355.8(1)-5.5(.)-554.3(H)-1.9(ence)-381(w)31.5(e)]TJ -29.6407 -1.1643 TD -0.0036 Tc [(ca)-5.9(n)-332.9(s)1.2(et)]TJ /F12 1 Tf 3.3324 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 138.36 266.22 Tm ()Tj /F5 1 Tf 0 -0.9034 TD (r)Tj /F6 1 Tf 11.9552 0 0 11.9552 146.28 261.9 Tm -0.0021 Tc [(=)-284.4(0)-4.4(,)-332.4(s)2.7(ince)-339.8(the)-329.8(upp)-30.3(er-)-6.9(l)-1.1(ef)-4(t)]TJ /F12 1 Tf 10.74 0 TD 0 Tc (M)Tj /F4 1 Tf 1.2747 0 TD ()Tj /F12 1 Tf 1.0037 0 TD (M)Tj /F6 1 Tf 1.0439 0 TD -0.0009 Tc [(-matrix)-337.3(i)0.1(s)-327.3(t)-1.5(he)-328.6(iden)31.2(tit)28.6(y)-347.3(matrix)4(.)-441.6(Since)-338.6(t)-1.5(his)]TJ -18.6093 -1.1644 TD -0.0021 Tc [(ho)-4.4(lds)-278.3(f)-4(o)-4.4(r)-263.7(ev)32.9(ery)-288.3(s)2.7(uc)31.6(h)-291.2(r)-2.7(o)25.8(w)]TJ /F12 1 Tf 10.4589 0 TD 0 Tc (r)Tj /F6 1 Tf 0.7327 0 TD 0.2791 Tc [(w)313.8(e)-8.4(e)282.7(n)281.1(d)10(u)281.1(pw)283.7(i)280.1(t)278.5(ha)15.9(n)281.1(e)282.7(w)-7.4(s)283.9(o)276.8(l)280.1(u)281.1(t)278.5(i)280.1(o)276.8(n)10(\()]TJ /F12 1 Tf 13.4701 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 393.36 252.3 Tm -0.0021 Tc ()Tj /F5 1 Tf 0 -0.9184 TD 0 Tc (i)Tj /F6 1 Tf 11.9552 0 0 11.9552 398.52 247.98 Tm 0.2723 Tc [(\)o)270(fe)275.9(q)277.2(u)274.3(.)-7.8(\()271.7(1)270(\))0.6(w)276.9(i)273.3(t)271.7(h)]TJ /F12 1 Tf 7.4477 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 494.16 252.3 Tm -0.0021 Tc ()Tj /F5 1 Tf 0 -0.9184 TD 0 Tc (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 497.64 243.9 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 505.68 247.98 Tm 0.2723 Tc (=1)Tj -34.6093 -1.2145 TD 0.0002 Tc (and)Tj /F12 1 Tf 1.877 0 TD 0 Tc (x)Tj /F8 1 Tf 7.9701 0 0 7.9701 120.96 237.9 Tm -0.0021 Tc ()Tj /F5 1 Tf 0 -0.9184 TD 0 Tc (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 124.44 229.5 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 132.48 233.46 Tm 0.0003 Tc [(=)-272(0)-2(.)-420.3(A)1.6(gain)-298.9(t)-0.3(he)-307.3(original)-299.8(s)5.1(o)-2(lution)-298.9(\()]TJ /F12 1 Tf 14.2831 0 TD 0 Tc (x)Tj /F5 1 Tf 7.9701 0 0 7.9701 309.84 231.66 Tm (i)Tj /F6 1 Tf 11.9552 0 0 11.9552 313.32 233.46 Tm -0.0003 Tc [(\))-302.1(w)34.4(as)-306.7(not)-292(m)2.6(inimal,)-320.5(a)-303.7(con)31.8(tradiction)-299.5(t)-0.9(o)-303.7(o)-2.6(ur)]TJ -18.5192 -1.1643 TD -0.0033 Tc [(a)-5.6(ssumpt)-3.9(i)-2.3(o)-5.6(n)-1.3(.)]TJ /T1 1 Tf 0.12 0 0 -0.12 514.679 219.54 Tm 0 Tc ()Tj /F1 1 Tf 11.9552 0 0 11.9552 91.92 192.42 Tm -0.0025 Tc [(Lemma)-419.2(4.)]TJ /F11 1 Tf 5.3098 0 TD 0.003 Tc [(If)-359.5(t)6.8(w)0.6(o)-370.3(c)1.3(y)6.2(c)1.3(les)]TJ /F12 1 Tf 5.7313 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 232.32 190.62 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 235.8 189.54 Tm (1)Tj /F11 1 Tf 11.9552 0 0 11.9552 244.8 192.42 Tm 0.002 Tc (and)Tj /F12 1 Tf 1.9171 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 276.12 190.62 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 279.6 189.54 Tm (2)Tj /F11 1 Tf 11.9552 0 0 11.9552 288.6 192.42 Tm 0.0014 Tc [(ar)52.6(e)-371.7(c)49.9(olor-r)52.6(elat)5.2(e)49.9(d)-361.8(w.r.t)5.2(.)-361.1(a)-371.9(g)-0.3(iven)-362(or)52.6(dering)-361.6(of)-361.1(t)5.2(h)-0.5(e)]TJ -16.4515 -1.1643 TD 0.001 Tc [(cycles,)-341.4(t)4.8(hen)]TJ /F12 1 Tf 5.2997 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 163.68 176.7 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 167.16 175.62 Tm (1)Tj /F11 1 Tf 11.9552 0 0 11.9552 175.92 178.5 Tm 0.002 Tc (and)Tj /F12 1 Tf 1.8971 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 207 176.7 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 210.48 175.62 Tm (2)Tj /F11 1 Tf 11.9552 0 0 11.9552 219.24 178.5 Tm 0.0018 Tc [(ar)53(e)-351.2(i)0.7(nt)5.6(er)53(change)50.3(able.)]TJ -10.6498 -2.2685 TD 0.0001 Tc [(Pr)51.3(o)48.4(o)-1.8(f.)]TJ /F6 1 Tf 2.9911 0 TD -0.0031 Tc (Assume)Tj /F12 1 Tf 3.6937 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 180.24 149.58 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 183.72 148.5 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 192.96 151.38 Tm 0.0002 Tc (and)Tj /F12 1 Tf 1.9774 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 225 149.58 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 228.48 148.5 Tm (2)Tj /F6 1 Tf 11.9552 0 0 11.9552 237.72 151.38 Tm 0.0013 Tc [(are)-386.6(c)4.9(olor-re)4.9(l)2.3(ate)4.9(d)-388.2(and)-388.2(l)2.3(e)4.9(t)]TJ /F4 1 Tf 10.8704 0 TD 0 Tc (J)Tj /F6 1 Tf 1.2547 0 TD -0.001 Tc [(denote)-398.9(the)-388.9(s)3.8(et)-403.1(of)-394.4(indices)-397.7(of)]TJ -24.3207 -1.1644 TD -0.0015 Tc [(the)-469.7(c)2.1(y)3.4(c)2.1(les)]TJ /F12 1 Tf 4.7778 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 157.44 135.66 Tm (i)Tj /F6 1 Tf 11.9552 0 0 11.9552 166.44 137.46 Tm 0.0009 Tc [(in)-468.9(the)-467.3(c)4.5(olor-e)4.5(q)5.8(u)2.9(iv)56(ale)4.5(n)2.9(c)4.5(e)-477.3(c)4.5(l)1.9(as)5.7(s)-476.1(o)-1.4(f)]TJ /F12 1 Tf 14.514 0 TD 0 Tc (C)Tj /F5 1 Tf 7.9701 0 0 7.9701 348.36 135.66 Tm (j)Tj /F16 1 Tf 5.9776 0 0 5.9776 351.84 134.58 Tm (1)Tj /F6 1 Tf 11.9552 0 0 11.9552 356.4 137.46 Tm -0.0024 Tc 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