5719:
5208:
5714:{\displaystyle {\begin{aligned}c_{j}\lambda _{j}^{t}+c_{j+1}\lambda _{j+1}^{t}&=M^{t}\left(c_{j}\left({\frac {\alpha }{M}}+{\frac {\beta }{M}}i\right)^{t}+c_{j+1}\left({\frac {\alpha }{M}}-{\frac {\beta }{M}}i\right)^{t}\right)\\&=M^{t}\left(c_{j}\left(\cos \theta +i\sin \theta \right)^{t}+c_{j+1}\left(\cos \theta -i\sin \theta \right)^{t}\right)\\&=M^{t}{\bigl (}c_{j}\left(\cos \theta t+i\sin \theta t\right)+c_{j+1}\left(\cos \theta t-i\sin \theta t\right){\bigr )}\end{aligned}}}
3574:
3326:
4197:
9318:; unlike in the stable case, this converged value depends on the initial conditions; different starting points lead to different points in the long run. If any root is −1, its term will contribute permanent fluctuations between two values. If any of the unit-magnitude roots are complex then constant-amplitude fluctuations of
6693:
9343:
If a non-homogeneous linear difference equation has been converted to homogeneous form which has been analyzed as above, then the stability and cyclicality properties of the original non-homogeneous equation will be the same as those of the derived homogeneous form, with convergence in the stable
9287:
grows if the root is +1 but will fluctuate between two values if the root is −1. If the absolute value of the root is greater than 1 the term will become larger and larger over time. A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the
3970:
6047:
Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicity involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate
3569:{\displaystyle {\begin{array}{lcl}M={\sqrt {\alpha ^{2}+\beta ^{2}}}&\cos(\theta )={\tfrac {\alpha }{M}}&\sin(\theta )={\tfrac {\beta }{M}}\\C,D=E\mp Fi&&\\G={\sqrt {E^{2}+F^{2}}}&\cos(\delta )={\tfrac {E}{G}}&\sin(\delta )={\tfrac {F}{G}}\end{array}}}
6416:
5126:
9174:
that lead to more convenient algebraic manipulations and more straightforward solutions. There are cases in which obtaining a direct solution would be all but impossible, yet solving the problem via a thoughtfully chosen integral transform is straightforward.
7521:
997:
1765:
are determined in order to fit the initial conditions of the recurrence. When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of
3317:
1524:
4656:
6819:
606:
9292:
of the roots is less than 1; if the modulus equals 1 then constant amplitude fluctuations in the combined terms will persist; and if the modulus is greater than 1, the combined terms will show fluctuations of ever-increasing magnitude.
7301:
3907:
7753:
2638:
6886:
5884:
5969:
1736:
9275:
4908:
3791:
8157:
6403:
5213:
4192:{\displaystyle {\begin{aligned}E&={\frac {-Aa_{1}+a_{2}}{B}}\\F&=-i{\frac {A^{2}a_{1}-Aa_{2}+2a_{1}B}{B{\sqrt {A^{2}+4B}}}}\\\theta &=\arccos \left({\frac {A}{2{\sqrt {-B}}}}\right)\end{aligned}}}
5028:
1330:
3098:
1955:
428:
6169:
7056:
318:
3169:
2793:
1211:
722:
7365:
830:
4815:
4526:
1092:
842:
3714:
6421:
5199:
3975:
2549:
2901:
8534:
8411:
3178:
8916:
1343:
7130:
7622:
1616:
8714:
6688:{\displaystyle {\begin{aligned}w_{1,t}&=a_{1}w_{1,t-1}+a_{2}w_{2,t-1}+\cdots +a_{n}w_{n,t-1}+b\\w_{2,t}&=w_{1,t-1}\\&\,\,\,\vdots \\w_{n,t}&=w_{n-1,t-1}.\end{aligned}}}
4535:
6713:
2355:
2472:
8849:
492:
7137:
4336:
2011:
8029:
7859:
3800:
8853:
This example shows how problems generally solved using the power series solution method taught in normal differential equation classes can be solved in a much easier way.
4253:
4226:
2714:
2687:
2127:
9049:
2397:
2244:
2051:
7633:
4414:
2837:
8959:
8290:
7891:
2544:
1823:
8049:
7983:
7947:
4372:
3656:
2537:
2160:
2084:
9101:
2501:
9148:
3963:
3936:
2997:
2970:
2196:
1763:
8233:
This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.
8208:
7813:
7784:
7552:
7332:
4923:
can be found by invoking the initial conditions. Specifically, for each time period for which an iterate value is known, this value and its corresponding value of
2724:
of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution
7358:
6828:
5816:
4697:. However, for use in a theoretical context it may be that the only information required about the roots is whether any of them are greater than or equal to 1 in
2270:
5893:
1621:
9188:
9121:
9071:
8228:
8179:
7911:
4280:
3636:
3616:
3596:
3040:
3020:
2943:
2923:
2660:
1843:
1784:
1554:
450:
9303:
If the largest root has absolute value 1, neither convergence to 0 nor divergence to infinity will occur. If all roots with magnitude 1 are real and positive,
4824:
3718:
8063:
6282:
9283:
grows indefinitely large if the absolute value of the characteristic root is less than 1. If the absolute value equals 1, the term will stay constant as
112:, and not of any iterate values, giving the value of the iterate at any time. To find the solution it is necessary to know the specific values (known as
1220:
1848:
1536:
The roots of the characteristic polynomial play a crucial role in finding and understanding the sequences satisfying the recurrence. If there are
327:
6095:
4962:
If there are complex roots, they come in conjugate pairs and so do the complex terms in the solution equation. If two of these complex terms are
730:. To solve this equation it is convenient to convert it to homogeneous form, with no constant term. This is done by first finding the equation's
6961:
223:
3105:
3042:), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form. In this case we can write the eigenvalues as
2729:
1119:
630:
761:
4729:
4440:
1006:
3661:
9370:
5151:
2844:
8423:
8295:
9443:
9408:
3045:
837:
Given the steady state, the difference equation can be rewritten in terms of deviations of the iterates from the steady state, as
8543:
9329:
will diverge to infinity as time goes to infinity, or will fluctuate between increasingly large positive and negative values.
2286:
7627:
2406:
8723:
6048:
characteristic roots. This can be seen in the trigonometric form of their contribution to the solution equation, involving
9549:
178:
5121:{\displaystyle \lambda _{j},\lambda _{j+1}=\alpha \pm \beta i=M\left({\frac {\alpha }{M}}\pm {\frac {\beta }{M}}i\right)}
9365:
4258:
In all cases—real distinct eigenvalues, real duplicated eigenvalues, and complex conjugate eigenvalues—the equation is
28:
132:
if from any set of initial conditions the variable's limit as time goes to infinity exists; this limit is called the
1967:
9539:
8864:
6938:
6183:
4936:
8236:
The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that:
8968:
7516:{\displaystyle Y(x)=\left({\frac {b}{1-x}}+p(x)\right)\cdot {\frac {1}{1-a_{1}x-a_{2}x^{2}-\cdots -a_{n}x^{n}}}.}
4954:
will also be real; but with non-real complex roots, in general some of these coefficients will also be non-real.
2163:
745:
successive iterates all had this value, so would all future values. This value is found by setting all values of
483:
7560:
7067:
4690:
2206:
Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that
1559:
992:{\displaystyle \left(y_{t}-y^{*}\right)=a_{1}\left(y_{t-1}-y^{*}\right)+\cdots +a_{n}\left(y_{t-n}-y^{*}\right)}
17:
8925:
8241:
9544:
5811:. Using this in the last equation gives this expression for the two complex terms in the solution equation:
4693:. If the solution is to be used numerically, all the roots of this characteristic equation can be found by
9534:
7994:
5772:
1528:
in which like terms can be combined to give a homogeneous equation of one order higher than the original.
146:
74:
66:
9336:
states that all roots have magnitude less than 1 (the stable case) if and only if a particular string of
3002:
In the case of complex eigenvalues (which also gives rise to complex values for the solution parameters
174:
3312:{\displaystyle a_{n}=2M^{n}\left(E\cos(\theta n)+F\sin(\theta n)\right)=2GM^{n}\cos(\theta n-\delta ),}
610:
whose roots play a crucial role in finding and understanding the sequences satisfying the recurrence.
4679:
4426:
Recurrence equation § Solving homogeneous linear recurrence relations with constant coefficients
4293:
1519:{\displaystyle y_{t}-a_{1}y_{t-1}-\cdots -a_{n}y_{t-n}=y_{t-1}-a_{1}y_{t-2}-\cdots -a_{n}y_{t-(n+1)}}
170:
9360:
8963:
The conversion of the differential equation to a difference equation of the Taylor coefficients is
8004:
7818:
7061:
4425:
2717:
2277:
2273:
4651:{\displaystyle \lambda ^{n}=a_{1}\lambda ^{n-1}+\cdots +a_{n-2}\lambda ^{2}+a_{n-1}\lambda +a_{n}}
4231:
4204:
2692:
2665:
2089:
9491:
Papanicolaou, Vassilis, "On the asymptotic stability of a class of linear difference equations,"
9171:
8051:
is a complex number that is determined by substituting the guess into the differential equation.
6814:{\displaystyle \mathbf {w} _{i}={\begin{bmatrix}w_{1,i}\\w_{2,i}\\\vdots \\w_{n,i}\end{bmatrix}}}
4290:. In this second-order case, this condition on the eigenvalues can be shown to be equivalent to
2362:
2209:
2016:
166:
128:
4724:
If all the characteristic roots are distinct, the solution of the homogeneous linear recurrence
4377:
2800:
8230:. The differential equation provides a linear difference equation relating these coefficients.
7864:
1789:
77:
0 or 1. A linear recurrence denotes the evolution of some variable over time, with the current
9439:
9404:
8034:
7952:
7916:
7555:
4713:
4694:
4341:
601:{\displaystyle p(\lambda )=\lambda ^{n}-a_{1}\lambda ^{n-1}-a_{2}\lambda ^{n-2}-\cdots -a_{n}}
114:
46:
3641:
2509:
2132:
2056:
9432:
9397:
9076:
4259:
2480:
456:
of the recurrence and denotes the longest time lag between iterates. The equation is called
104:
9506:
9126:
7296:{\displaystyle Y(x)=a_{1}xY(x)+a_{2}x^{2}Y(x)+\cdots +a_{n}x^{n}Y(x)+{\frac {b}{1-x}}+p(x)}
3941:
3914:
2975:
2948:
2169:
1741:
8184:
7789:
7760:
7528:
7308:
7337:
2249:
4943:
parameter values. If all characteristic roots are real, then all the coefficient values
9427:
9375:
9106:
9056:
8213:
8164:
7896:
5143:
5135:
4709:
4698:
4287:
4265:
3621:
3601:
3581:
3025:
3005:
2928:
2908:
2645:
1828:
1769:
1539:
435:
150:
42:
3902:{\displaystyle a_{n}=(-B)^{\frac {n}{2}}\left(E\cos(\theta n)+F\sin(\theta n)\right),}
9528:
8055:
3331:
2272:
is a root of the characteristic polynomial. This can be approached directly or using
154:
78:
38:
9507:"Parapermanent of triangular matrices and some general theorems on number sequences"
9460:
9300:
will converge to 0 if all of the characteristic roots have magnitude less than 1.
7748:{\displaystyle {\frac {f(x)}{g(x)}}=\sum _{i}{\frac {f_{i}(x)}{(x-r_{i})^{m_{i}}}}}
7132:. The recurrence is then equivalent to the following generating function equation:
834:
assuming the denominator is not 0. If it is zero, the steady state does not exist.
162:
134:
2633:{\displaystyle {\begin{aligned}r^{2}&=Ar+B,\\r^{2}-Ar-B&=0,\end{aligned}}}
4927:
can be substituted into the solution equation to obtain a linear equation in the
9337:
9333:
9163:
9159:
8418:
The recurrence relationship for the Taylor series coefficients of the equation:
4705:
158:
34:
161:
because values of these variables are only measured at discrete intervals. In
62:
6881:{\displaystyle \mathbf {w} _{t}=\mathbf {A} \mathbf {w} _{t-1}+\mathbf {b} }
5879:{\displaystyle 2M^{t}\left(\gamma \cos \theta t-\delta \sin \theta t\right)}
2642:
which is the characteristic equation of the recurrence relation. Solve for
194:
142:
5964:{\displaystyle 2{\sqrt {\gamma ^{2}+\delta ^{2}}}M^{t}\cos(\theta t+\psi )}
1731:{\displaystyle a_{n}=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+\cdots +k_{d}r_{d}^{n},}
9270:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{n}\lambda _{n}^{t},}
5801:
guarantees that they are also complex conjugates, which can be written as
4903:{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{n}\lambda _{n}^{t}}
3786:{\displaystyle \lambda _{1}\cdot \lambda _{2}=\alpha ^{2}+\beta ^{2}=-B,}
70:
6924:
and all other rows have a single 1 with all other elements being 0, and
1001:
which has no constant term, and which can be written more succinctly as
9325:
Finally, if any characteristic root has magnitude greater than 1, then
8152:{\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}
6398:{\displaystyle y_{t}=a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots +a_{n}y_{t-n}+b}
5203:
Then the two complex terms in the solution equation can be written as
7998:
7360:
correcting the initial terms. From this equation we can solve to get
3658:) are real constants which depend on the initial conditions. Using
2400:
126:
iterates that are oldest. The equation or its variable is said to be
8161:
it can be seen that the coefficients of the series are given by the
6937:
This matrix equation can be solved using the methods in the article
1786:. For instance, if the characteristic polynomial can be factored as
141:
Difference equations are used in a variety of contexts, such as in
9463:(1982), "2.1.1 Constant coefficients – A) Homogeneous equations",
8001:) for linear differential equations with constant coefficients is
1325:{\displaystyle y_{t-1}=a_{1}y_{t-2}+\cdots +a_{n}y_{t-(n+1)}+b}
481:
If the equation is homogeneous, the coefficients determine the
73:. The polynomial's linearity means that each of its terms has
3093:{\displaystyle \lambda _{1},\lambda _{2}=\alpha \pm \beta i.}
1950:{\displaystyle a_{n}=k_{1}r^{n}+k_{2}nr^{n}+k_{3}n^{2}r^{n}.}
1845:
occurring three times, then the solution would take the form
2539:, we get that all these equations reduce to the same thing:
423:{\displaystyle y_{t+n}=a_{1}y_{t+n-1}+\cdots +a_{n}y_{t}+b.}
7525:
In other words, not worrying about the exact coefficients,
6164:{\displaystyle x_{t}=c_{1}\lambda ^{t}+c_{2}t\lambda ^{t}.}
165:
applications, linear difference equations are modeled with
7051:{\displaystyle y_{t}=a_{1}y_{t-1}+\cdots +a_{n}y_{t-n}+b,}
6070:
In the second-order case, if the two roots are identical (
313:{\displaystyle y_{t}=a_{1}y_{t-1}+\cdots +a_{n}y_{t-n}+b,}
193:
is an equation of the following form, written in terms of
7630:. Specifically, if the generating function is written as
3164:{\displaystyle a_{n}=C\lambda _{1}^{n}+D\lambda _{2}^{n}}
2788:{\displaystyle a_{n}=C\lambda _{1}^{n}+D\lambda _{2}^{n}}
2359:
When does it have a solution of the same general form as
2283:
Consider, for example, a recurrence relation of the form
1206:{\displaystyle y_{t}=a_{1}y_{t-1}+\cdots +a_{n}y_{t-n}+b}
717:{\displaystyle y_{t}=a_{1}y_{t-1}+\cdots +a_{n}y_{t-n}+b}
145:
to model the evolution through time of variables such as
9279:
a term with real characteristic roots converges to 0 as
7997:
is similar to the method above—the "intelligent guess" (
825:{\displaystyle y^{*}={\frac {b}{1-a_{1}-\cdots -a_{n}}}}
756:
in the difference equation, and solving, thus obtaining
487:(also "auxiliary polynomial" or "companion polynomial")
6178:
An alternative solution method involves converting the
4935:
such equations, one for each initial condition, can be
4819:
can be written in terms of the characteristic roots as
4810:{\displaystyle x_{t}=a_{1}x_{t-1}+\cdots +a_{n}x_{t-n}}
4521:{\displaystyle x_{t}=a_{1}x_{t-1}+\cdots +a_{n}x_{t-n}}
1087:{\displaystyle x_{t}=a_{1}x_{t-1}+\cdots +a_{n}x_{t-n}}
7070:
6737:
3551:
3519:
3420:
3388:
2162:. The characteristic polynomial equated to zero (the
9191:
9129:
9109:
9079:
9059:
8971:
8928:
8867:
8726:
8546:
8426:
8298:
8244:
8216:
8187:
8167:
8066:
8037:
8007:
7955:
7919:
7899:
7867:
7821:
7792:
7763:
7636:
7563:
7531:
7368:
7340:
7311:
7140:
6964:
6831:
6716:
6419:
6285:
6098:
5896:
5819:
5211:
5154:
5031:
4827:
4732:
4538:
4530:
involves first solving its characteristic polynomial
4443:
4380:
4344:
4296:
4268:
4234:
4207:
3973:
3944:
3917:
3803:
3721:
3709:{\displaystyle \lambda _{1}+\lambda _{2}=2\alpha =A,}
3664:
3644:
3624:
3604:
3584:
3329:
3181:
3108:
3048:
3028:
3008:
2978:
2951:
2931:
2911:
2905:
This is the most general solution; the two constants
2847:
2803:
2732:
2695:
2668:
2648:
2547:
2512:
2483:
2409:
2365:
2289:
2252:
2212:
2172:
2135:
2092:
2059:
2019:
1970:
1851:
1831:
1792:
1772:
1744:
1624:
1562:
1542:
1346:
1223:
1122:
1114:
If there is no steady state, the difference equation
1009:
845:
764:
633:
495:
438:
330:
226:
4712:. In the latter case, all the complex roots come in
2945:
can be chosen based on two given initial conditions
1618:
then each solution to the recurrence takes the form
8058:of the solution to a linear differential equation:
5194:{\displaystyle M={\sqrt {\alpha ^{2}+\beta ^{2}}}.}
181:(ARMA) models that combine AR with other features.
9431:
9396:
9269:
9142:
9115:
9095:
9065:
9043:
8953:
8910:
8843:
8708:
8528:
8405:
8284:
8222:
8202:
8173:
8151:
8043:
8023:
7977:
7941:
7905:
7885:
7853:
7807:
7778:
7747:
7616:
7546:
7515:
7352:
7326:
7295:
7124:
7050:
6948:is a para-permanent of a lower triangular matrix
6880:
6813:
6687:
6397:
6163:
5963:
5878:
5713:
5193:
5120:
4902:
4809:
4650:
4520:
4408:
4366:
4330:
4274:
4247:
4220:
4191:
3957:
3930:
3901:
3785:
3708:
3650:
3630:
3610:
3590:
3568:
3311:
3163:
3092:
3034:
3014:
2991:
2964:
2937:
2917:
2896:{\displaystyle a_{n}=C\lambda ^{n}+Dn\lambda ^{n}}
2895:
2831:
2787:
2708:
2681:
2654:
2632:
2531:
2495:
2466:
2391:
2349:
2264:
2238:
2190:
2154:
2121:
2078:
2045:
2005:
1949:
1837:
1817:
1778:
1757:
1730:
1610:
1548:
1518:
1324:
1205:
1086:
991:
824:
716:
600:
444:
422:
312:
9307:will converge to the sum of their constant terms
4958:Converting complex solution to trigonometric form
8529:{\displaystyle (x^{2}+3x-4)y^{}-(3x+1)y^{}+2y=0}
18:Characteristic equation (of difference equation)
8406:{\displaystyle x^{m}*y^{}\to n(n-1)...(n-m+1)f}
9158:Certain difference equations - in particular,
7989:Relation to solution to differential equations
6182:th order difference equation to a first-order
2246:is a solution for the recurrence exactly when
9480:Fundamental Methods of Mathematical Economics
9399:Fundamental Methods of Mathematical Economics
5702:
5586:
3795:one may simplify the solution given above as
69:—that is, in the values of the elements of a
8:
8054:This is not a coincidence. Considering the
6066:Solution with duplicate characteristic roots
5778:Now the process of finding the coefficients
4282:converges to a fixed value ) if and only if
191:linear recurrence with constant coefficients
122:of the iterates, and normally these are the
65:that is linear in the various iterates of a
51:linear recurrence with constant coefficients
9162:difference equations - can be solved using
7125:{\textstyle Y(x)=\sum _{t\geq 0}y_{t}x^{t}}
6934:and with the rest of its elements being 0.
4720:Solution with distinct characteristic roots
2403:) in the recurrence relation, we find that
9465:Mathematics for the Analysis of Algorithms
7786:determines the initial set of corrections
7617:{\displaystyle Y(x)={\frac {f(x)}{g(x)}}.}
4201:In this way there is no need to solve for
1611:{\displaystyle r_{1},r_{2},\ldots ,r_{d},}
9403:(Third ed.). New York: McGraw-Hill.
9258:
9253:
9243:
9224:
9219:
9209:
9196:
9190:
9134:
9128:
9108:
9084:
9078:
9058:
8970:
8939:
8927:
8866:
8725:
8709:{\displaystyle n(n-1)f+3nf-4f-3nf-f+2f=0}
8545:
8499:
8462:
8434:
8425:
8316:
8303:
8297:
8249:
8243:
8215:
8186:
8166:
8143:
8095:
8088:
8082:
8071:
8065:
8036:
8012:
8006:
7960:
7954:
7924:
7918:
7898:
7877:
7872:
7866:
7845:
7835:
7820:
7791:
7762:
7734:
7729:
7719:
7689:
7682:
7676:
7637:
7635:
7579:
7562:
7530:
7501:
7491:
7472:
7462:
7446:
7430:
7389:
7367:
7339:
7310:
7260:
7239:
7229:
7198:
7188:
7160:
7139:
7116:
7106:
7090:
7069:
7027:
7017:
6992:
6982:
6969:
6963:
6873:
6858:
6853:
6847:
6838:
6833:
6830:
6791:
6764:
6744:
6732:
6723:
6718:
6715:
6654:
6631:
6619:
6618:
6617:
6592:
6569:
6537:
6527:
6496:
6486:
6461:
6451:
6428:
6420:
6418:
6377:
6367:
6342:
6332:
6313:
6303:
6290:
6284:
6152:
6139:
6126:
6116:
6103:
6097:
5931:
5919:
5906:
5900:
5895:
5827:
5818:
5701:
5700:
5648:
5595:
5585:
5584:
5578:
5553:
5506:
5493:
5452:
5437:
5412:
5394:
5381:
5363:
5350:
5332:
5319:
5307:
5292:
5275:
5264:
5248:
5235:
5230:
5220:
5212:
5210:
5180:
5167:
5161:
5153:
5100:
5087:
5049:
5036:
5030:
4894:
4889:
4879:
4860:
4855:
4845:
4832:
4826:
4795:
4785:
4760:
4750:
4737:
4731:
4642:
4620:
4607:
4591:
4566:
4556:
4543:
4537:
4506:
4496:
4471:
4461:
4448:
4442:
4389:
4381:
4379:
4353:
4345:
4343:
4305:
4297:
4295:
4267:
4239:
4233:
4212:
4206:
4168:
4159:
4115:
4109:
4095:
4079:
4063:
4053:
4046:
4014:
4001:
3988:
3974:
3972:
3949:
3943:
3922:
3916:
3830:
3808:
3802:
3765:
3752:
3739:
3726:
3720:
3682:
3669:
3663:
3643:
3623:
3603:
3583:
3550:
3518:
3490:
3477:
3471:
3419:
3387:
3359:
3346:
3340:
3330:
3328:
3276:
3202:
3186:
3180:
3155:
3150:
3134:
3129:
3113:
3107:
3066:
3053:
3047:
3027:
3007:
2983:
2977:
2956:
2950:
2930:
2910:
2887:
2868:
2852:
2846:
2808:
2802:
2779:
2774:
2758:
2753:
2737:
2731:
2700:
2694:
2673:
2667:
2647:
2592:
2556:
2548:
2546:
2517:
2511:
2482:
2452:
2430:
2414:
2408:
2383:
2370:
2364:
2332:
2310:
2294:
2288:
2251:
2230:
2217:
2211:
2171:
2140:
2134:
2113:
2097:
2091:
2064:
2058:
2037:
2024:
2018:
1991:
1975:
1969:
1938:
1928:
1918:
1905:
1892:
1879:
1869:
1856:
1850:
1830:
1809:
1791:
1771:
1749:
1743:
1719:
1714:
1704:
1685:
1680:
1670:
1657:
1652:
1642:
1629:
1623:
1599:
1580:
1567:
1561:
1541:
1492:
1482:
1457:
1447:
1428:
1409:
1399:
1374:
1364:
1351:
1345:
1292:
1282:
1257:
1247:
1228:
1222:
1215:can be combined with its equivalent form
1185:
1175:
1150:
1140:
1127:
1121:
1072:
1062:
1037:
1027:
1014:
1008:
978:
959:
944:
920:
901:
886:
868:
855:
844:
813:
794:
778:
769:
763:
696:
686:
661:
651:
638:
632:
592:
567:
557:
538:
528:
515:
494:
437:
405:
395:
364:
354:
335:
329:
289:
279:
254:
244:
231:
225:
9422:
9420:
7626:The closed form can then be derived via
2350:{\displaystyle a_{n}=Aa_{n-1}+Ba_{n-2}.}
9438:(Third ed.). New York: Macmillan.
9387:
6906:matrix in which the first row contains
2467:{\displaystyle r^{n}=Ar^{n-1}+Br^{n-2}}
9467:(2nd ed.), Birkhäuser, p. 17
6930:is a column vector with first element
6273:, and so on. Then the original single
4708:or instead there may be some that are
157:, etc. They are used in modeling such
81:or discrete moment in time denoted as
9344:case being to the steady-state value
8844:{\displaystyle -4f+2nf+n(n-4)f+2f=0.}
7949:determine the polynomial coefficient
6174:Solution by conversion to matrix form
5771:; the last equality here made use of
108:of such an equation is a function of
7:
9505:Zatorsky, Roman; Goy, Taras (2016).
4286:eigenvalues are smaller than one in
9482:, third edition, McGraw-Hill, 1984.
6952:Solution using generating functions
4431:Characteristic polynomial and roots
9495:69(1), February 1996, 34–43.
8083:
7334:is a polynomial of degree at most
7060:can be solved using the theory of
6823:this can be put in matrix form as
6186:. This is accomplished by writing
6090:and a solution may be of the form
2797:while if they are identical (when
25:
6407:can be replaced by the following
4704:It may be that all the roots are
4435:Solving the homogeneous equation
2086:and the most general solution is
1532:Solution example for small orders
7861:determines the exponential term
6874:
6854:
6848:
6834:
6719:
4678:. These roots can be solved for
2999:to produce a specific solution.
1111:. This is the homogeneous form.
85:, one period earlier denoted as
6086:), they can both be denoted as
4331:{\displaystyle |A|<1-B<2}
3965:are the initial conditions and
2716:: these roots are known as the
1334:to obtain (by solving both for
9288:absolute value of the modulus
9032:
9026:
9014:
9002:
8990:
8978:
8832:
8826:
8814:
8802:
8796:
8784:
8775:
8763:
8748:
8736:
8697:
8691:
8679:
8667:
8658:
8646:
8631:
8619:
8607:
8595:
8580:
8568:
8562:
8550:
8506:
8500:
8492:
8477:
8469:
8463:
8455:
8427:
8400:
8382:
8376:
8358:
8346:
8334:
8328:
8323:
8317:
8279:
8267:
8261:
8256:
8250:
8197:
8191:
8140:
8127:
8113:
8107:
8102:
8096:
7993:The method for solving linear
7972:
7966:
7936:
7930:
7842:
7822:
7802:
7796:
7773:
7767:
7726:
7706:
7701:
7695:
7663:
7657:
7649:
7643:
7628:partial fraction decomposition
7605:
7599:
7591:
7585:
7573:
7567:
7541:
7535:
7419:
7413:
7378:
7372:
7321:
7315:
7290:
7284:
7254:
7248:
7213:
7207:
7178:
7172:
7150:
7144:
7080:
7074:
5958:
5943:
4390:
4382:
4354:
4346:
4306:
4298:
3888:
3879:
3864:
3855:
3827:
3817:
3544:
3538:
3512:
3506:
3413:
3407:
3381:
3375:
3303:
3288:
3255:
3246:
3231:
3222:
2006:{\displaystyle a_{n}=ra_{n-1}}
1806:
1793:
1511:
1499:
1311:
1299:
614:Conversion to homogeneous form
505:
499:
1:
8911:{\displaystyle ay''+by'+cy=0}
8024:{\displaystyle e^{\lambda x}}
7854:{\displaystyle (x-r_{i})^{m}}
5977:is the angle whose cosine is
5888:which can also be written as
5727:is the angle whose cosine is
4660:for its characteristic roots
179:autoregressive moving average
9366:Linear differential equation
7913:together with the numerator
4248:{\displaystyle \lambda _{2}}
4221:{\displaystyle \lambda _{1}}
2709:{\displaystyle \lambda _{2}}
2682:{\displaystyle \lambda _{1}}
2122:{\displaystyle a_{n}=kr^{n}}
1964:For order 1, the recurrence
9296:Thus the evolving variable
9170:-transforms are a class of
9160:linear constant coefficient
9053:It is easy to see that the
9044:{\displaystyle af+bf+cf=0.}
4931:as-yet-unknown parameters;
2399:? Substituting this guess (
2392:{\displaystyle a_{n}=r^{n}}
2239:{\displaystyle a_{n}=r^{n}}
2046:{\displaystyle a_{n}=r^{n}}
29:Constant-recursive sequence
9566:
9371:Skolem–Mahler–Lech theorem
8859:The differential equation
6941:. In the homogeneous case
6939:Matrix difference equation
6184:matrix difference equation
4423:
4409:{\displaystyle |A|<1-B}
3100:Then it can be shown that
2832:{\displaystyle A^{2}+4B=0}
171:autoregressive (AR) models
59:linear difference equation
55:linear recurrence relation
26:
9183:In the solution equation
9154:Solving with z-transforms
8954:{\displaystyle y=e^{ax}.}
8285:{\displaystyle y^{}\to f}
7886:{\displaystyle r_{i}^{n}}
4691:not necessarily otherwise
4338:, which is equivalent to
1818:{\displaystyle (x-r)^{3}}
484:characteristic polynomial
8044:{\displaystyle \lambda }
7978:{\displaystyle k_{i}(n)}
7942:{\displaystyle f_{i}(x)}
4367:{\displaystyle |B|<1}
2662:to obtain the two roots
8210:evaluated at the point
6411:first-order equations:
4912:where the coefficients
4262:(that is, the variable
3651:{\displaystyle \delta }
2532:{\displaystyle r^{n-2}}
2164:characteristic equation
2155:{\displaystyle a_{0}=k}
2079:{\displaystyle a_{0}=1}
1738:where the coefficients
9395:Chiang, Alpha (1984).
9271:
9144:
9117:
9097:
9096:{\displaystyle e^{ax}}
9067:
9045:
8955:
8912:
8845:
8710:
8530:
8407:
8286:
8224:
8204:
8175:
8153:
8087:
8045:
8025:
7995:differential equations
7979:
7943:
7907:
7887:
7855:
7809:
7780:
7749:
7618:
7554:can be expressed as a
7548:
7517:
7354:
7328:
7297:
7126:
7052:
6882:
6815:
6689:
6399:
6165:
5965:
5880:
5715:
5195:
5122:
4904:
4811:
4652:
4522:
4410:
4368:
4332:
4276:
4249:
4222:
4193:
3959:
3932:
3903:
3787:
3710:
3652:
3632:
3612:
3592:
3570:
3313:
3165:
3094:
3036:
3016:
2993:
2966:
2939:
2919:
2897:
2833:
2789:
2710:
2683:
2656:
2634:
2533:
2497:
2496:{\displaystyle n>1}
2468:
2393:
2351:
2266:
2240:
2192:
2156:
2123:
2080:
2047:
2007:
1951:
1839:
1819:
1780:
1759:
1732:
1612:
1550:
1520:
1326:
1207:
1088:
993:
826:
718:
602:
446:
424:
314:
173:and in models such as
147:gross domestic product
92:, one period later as
9272:
9145:
9143:{\displaystyle a^{n}}
9118:
9098:
9068:
9046:
8956:
8913:
8846:
8711:
8531:
8408:
8287:
8225:
8205:
8176:
8154:
8067:
8046:
8026:
7980:
7944:
7908:
7888:
7856:
7810:
7781:
7750:
7619:
7549:
7518:
7355:
7329:
7298:
7127:
7053:
6883:
6816:
6690:
6400:
6166:
5966:
5881:
5716:
5196:
5123:
4937:solved simultaneously
4905:
4812:
4653:
4523:
4411:
4369:
4333:
4277:
4250:
4223:
4194:
3960:
3958:{\displaystyle a_{2}}
3933:
3931:{\displaystyle a_{1}}
3904:
3788:
3711:
3653:
3633:
3613:
3593:
3571:
3314:
3166:
3095:
3037:
3017:
2994:
2992:{\displaystyle a_{1}}
2967:
2965:{\displaystyle a_{0}}
2940:
2920:
2898:
2834:
2790:
2711:
2684:
2657:
2635:
2534:
2498:
2469:
2394:
2352:
2267:
2241:
2193:
2191:{\displaystyle t-r=0}
2157:
2124:
2081:
2048:
2008:
1952:
1840:
1825:, with the same root
1820:
1781:
1760:
1758:{\displaystyle k_{i}}
1733:
1613:
1551:
1521:
1327:
1208:
1089:
994:
827:
719:
603:
447:
432:The positive integer
425:
315:
175:vector autoregression
27:Further information:
9550:Recurrence relations
9493:Mathematics Magazine
9189:
9127:
9107:
9077:
9057:
8969:
8926:
8865:
8724:
8544:
8424:
8296:
8242:
8214:
8203:{\displaystyle f(x)}
8185:
8165:
8064:
8035:
8005:
7953:
7917:
7897:
7865:
7819:
7808:{\displaystyle z(n)}
7790:
7779:{\displaystyle p(x)}
7761:
7757:then the polynomial
7634:
7561:
7547:{\displaystyle Y(x)}
7529:
7366:
7338:
7327:{\displaystyle p(x)}
7309:
7138:
7068:
7062:generating functions
6962:
6829:
6714:
6697:Defining the vector
6417:
6283:
6096:
5894:
5817:
5209:
5152:
5029:
4825:
4730:
4536:
4441:
4378:
4342:
4294:
4266:
4232:
4205:
3971:
3942:
3915:
3801:
3719:
3662:
3642:
3622:
3602:
3582:
3327:
3179:
3173:can be rewritten as
3106:
3046:
3026:
3006:
2976:
2949:
2929:
2909:
2845:
2801:
2730:
2718:characteristic roots
2693:
2666:
2646:
2545:
2510:
2506:Dividing through by
2481:
2407:
2363:
2287:
2274:generating functions
2250:
2210:
2170:
2133:
2090:
2057:
2017:
1968:
1849:
1829:
1790:
1770:
1742:
1622:
1560:
1540:
1344:
1221:
1120:
1007:
843:
762:
631:
493:
436:
328:
224:
61:) sets equal to 0 a
9459:Greene, Daniel H.;
9361:Recurrence relation
9263:
9229:
9172:integral transforms
8292:and more generally
7882:
7353:{\displaystyle n-1}
5773:de Moivre's formula
5280:
5240:
4899:
4865:
3160:
3139:
2784:
2763:
2278:formal power series
2265:{\displaystyle t=r}
1724:
1690:
1662:
322:or equivalently as
9478:Chiang, Alpha C.,
9340:are all positive.
9267:
9249:
9215:
9140:
9113:
9093:
9073:-th derivative of
9063:
9041:
8951:
8908:
8841:
8706:
8526:
8403:
8282:
8220:
8200:
8181:-th derivative of
8171:
8149:
8041:
8021:
7975:
7939:
7903:
7883:
7868:
7851:
7815:, the denominator
7805:
7776:
7745:
7681:
7614:
7544:
7513:
7350:
7324:
7293:
7122:
7101:
7064:. First, we write
7048:
6878:
6811:
6805:
6685:
6683:
6395:
6277:th-order equation
6161:
6008:and whose sine is
5961:
5876:
5749:and whose sine is
5711:
5709:
5260:
5226:
5191:
5118:
5023:can be written as
4900:
4885:
4851:
4807:
4648:
4518:
4406:
4364:
4328:
4272:
4245:
4218:
4189:
4187:
3955:
3928:
3899:
3783:
3706:
3648:
3628:
3618:(or equivalently,
3608:
3588:
3566:
3564:
3560:
3528:
3429:
3397:
3309:
3161:
3146:
3125:
3090:
3032:
3012:
2989:
2962:
2935:
2915:
2893:
2829:
2785:
2770:
2749:
2706:
2679:
2652:
2630:
2628:
2529:
2493:
2464:
2389:
2347:
2262:
2236:
2188:
2152:
2119:
2076:
2043:
2003:
1947:
1835:
1815:
1776:
1755:
1728:
1710:
1676:
1648:
1608:
1546:
1516:
1322:
1203:
1084:
989:
822:
732:steady state value
714:
598:
442:
420:
310:
115:initial conditions
9540:Dynamical systems
9434:Economic Dynamics
9351:instead of to 0.
9116:{\displaystyle 0}
9066:{\displaystyle n}
8223:{\displaystyle a}
8174:{\displaystyle n}
8125:
7906:{\displaystyle m}
7893:, and the degree
7743:
7672:
7667:
7609:
7556:rational function
7508:
7405:
7276:
7086:
5925:
5402:
5389:
5340:
5327:
5186:
5108:
5095:
4714:complex conjugate
4695:numerical methods
4275:{\displaystyle a}
4179:
4176:
4133:
4130:
4024:
3838:
3631:{\displaystyle G}
3611:{\displaystyle F}
3591:{\displaystyle E}
3559:
3527:
3496:
3428:
3396:
3365:
3035:{\displaystyle D}
3015:{\displaystyle C}
2938:{\displaystyle D}
2918:{\displaystyle C}
2655:{\displaystyle r}
2474:must be true for
2013:has the solution
1838:{\displaystyle r}
1779:{\displaystyle n}
1549:{\displaystyle d}
820:
445:{\displaystyle n}
53:(also known as a
47:dynamical systems
16:(Redirected from
9557:
9519:
9518:
9502:
9496:
9489:
9483:
9476:
9470:
9468:
9461:Knuth, Donald E.
9456:
9450:
9449:
9437:
9424:
9415:
9414:
9402:
9392:
9350:
9328:
9321:
9317:
9306:
9299:
9291:
9286:
9282:
9276:
9274:
9273:
9268:
9262:
9257:
9248:
9247:
9228:
9223:
9214:
9213:
9201:
9200:
9149:
9147:
9146:
9141:
9139:
9138:
9122:
9120:
9119:
9114:
9102:
9100:
9099:
9094:
9092:
9091:
9072:
9070:
9069:
9064:
9050:
9048:
9047:
9042:
8960:
8958:
8957:
8952:
8947:
8946:
8917:
8915:
8914:
8909:
8892:
8878:
8850:
8848:
8847:
8842:
8715:
8713:
8712:
8707:
8535:
8533:
8532:
8527:
8510:
8509:
8473:
8472:
8439:
8438:
8412:
8410:
8409:
8404:
8327:
8326:
8308:
8307:
8291:
8289:
8288:
8283:
8260:
8259:
8229:
8227:
8226:
8221:
8209:
8207:
8206:
8201:
8180:
8178:
8177:
8172:
8158:
8156:
8155:
8150:
8148:
8147:
8126:
8124:
8116:
8106:
8105:
8089:
8086:
8081:
8050:
8048:
8047:
8042:
8030:
8028:
8027:
8022:
8020:
8019:
7984:
7982:
7981:
7976:
7965:
7964:
7948:
7946:
7945:
7940:
7929:
7928:
7912:
7910:
7909:
7904:
7892:
7890:
7889:
7884:
7881:
7876:
7860:
7858:
7857:
7852:
7850:
7849:
7840:
7839:
7814:
7812:
7811:
7806:
7785:
7783:
7782:
7777:
7754:
7752:
7751:
7746:
7744:
7742:
7741:
7740:
7739:
7738:
7724:
7723:
7704:
7694:
7693:
7683:
7680:
7668:
7666:
7652:
7638:
7623:
7621:
7620:
7615:
7610:
7608:
7594:
7580:
7553:
7551:
7550:
7545:
7522:
7520:
7519:
7514:
7509:
7507:
7506:
7505:
7496:
7495:
7477:
7476:
7467:
7466:
7451:
7450:
7431:
7426:
7422:
7406:
7404:
7390:
7359:
7357:
7356:
7351:
7333:
7331:
7330:
7325:
7302:
7300:
7299:
7294:
7277:
7275:
7261:
7244:
7243:
7234:
7233:
7203:
7202:
7193:
7192:
7165:
7164:
7131:
7129:
7128:
7123:
7121:
7120:
7111:
7110:
7100:
7057:
7055:
7054:
7049:
7038:
7037:
7022:
7021:
7003:
7002:
6987:
6986:
6974:
6973:
6947:
6933:
6929:
6923:
6905:
6895:
6887:
6885:
6884:
6879:
6877:
6869:
6868:
6857:
6851:
6843:
6842:
6837:
6820:
6818:
6817:
6812:
6810:
6809:
6802:
6801:
6775:
6774:
6755:
6754:
6728:
6727:
6722:
6707:
6694:
6692:
6691:
6686:
6684:
6677:
6676:
6642:
6641:
6613:
6609:
6608:
6580:
6579:
6554:
6553:
6532:
6531:
6513:
6512:
6491:
6490:
6478:
6477:
6456:
6455:
6439:
6438:
6410:
6404:
6402:
6401:
6396:
6388:
6387:
6372:
6371:
6353:
6352:
6337:
6336:
6324:
6323:
6308:
6307:
6295:
6294:
6276:
6272:
6239:
6206:
6181:
6170:
6168:
6167:
6162:
6157:
6156:
6144:
6143:
6131:
6130:
6121:
6120:
6108:
6107:
6089:
6085:
6061:
6054:
6038:
6037:
6035:
6034:
6033:
6032:
6020:
6017:
6007:
6006:
6004:
6003:
6002:
6001:
5989:
5986:
5976:
5970:
5968:
5967:
5962:
5936:
5935:
5926:
5924:
5923:
5911:
5910:
5901:
5885:
5883:
5882:
5877:
5875:
5871:
5832:
5831:
5810:
5800:
5788:
5770:
5769:
5767:
5766:
5761:
5758:
5748:
5747:
5745:
5744:
5739:
5736:
5726:
5720:
5718:
5717:
5712:
5710:
5706:
5705:
5699:
5695:
5659:
5658:
5640:
5636:
5600:
5599:
5590:
5589:
5583:
5582:
5567:
5563:
5559:
5558:
5557:
5552:
5548:
5517:
5516:
5498:
5497:
5492:
5488:
5457:
5456:
5442:
5441:
5426:
5422:
5418:
5417:
5416:
5411:
5407:
5403:
5395:
5390:
5382:
5374:
5373:
5355:
5354:
5349:
5345:
5341:
5333:
5328:
5320:
5312:
5311:
5297:
5296:
5279:
5274:
5259:
5258:
5239:
5234:
5225:
5224:
5200:
5198:
5197:
5192:
5187:
5185:
5184:
5172:
5171:
5162:
5141:
5133:
5127:
5125:
5124:
5119:
5117:
5113:
5109:
5101:
5096:
5088:
5060:
5059:
5041:
5040:
5022:
5011:
5010:
5009:
4985:
4984:
4983:
4953:
4942:
4934:
4930:
4926:
4922:
4909:
4907:
4906:
4901:
4898:
4893:
4884:
4883:
4864:
4859:
4850:
4849:
4837:
4836:
4816:
4814:
4813:
4808:
4806:
4805:
4790:
4789:
4771:
4770:
4755:
4754:
4742:
4741:
4688:
4677:
4657:
4655:
4654:
4649:
4647:
4646:
4631:
4630:
4612:
4611:
4602:
4601:
4577:
4576:
4561:
4560:
4548:
4547:
4527:
4525:
4524:
4519:
4517:
4516:
4501:
4500:
4482:
4481:
4466:
4465:
4453:
4452:
4420:General solution
4415:
4413:
4412:
4407:
4393:
4385:
4373:
4371:
4370:
4365:
4357:
4349:
4337:
4335:
4334:
4329:
4309:
4301:
4281:
4279:
4278:
4273:
4254:
4252:
4251:
4246:
4244:
4243:
4227:
4225:
4224:
4219:
4217:
4216:
4198:
4196:
4195:
4190:
4188:
4184:
4180:
4178:
4177:
4169:
4160:
4134:
4132:
4131:
4120:
4119:
4110:
4104:
4100:
4099:
4084:
4083:
4068:
4067:
4058:
4057:
4047:
4025:
4020:
4019:
4018:
4006:
4005:
3989:
3964:
3962:
3961:
3956:
3954:
3953:
3937:
3935:
3934:
3929:
3927:
3926:
3908:
3906:
3905:
3900:
3895:
3891:
3840:
3839:
3831:
3813:
3812:
3792:
3790:
3789:
3784:
3770:
3769:
3757:
3756:
3744:
3743:
3731:
3730:
3715:
3713:
3712:
3707:
3687:
3686:
3674:
3673:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3617:
3615:
3614:
3609:
3597:
3595:
3594:
3589:
3575:
3573:
3572:
3567:
3565:
3561:
3552:
3529:
3520:
3497:
3495:
3494:
3482:
3481:
3472:
3461:
3460:
3430:
3421:
3398:
3389:
3366:
3364:
3363:
3351:
3350:
3341:
3318:
3316:
3315:
3310:
3281:
3280:
3262:
3258:
3207:
3206:
3191:
3190:
3170:
3168:
3167:
3162:
3159:
3154:
3138:
3133:
3118:
3117:
3099:
3097:
3096:
3091:
3071:
3070:
3058:
3057:
3041:
3039:
3038:
3033:
3021:
3019:
3018:
3013:
2998:
2996:
2995:
2990:
2988:
2987:
2971:
2969:
2968:
2963:
2961:
2960:
2944:
2942:
2941:
2936:
2924:
2922:
2921:
2916:
2902:
2900:
2899:
2894:
2892:
2891:
2873:
2872:
2857:
2856:
2838:
2836:
2835:
2830:
2813:
2812:
2794:
2792:
2791:
2786:
2783:
2778:
2762:
2757:
2742:
2741:
2715:
2713:
2712:
2707:
2705:
2704:
2688:
2686:
2685:
2680:
2678:
2677:
2661:
2659:
2658:
2653:
2639:
2637:
2636:
2631:
2629:
2597:
2596:
2561:
2560:
2538:
2536:
2535:
2530:
2528:
2527:
2502:
2500:
2499:
2494:
2473:
2471:
2470:
2465:
2463:
2462:
2441:
2440:
2419:
2418:
2398:
2396:
2395:
2390:
2388:
2387:
2375:
2374:
2356:
2354:
2353:
2348:
2343:
2342:
2321:
2320:
2299:
2298:
2271:
2269:
2268:
2263:
2245:
2243:
2242:
2237:
2235:
2234:
2222:
2221:
2197:
2195:
2194:
2189:
2161:
2159:
2158:
2153:
2145:
2144:
2128:
2126:
2125:
2120:
2118:
2117:
2102:
2101:
2085:
2083:
2082:
2077:
2069:
2068:
2052:
2050:
2049:
2044:
2042:
2041:
2029:
2028:
2012:
2010:
2009:
2004:
2002:
2001:
1980:
1979:
1956:
1954:
1953:
1948:
1943:
1942:
1933:
1932:
1923:
1922:
1910:
1909:
1897:
1896:
1884:
1883:
1874:
1873:
1861:
1860:
1844:
1842:
1841:
1836:
1824:
1822:
1821:
1816:
1814:
1813:
1785:
1783:
1782:
1777:
1764:
1762:
1761:
1756:
1754:
1753:
1737:
1735:
1734:
1729:
1723:
1718:
1709:
1708:
1689:
1684:
1675:
1674:
1661:
1656:
1647:
1646:
1634:
1633:
1617:
1615:
1614:
1609:
1604:
1603:
1585:
1584:
1572:
1571:
1555:
1553:
1552:
1547:
1525:
1523:
1522:
1517:
1515:
1514:
1487:
1486:
1468:
1467:
1452:
1451:
1439:
1438:
1420:
1419:
1404:
1403:
1385:
1384:
1369:
1368:
1356:
1355:
1337:
1331:
1329:
1328:
1323:
1315:
1314:
1287:
1286:
1268:
1267:
1252:
1251:
1239:
1238:
1212:
1210:
1209:
1204:
1196:
1195:
1180:
1179:
1161:
1160:
1145:
1144:
1132:
1131:
1110:
1099:
1093:
1091:
1090:
1085:
1083:
1082:
1067:
1066:
1048:
1047:
1032:
1031:
1019:
1018:
998:
996:
995:
990:
988:
984:
983:
982:
970:
969:
949:
948:
930:
926:
925:
924:
912:
911:
891:
890:
878:
874:
873:
872:
860:
859:
831:
829:
828:
823:
821:
819:
818:
817:
799:
798:
779:
774:
773:
755:
748:
744:
740:
723:
721:
720:
715:
707:
706:
691:
690:
672:
671:
656:
655:
643:
642:
624:
607:
605:
604:
599:
597:
596:
578:
577:
562:
561:
549:
548:
533:
532:
520:
519:
477:
466:
451:
449:
448:
443:
429:
427:
426:
421:
410:
409:
400:
399:
381:
380:
359:
358:
346:
345:
319:
317:
316:
311:
300:
299:
284:
283:
265:
264:
249:
248:
236:
235:
217:
213:
167:stochastic terms
125:
121:
111:
98:
91:
84:
21:
9565:
9564:
9560:
9559:
9558:
9556:
9555:
9554:
9525:
9524:
9523:
9522:
9504:
9503:
9499:
9490:
9486:
9477:
9473:
9458:
9457:
9453:
9446:
9428:Baumol, William
9426:
9425:
9418:
9411:
9394:
9393:
9389:
9384:
9357:
9345:
9326:
9322:will persist.
9319:
9316:
9308:
9304:
9297:
9289:
9284:
9280:
9239:
9205:
9192:
9187:
9186:
9181:
9156:
9130:
9125:
9124:
9105:
9104:
9080:
9075:
9074:
9055:
9054:
8967:
8966:
8935:
8924:
8923:
8885:
8871:
8863:
8862:
8722:
8721:
8542:
8541:
8495:
8458:
8430:
8422:
8421:
8312:
8299:
8294:
8293:
8245:
8240:
8239:
8212:
8211:
8183:
8182:
8163:
8162:
8139:
8117:
8091:
8090:
8062:
8061:
8033:
8032:
8008:
8003:
8002:
7991:
7956:
7951:
7950:
7920:
7915:
7914:
7895:
7894:
7863:
7862:
7841:
7831:
7817:
7816:
7788:
7787:
7759:
7758:
7730:
7725:
7715:
7705:
7685:
7684:
7653:
7639:
7632:
7631:
7595:
7581:
7559:
7558:
7527:
7526:
7497:
7487:
7468:
7458:
7442:
7435:
7394:
7388:
7384:
7364:
7363:
7336:
7335:
7307:
7306:
7265:
7235:
7225:
7194:
7184:
7156:
7136:
7135:
7112:
7102:
7066:
7065:
7023:
7013:
6988:
6978:
6965:
6960:
6959:
6956:The recurrence
6954:
6946:
6942:
6931:
6925:
6922:
6913:
6907:
6901: ×
6897:
6891:
6852:
6832:
6827:
6826:
6804:
6803:
6787:
6784:
6783:
6777:
6776:
6760:
6757:
6756:
6740:
6733:
6717:
6712:
6711:
6706:
6698:
6682:
6681:
6650:
6643:
6627:
6624:
6623:
6611:
6610:
6588:
6581:
6565:
6562:
6561:
6533:
6523:
6492:
6482:
6457:
6447:
6440:
6424:
6415:
6414:
6408:
6373:
6363:
6338:
6328:
6309:
6299:
6286:
6281:
6280:
6274:
6271:
6260:
6250:
6241:
6238:
6227:
6217:
6208:
6205:
6196:
6187:
6179:
6176:
6148:
6135:
6122:
6112:
6099:
6094:
6093:
6087:
6084:
6077:
6071:
6068:
6056:
6049:
6045:
6024:
6022:
6021:
6018:
6013:
6012:
6010:
6009:
5993:
5991:
5990:
5987:
5982:
5981:
5979:
5978:
5974:
5927:
5915:
5902:
5892:
5891:
5837:
5833:
5823:
5815:
5814:
5802:
5799:
5790:
5787:
5779:
5762:
5759:
5754:
5753:
5751:
5750:
5740:
5737:
5732:
5731:
5729:
5728:
5724:
5708:
5707:
5664:
5660:
5644:
5605:
5601:
5591:
5574:
5565:
5564:
5523:
5519:
5518:
5502:
5463:
5459:
5458:
5448:
5447:
5443:
5433:
5424:
5423:
5380:
5376:
5375:
5359:
5318:
5314:
5313:
5303:
5302:
5298:
5288:
5281:
5244:
5216:
5207:
5206:
5176:
5163:
5150:
5149:
5139:
5131:
5086:
5082:
5045:
5032:
5027:
5026:
5021:
5013:
5008:
5002:
5001:
5000:
4996:
4987:
4982:
4977:
4976:
4975:
4971:
4963:
4960:
4952:
4944:
4940:
4932:
4928:
4924:
4921:
4913:
4875:
4841:
4828:
4823:
4822:
4791:
4781:
4756:
4746:
4733:
4728:
4727:
4722:
4710:complex numbers
4683:
4676:
4667:
4661:
4638:
4616:
4603:
4587:
4562:
4552:
4539:
4534:
4533:
4502:
4492:
4467:
4457:
4444:
4439:
4438:
4433:
4428:
4422:
4376:
4375:
4340:
4339:
4292:
4291:
4264:
4263:
4235:
4230:
4229:
4208:
4203:
4202:
4186:
4185:
4164:
4155:
4142:
4136:
4135:
4111:
4105:
4091:
4075:
4059:
4049:
4048:
4033:
4027:
4026:
4010:
3997:
3990:
3981:
3969:
3968:
3945:
3940:
3939:
3918:
3913:
3912:
3845:
3841:
3826:
3804:
3799:
3798:
3761:
3748:
3735:
3722:
3717:
3716:
3678:
3665:
3660:
3659:
3640:
3639:
3620:
3619:
3600:
3599:
3580:
3579:
3563:
3562:
3530:
3498:
3486:
3473:
3462:
3459:
3432:
3431:
3399:
3367:
3355:
3342:
3325:
3324:
3272:
3212:
3208:
3198:
3182:
3177:
3176:
3109:
3104:
3103:
3062:
3049:
3044:
3043:
3024:
3023:
3004:
3003:
2979:
2974:
2973:
2952:
2947:
2946:
2927:
2926:
2907:
2906:
2883:
2864:
2848:
2843:
2842:
2804:
2799:
2798:
2733:
2728:
2727:
2696:
2691:
2690:
2669:
2664:
2663:
2644:
2643:
2627:
2626:
2613:
2588:
2585:
2584:
2562:
2552:
2543:
2542:
2513:
2508:
2507:
2479:
2478:
2448:
2426:
2410:
2405:
2404:
2379:
2366:
2361:
2360:
2328:
2306:
2290:
2285:
2284:
2280:) or matrices.
2248:
2247:
2226:
2213:
2208:
2207:
2204:
2168:
2167:
2136:
2131:
2130:
2109:
2093:
2088:
2087:
2060:
2055:
2054:
2033:
2020:
2015:
2014:
1987:
1971:
1966:
1965:
1962:
1934:
1924:
1914:
1901:
1888:
1875:
1865:
1852:
1847:
1846:
1827:
1826:
1805:
1788:
1787:
1768:
1767:
1745:
1740:
1739:
1700:
1666:
1638:
1625:
1620:
1619:
1595:
1576:
1563:
1558:
1557:
1556:distinct roots
1538:
1537:
1534:
1488:
1478:
1453:
1443:
1424:
1405:
1395:
1370:
1360:
1347:
1342:
1341:
1335:
1288:
1278:
1253:
1243:
1224:
1219:
1218:
1181:
1171:
1146:
1136:
1123:
1118:
1117:
1101:
1097:
1068:
1058:
1033:
1023:
1010:
1005:
1004:
974:
955:
954:
950:
940:
916:
897:
896:
892:
882:
864:
851:
850:
846:
841:
840:
809:
790:
783:
765:
760:
759:
750:
746:
742:
735:
692:
682:
657:
647:
634:
629:
628:
625:, the equation
619:
616:
588:
563:
553:
534:
524:
511:
491:
490:
472:
461:
434:
433:
401:
391:
360:
350:
331:
326:
325:
285:
275:
250:
240:
227:
222:
221:
215:
212:
203:
197:
187:
169:in the form of
123:
119:
109:
93:
86:
82:
31:
23:
22:
15:
12:
11:
5:
9563:
9561:
9553:
9552:
9547:
9545:Linear algebra
9542:
9537:
9527:
9526:
9521:
9520:
9497:
9484:
9471:
9451:
9444:
9416:
9409:
9386:
9385:
9383:
9380:
9379:
9378:
9376:Skolem problem
9373:
9368:
9363:
9356:
9353:
9312:
9266:
9261:
9256:
9252:
9246:
9242:
9238:
9235:
9232:
9227:
9222:
9218:
9212:
9208:
9204:
9199:
9195:
9180:
9177:
9155:
9152:
9137:
9133:
9112:
9090:
9087:
9083:
9062:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
9016:
9013:
9010:
9007:
9004:
9001:
8998:
8995:
8992:
8989:
8986:
8983:
8980:
8977:
8974:
8950:
8945:
8942:
8938:
8934:
8931:
8907:
8904:
8901:
8898:
8895:
8891:
8888:
8884:
8881:
8877:
8874:
8870:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8795:
8792:
8789:
8786:
8783:
8780:
8777:
8774:
8771:
8768:
8765:
8762:
8759:
8756:
8753:
8750:
8747:
8744:
8741:
8738:
8735:
8732:
8729:
8705:
8702:
8699:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8669:
8666:
8663:
8660:
8657:
8654:
8651:
8648:
8645:
8642:
8639:
8636:
8633:
8630:
8627:
8624:
8621:
8618:
8615:
8612:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8585:
8582:
8579:
8576:
8573:
8570:
8567:
8564:
8561:
8558:
8555:
8552:
8549:
8525:
8522:
8519:
8516:
8513:
8508:
8505:
8502:
8498:
8494:
8491:
8488:
8485:
8482:
8479:
8476:
8471:
8468:
8465:
8461:
8457:
8454:
8451:
8448:
8445:
8442:
8437:
8433:
8429:
8402:
8399:
8396:
8393:
8390:
8387:
8384:
8381:
8378:
8375:
8372:
8369:
8366:
8363:
8360:
8357:
8354:
8351:
8348:
8345:
8342:
8339:
8336:
8333:
8330:
8325:
8322:
8319:
8315:
8311:
8306:
8302:
8281:
8278:
8275:
8272:
8269:
8266:
8263:
8258:
8255:
8252:
8248:
8219:
8199:
8196:
8193:
8190:
8170:
8146:
8142:
8138:
8135:
8132:
8129:
8123:
8120:
8115:
8112:
8109:
8104:
8101:
8098:
8094:
8085:
8080:
8077:
8074:
8070:
8040:
8018:
8015:
8011:
7990:
7987:
7974:
7971:
7968:
7963:
7959:
7938:
7935:
7932:
7927:
7923:
7902:
7880:
7875:
7871:
7848:
7844:
7838:
7834:
7830:
7827:
7824:
7804:
7801:
7798:
7795:
7775:
7772:
7769:
7766:
7737:
7733:
7728:
7722:
7718:
7714:
7711:
7708:
7703:
7700:
7697:
7692:
7688:
7679:
7675:
7671:
7665:
7662:
7659:
7656:
7651:
7648:
7645:
7642:
7613:
7607:
7604:
7601:
7598:
7593:
7590:
7587:
7584:
7578:
7575:
7572:
7569:
7566:
7543:
7540:
7537:
7534:
7512:
7504:
7500:
7494:
7490:
7486:
7483:
7480:
7475:
7471:
7465:
7461:
7457:
7454:
7449:
7445:
7441:
7438:
7434:
7429:
7425:
7421:
7418:
7415:
7412:
7409:
7403:
7400:
7397:
7393:
7387:
7383:
7380:
7377:
7374:
7371:
7349:
7346:
7343:
7323:
7320:
7317:
7314:
7292:
7289:
7286:
7283:
7280:
7274:
7271:
7268:
7264:
7259:
7256:
7253:
7250:
7247:
7242:
7238:
7232:
7228:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7201:
7197:
7191:
7187:
7183:
7180:
7177:
7174:
7171:
7168:
7163:
7159:
7155:
7152:
7149:
7146:
7143:
7119:
7115:
7109:
7105:
7099:
7096:
7093:
7089:
7085:
7082:
7079:
7076:
7073:
7047:
7044:
7041:
7036:
7033:
7030:
7026:
7020:
7016:
7012:
7009:
7006:
7001:
6998:
6995:
6991:
6985:
6981:
6977:
6972:
6968:
6953:
6950:
6944:
6918:
6911:
6876:
6872:
6867:
6864:
6861:
6856:
6850:
6846:
6841:
6836:
6808:
6800:
6797:
6794:
6790:
6786:
6785:
6782:
6779:
6778:
6773:
6770:
6767:
6763:
6759:
6758:
6753:
6750:
6747:
6743:
6739:
6738:
6736:
6731:
6726:
6721:
6702:
6680:
6675:
6672:
6669:
6666:
6663:
6660:
6657:
6653:
6649:
6646:
6644:
6640:
6637:
6634:
6630:
6626:
6625:
6622:
6616:
6614:
6612:
6607:
6604:
6601:
6598:
6595:
6591:
6587:
6584:
6582:
6578:
6575:
6572:
6568:
6564:
6563:
6560:
6557:
6552:
6549:
6546:
6543:
6540:
6536:
6530:
6526:
6522:
6519:
6516:
6511:
6508:
6505:
6502:
6499:
6495:
6489:
6485:
6481:
6476:
6473:
6470:
6467:
6464:
6460:
6454:
6450:
6446:
6443:
6441:
6437:
6434:
6431:
6427:
6423:
6422:
6394:
6391:
6386:
6383:
6380:
6376:
6370:
6366:
6362:
6359:
6356:
6351:
6348:
6345:
6341:
6335:
6331:
6327:
6322:
6319:
6316:
6312:
6306:
6302:
6298:
6293:
6289:
6265:
6255:
6245:
6232:
6222:
6212:
6201:
6191:
6175:
6172:
6160:
6155:
6151:
6147:
6142:
6138:
6134:
6129:
6125:
6119:
6115:
6111:
6106:
6102:
6082:
6075:
6067:
6064:
6044:
6041:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5934:
5930:
5922:
5918:
5914:
5909:
5905:
5899:
5874:
5870:
5867:
5864:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5836:
5830:
5826:
5822:
5794:
5783:
5704:
5698:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5663:
5657:
5654:
5651:
5647:
5643:
5639:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5604:
5598:
5594:
5588:
5581:
5577:
5573:
5570:
5568:
5566:
5562:
5556:
5551:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5526:
5522:
5515:
5512:
5509:
5505:
5501:
5496:
5491:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5466:
5462:
5455:
5451:
5446:
5440:
5436:
5432:
5429:
5427:
5425:
5421:
5415:
5410:
5406:
5401:
5398:
5393:
5388:
5385:
5379:
5372:
5369:
5366:
5362:
5358:
5353:
5348:
5344:
5339:
5336:
5331:
5326:
5323:
5317:
5310:
5306:
5301:
5295:
5291:
5287:
5284:
5282:
5278:
5273:
5270:
5267:
5263:
5257:
5254:
5251:
5247:
5243:
5238:
5233:
5229:
5223:
5219:
5215:
5214:
5190:
5183:
5179:
5175:
5170:
5166:
5160:
5157:
5146:of the roots:
5136:imaginary unit
5116:
5112:
5107:
5104:
5099:
5094:
5091:
5085:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5058:
5055:
5052:
5048:
5044:
5039:
5035:
5017:
5003:
4991:
4978:
4967:
4959:
4956:
4948:
4917:
4897:
4892:
4888:
4882:
4878:
4874:
4871:
4868:
4863:
4858:
4854:
4848:
4844:
4840:
4835:
4831:
4804:
4801:
4798:
4794:
4788:
4784:
4780:
4777:
4774:
4769:
4766:
4763:
4759:
4753:
4749:
4745:
4740:
4736:
4721:
4718:
4699:absolute value
4672:
4665:
4645:
4641:
4637:
4634:
4629:
4626:
4623:
4619:
4615:
4610:
4606:
4600:
4597:
4594:
4590:
4586:
4583:
4580:
4575:
4572:
4569:
4565:
4559:
4555:
4551:
4546:
4542:
4515:
4512:
4509:
4505:
4499:
4495:
4491:
4488:
4485:
4480:
4477:
4474:
4470:
4464:
4460:
4456:
4451:
4447:
4432:
4429:
4421:
4418:
4405:
4402:
4399:
4396:
4392:
4388:
4384:
4363:
4360:
4356:
4352:
4348:
4327:
4324:
4321:
4318:
4315:
4312:
4308:
4304:
4300:
4288:absolute value
4271:
4242:
4238:
4215:
4211:
4183:
4175:
4172:
4167:
4163:
4158:
4154:
4151:
4148:
4145:
4143:
4141:
4138:
4137:
4129:
4126:
4123:
4118:
4114:
4108:
4103:
4098:
4094:
4090:
4087:
4082:
4078:
4074:
4071:
4066:
4062:
4056:
4052:
4045:
4042:
4039:
4036:
4034:
4032:
4029:
4028:
4023:
4017:
4013:
4009:
4004:
4000:
3996:
3993:
3987:
3984:
3982:
3980:
3977:
3976:
3952:
3948:
3925:
3921:
3898:
3894:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3844:
3837:
3834:
3829:
3825:
3822:
3819:
3816:
3811:
3807:
3782:
3779:
3776:
3773:
3768:
3764:
3760:
3755:
3751:
3747:
3742:
3738:
3734:
3729:
3725:
3705:
3702:
3699:
3696:
3693:
3690:
3685:
3681:
3677:
3672:
3668:
3647:
3627:
3607:
3587:
3558:
3555:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3526:
3523:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3493:
3489:
3485:
3480:
3476:
3470:
3467:
3464:
3463:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3433:
3427:
3424:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3395:
3392:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3362:
3358:
3354:
3349:
3345:
3339:
3336:
3333:
3332:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3279:
3275:
3271:
3268:
3265:
3261:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3211:
3205:
3201:
3197:
3194:
3189:
3185:
3158:
3153:
3149:
3145:
3142:
3137:
3132:
3128:
3124:
3121:
3116:
3112:
3089:
3086:
3083:
3080:
3077:
3074:
3069:
3065:
3061:
3056:
3052:
3031:
3011:
2986:
2982:
2959:
2955:
2934:
2914:
2890:
2886:
2882:
2879:
2876:
2871:
2867:
2863:
2860:
2855:
2851:
2828:
2825:
2822:
2819:
2816:
2811:
2807:
2782:
2777:
2773:
2769:
2766:
2761:
2756:
2752:
2748:
2745:
2740:
2736:
2703:
2699:
2676:
2672:
2651:
2625:
2622:
2619:
2616:
2614:
2612:
2609:
2606:
2603:
2600:
2595:
2591:
2587:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2563:
2559:
2555:
2551:
2550:
2526:
2523:
2520:
2516:
2492:
2489:
2486:
2461:
2458:
2455:
2451:
2447:
2444:
2439:
2436:
2433:
2429:
2425:
2422:
2417:
2413:
2386:
2382:
2378:
2373:
2369:
2346:
2341:
2338:
2335:
2331:
2327:
2324:
2319:
2316:
2313:
2309:
2305:
2302:
2297:
2293:
2261:
2258:
2255:
2233:
2229:
2225:
2220:
2216:
2203:
2200:
2187:
2184:
2181:
2178:
2175:
2151:
2148:
2143:
2139:
2116:
2112:
2108:
2105:
2100:
2096:
2075:
2072:
2067:
2063:
2040:
2036:
2032:
2027:
2023:
2000:
1997:
1994:
1990:
1986:
1983:
1978:
1974:
1961:
1958:
1946:
1941:
1937:
1931:
1927:
1921:
1917:
1913:
1908:
1904:
1900:
1895:
1891:
1887:
1882:
1878:
1872:
1868:
1864:
1859:
1855:
1834:
1812:
1808:
1804:
1801:
1798:
1795:
1775:
1752:
1748:
1727:
1722:
1717:
1713:
1707:
1703:
1699:
1696:
1693:
1688:
1683:
1679:
1673:
1669:
1665:
1660:
1655:
1651:
1645:
1641:
1637:
1632:
1628:
1607:
1602:
1598:
1594:
1591:
1588:
1583:
1579:
1575:
1570:
1566:
1545:
1533:
1530:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1491:
1485:
1481:
1477:
1474:
1471:
1466:
1463:
1460:
1456:
1450:
1446:
1442:
1437:
1434:
1431:
1427:
1423:
1418:
1415:
1412:
1408:
1402:
1398:
1394:
1391:
1388:
1383:
1380:
1377:
1373:
1367:
1363:
1359:
1354:
1350:
1321:
1318:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1291:
1285:
1281:
1277:
1274:
1271:
1266:
1263:
1260:
1256:
1250:
1246:
1242:
1237:
1234:
1231:
1227:
1202:
1199:
1194:
1191:
1188:
1184:
1178:
1174:
1170:
1167:
1164:
1159:
1156:
1153:
1149:
1143:
1139:
1135:
1130:
1126:
1081:
1078:
1075:
1071:
1065:
1061:
1057:
1054:
1051:
1046:
1043:
1040:
1036:
1030:
1026:
1022:
1017:
1013:
987:
981:
977:
973:
968:
965:
962:
958:
953:
947:
943:
939:
936:
933:
929:
923:
919:
915:
910:
907:
904:
900:
895:
889:
885:
881:
877:
871:
867:
863:
858:
854:
849:
816:
812:
808:
805:
802:
797:
793:
789:
786:
782:
777:
772:
768:
741:such that, if
728:nonhomogeneous
726:is said to be
713:
710:
705:
702:
699:
695:
689:
685:
681:
678:
675:
670:
667:
664:
660:
654:
650:
646:
641:
637:
615:
612:
595:
591:
587:
584:
581:
576:
573:
570:
566:
560:
556:
552:
547:
544:
541:
537:
531:
527:
523:
518:
514:
510:
507:
504:
501:
498:
469:nonhomogeneous
452:is called the
441:
419:
416:
413:
408:
404:
398:
394:
390:
387:
384:
379:
376:
373:
370:
367:
363:
357:
353:
349:
344:
341:
338:
334:
309:
306:
303:
298:
295:
292:
288:
282:
278:
274:
271:
268:
263:
260:
257:
253:
247:
243:
239:
234:
230:
208:
201:
186:
183:
151:inflation rate
43:linear algebra
24:
14:
13:
10:
9:
6:
4:
3:
2:
9562:
9551:
9548:
9546:
9543:
9541:
9538:
9536:
9535:Combinatorics
9533:
9532:
9530:
9516:
9512:
9508:
9501:
9498:
9494:
9488:
9485:
9481:
9475:
9472:
9466:
9462:
9455:
9452:
9447:
9445:0-02-306660-1
9441:
9436:
9435:
9429:
9423:
9421:
9417:
9412:
9410:0-07-010813-7
9406:
9401:
9400:
9391:
9388:
9381:
9377:
9374:
9372:
9369:
9367:
9364:
9362:
9359:
9358:
9354:
9352:
9348:
9341:
9339:
9335:
9332:A theorem of
9330:
9323:
9315:
9311:
9301:
9294:
9277:
9264:
9259:
9254:
9250:
9244:
9240:
9236:
9233:
9230:
9225:
9220:
9216:
9210:
9206:
9202:
9197:
9193:
9184:
9178:
9176:
9173:
9169:
9165:
9161:
9153:
9151:
9135:
9131:
9110:
9103:evaluated at
9088:
9085:
9081:
9060:
9051:
9038:
9035:
9029:
9023:
9020:
9017:
9011:
9008:
9005:
8999:
8996:
8993:
8987:
8984:
8981:
8975:
8972:
8964:
8961:
8948:
8943:
8940:
8936:
8932:
8929:
8921:
8920:has solution
8918:
8905:
8902:
8899:
8896:
8893:
8889:
8886:
8882:
8879:
8875:
8872:
8868:
8860:
8858:
8854:
8851:
8838:
8835:
8829:
8823:
8820:
8817:
8811:
8808:
8805:
8799:
8793:
8790:
8787:
8781:
8778:
8772:
8769:
8766:
8760:
8757:
8754:
8751:
8745:
8742:
8739:
8733:
8730:
8727:
8719:
8716:
8703:
8700:
8694:
8688:
8685:
8682:
8676:
8673:
8670:
8664:
8661:
8655:
8652:
8649:
8643:
8640:
8637:
8634:
8628:
8625:
8622:
8616:
8613:
8610:
8604:
8601:
8598:
8592:
8589:
8586:
8583:
8577:
8574:
8571:
8565:
8559:
8556:
8553:
8547:
8539:
8536:
8523:
8520:
8517:
8514:
8511:
8503:
8496:
8489:
8486:
8483:
8480:
8474:
8466:
8459:
8452:
8449:
8446:
8443:
8440:
8435:
8431:
8419:
8417:
8413:
8397:
8394:
8391:
8388:
8385:
8379:
8373:
8370:
8367:
8364:
8361:
8355:
8352:
8349:
8343:
8340:
8337:
8331:
8320:
8313:
8309:
8304:
8300:
8276:
8273:
8270:
8264:
8253:
8246:
8237:
8234:
8231:
8217:
8194:
8188:
8168:
8159:
8144:
8136:
8133:
8130:
8121:
8118:
8110:
8099:
8092:
8078:
8075:
8072:
8068:
8059:
8057:
8056:Taylor series
8052:
8038:
8016:
8013:
8009:
8000:
7996:
7988:
7986:
7969:
7961:
7957:
7933:
7925:
7921:
7900:
7878:
7873:
7869:
7846:
7836:
7832:
7828:
7825:
7799:
7793:
7770:
7764:
7755:
7735:
7731:
7720:
7716:
7712:
7709:
7698:
7690:
7686:
7677:
7673:
7669:
7660:
7654:
7646:
7640:
7629:
7624:
7611:
7602:
7596:
7588:
7582:
7576:
7570:
7564:
7557:
7538:
7532:
7523:
7510:
7502:
7498:
7492:
7488:
7484:
7481:
7478:
7473:
7469:
7463:
7459:
7455:
7452:
7447:
7443:
7439:
7436:
7432:
7427:
7423:
7416:
7410:
7407:
7401:
7398:
7395:
7391:
7385:
7381:
7375:
7369:
7361:
7347:
7344:
7341:
7318:
7312:
7303:
7287:
7281:
7278:
7272:
7269:
7266:
7262:
7257:
7251:
7245:
7240:
7236:
7230:
7226:
7222:
7219:
7216:
7210:
7204:
7199:
7195:
7189:
7185:
7181:
7175:
7169:
7166:
7161:
7157:
7153:
7147:
7141:
7133:
7117:
7113:
7107:
7103:
7097:
7094:
7091:
7087:
7083:
7077:
7071:
7063:
7058:
7045:
7042:
7039:
7034:
7031:
7028:
7024:
7018:
7014:
7010:
7007:
7004:
6999:
6996:
6993:
6989:
6983:
6979:
6975:
6970:
6966:
6957:
6951:
6949:
6940:
6935:
6928:
6921:
6917:
6910:
6904:
6900:
6894:
6888:
6870:
6865:
6862:
6859:
6844:
6839:
6824:
6821:
6806:
6798:
6795:
6792:
6788:
6780:
6771:
6768:
6765:
6761:
6751:
6748:
6745:
6741:
6734:
6729:
6724:
6709:
6705:
6701:
6695:
6678:
6673:
6670:
6667:
6664:
6661:
6658:
6655:
6651:
6647:
6645:
6638:
6635:
6632:
6628:
6620:
6615:
6605:
6602:
6599:
6596:
6593:
6589:
6585:
6583:
6576:
6573:
6570:
6566:
6558:
6555:
6550:
6547:
6544:
6541:
6538:
6534:
6528:
6524:
6520:
6517:
6514:
6509:
6506:
6503:
6500:
6497:
6493:
6487:
6483:
6479:
6474:
6471:
6468:
6465:
6462:
6458:
6452:
6448:
6444:
6442:
6435:
6432:
6429:
6425:
6412:
6405:
6392:
6389:
6384:
6381:
6378:
6374:
6368:
6364:
6360:
6357:
6354:
6349:
6346:
6343:
6339:
6333:
6329:
6325:
6320:
6317:
6314:
6310:
6304:
6300:
6296:
6291:
6287:
6278:
6269:
6264:
6258:
6254:
6249:
6244:
6236:
6231:
6225:
6221:
6216:
6211:
6204:
6200:
6195:
6190:
6185:
6173:
6171:
6158:
6153:
6149:
6145:
6140:
6136:
6132:
6127:
6123:
6117:
6113:
6109:
6104:
6100:
6091:
6081:
6074:
6065:
6063:
6060:
6053:
6042:
6040:
6031:
6027:
6016:
6000:
5996:
5985:
5971:
5955:
5952:
5949:
5946:
5940:
5937:
5932:
5928:
5920:
5916:
5912:
5907:
5903:
5897:
5889:
5886:
5872:
5868:
5865:
5862:
5859:
5856:
5853:
5850:
5847:
5844:
5841:
5838:
5834:
5828:
5824:
5820:
5812:
5809:
5805:
5797:
5793:
5786:
5782:
5776:
5774:
5765:
5757:
5743:
5735:
5721:
5696:
5692:
5689:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5665:
5661:
5655:
5652:
5649:
5645:
5641:
5637:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5602:
5596:
5592:
5579:
5575:
5571:
5569:
5560:
5554:
5549:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5520:
5513:
5510:
5507:
5503:
5499:
5494:
5489:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5460:
5453:
5449:
5444:
5438:
5434:
5430:
5428:
5419:
5413:
5408:
5404:
5399:
5396:
5391:
5386:
5383:
5377:
5370:
5367:
5364:
5360:
5356:
5351:
5346:
5342:
5337:
5334:
5329:
5324:
5321:
5315:
5308:
5304:
5299:
5293:
5289:
5285:
5283:
5276:
5271:
5268:
5265:
5261:
5255:
5252:
5249:
5245:
5241:
5236:
5231:
5227:
5221:
5217:
5204:
5201:
5188:
5181:
5177:
5173:
5168:
5164:
5158:
5155:
5147:
5145:
5137:
5128:
5114:
5110:
5105:
5102:
5097:
5092:
5089:
5083:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5056:
5053:
5050:
5046:
5042:
5037:
5033:
5024:
5020:
5016:
5006:
4999:
4994:
4990:
4981:
4974:
4970:
4966:
4957:
4955:
4951:
4947:
4938:
4920:
4916:
4910:
4895:
4890:
4886:
4880:
4876:
4872:
4869:
4866:
4861:
4856:
4852:
4846:
4842:
4838:
4833:
4829:
4820:
4817:
4802:
4799:
4796:
4792:
4786:
4782:
4778:
4775:
4772:
4767:
4764:
4761:
4757:
4751:
4747:
4743:
4738:
4734:
4725:
4719:
4717:
4715:
4711:
4707:
4702:
4700:
4696:
4692:
4686:
4681:
4680:algebraically
4675:
4671:
4664:
4658:
4643:
4639:
4635:
4632:
4627:
4624:
4621:
4617:
4613:
4608:
4604:
4598:
4595:
4592:
4588:
4584:
4581:
4578:
4573:
4570:
4567:
4563:
4557:
4553:
4549:
4544:
4540:
4531:
4528:
4513:
4510:
4507:
4503:
4497:
4493:
4489:
4486:
4483:
4478:
4475:
4472:
4468:
4462:
4458:
4454:
4449:
4445:
4436:
4430:
4427:
4419:
4417:
4403:
4400:
4397:
4394:
4386:
4361:
4358:
4350:
4325:
4322:
4319:
4316:
4313:
4310:
4302:
4289:
4285:
4269:
4261:
4256:
4240:
4236:
4213:
4209:
4199:
4181:
4173:
4170:
4165:
4161:
4156:
4152:
4149:
4146:
4144:
4139:
4127:
4124:
4121:
4116:
4112:
4106:
4101:
4096:
4092:
4088:
4085:
4080:
4076:
4072:
4069:
4064:
4060:
4054:
4050:
4043:
4040:
4037:
4035:
4030:
4021:
4015:
4011:
4007:
4002:
3998:
3994:
3991:
3985:
3983:
3978:
3966:
3950:
3946:
3923:
3919:
3909:
3896:
3892:
3885:
3882:
3876:
3873:
3870:
3867:
3861:
3858:
3852:
3849:
3846:
3842:
3835:
3832:
3823:
3820:
3814:
3809:
3805:
3796:
3793:
3780:
3777:
3774:
3771:
3766:
3762:
3758:
3753:
3749:
3745:
3740:
3736:
3732:
3727:
3723:
3703:
3700:
3697:
3694:
3691:
3688:
3683:
3679:
3675:
3670:
3666:
3645:
3625:
3605:
3585:
3576:
3556:
3553:
3547:
3541:
3535:
3532:
3524:
3521:
3515:
3509:
3503:
3500:
3491:
3487:
3483:
3478:
3474:
3468:
3465:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3425:
3422:
3416:
3410:
3404:
3401:
3393:
3390:
3384:
3378:
3372:
3369:
3360:
3356:
3352:
3347:
3343:
3337:
3334:
3322:
3319:
3306:
3300:
3297:
3294:
3291:
3285:
3282:
3277:
3273:
3269:
3266:
3263:
3259:
3252:
3249:
3243:
3240:
3237:
3234:
3228:
3225:
3219:
3216:
3213:
3209:
3203:
3199:
3195:
3192:
3187:
3183:
3174:
3171:
3156:
3151:
3147:
3143:
3140:
3135:
3130:
3126:
3122:
3119:
3114:
3110:
3101:
3087:
3084:
3081:
3078:
3075:
3072:
3067:
3063:
3059:
3054:
3050:
3029:
3009:
3000:
2984:
2980:
2957:
2953:
2932:
2912:
2903:
2888:
2884:
2880:
2877:
2874:
2869:
2865:
2861:
2858:
2853:
2849:
2840:
2826:
2823:
2820:
2817:
2814:
2809:
2805:
2795:
2780:
2775:
2771:
2767:
2764:
2759:
2754:
2750:
2746:
2743:
2738:
2734:
2725:
2723:
2719:
2701:
2697:
2674:
2670:
2649:
2640:
2623:
2620:
2617:
2615:
2610:
2607:
2604:
2601:
2598:
2593:
2589:
2581:
2578:
2575:
2572:
2569:
2566:
2564:
2557:
2553:
2540:
2524:
2521:
2518:
2514:
2504:
2490:
2487:
2484:
2477:
2459:
2456:
2453:
2449:
2445:
2442:
2437:
2434:
2431:
2427:
2423:
2420:
2415:
2411:
2402:
2384:
2380:
2376:
2371:
2367:
2357:
2344:
2339:
2336:
2333:
2329:
2325:
2322:
2317:
2314:
2311:
2307:
2303:
2300:
2295:
2291:
2281:
2279:
2275:
2259:
2256:
2253:
2231:
2227:
2223:
2218:
2214:
2201:
2199:
2185:
2182:
2179:
2176:
2173:
2165:
2149:
2146:
2141:
2137:
2114:
2110:
2106:
2103:
2098:
2094:
2073:
2070:
2065:
2061:
2038:
2034:
2030:
2025:
2021:
1998:
1995:
1992:
1988:
1984:
1981:
1976:
1972:
1959:
1957:
1944:
1939:
1935:
1929:
1925:
1919:
1915:
1911:
1906:
1902:
1898:
1893:
1889:
1885:
1880:
1876:
1870:
1866:
1862:
1857:
1853:
1832:
1810:
1802:
1799:
1796:
1773:
1750:
1746:
1725:
1720:
1715:
1711:
1705:
1701:
1697:
1694:
1691:
1686:
1681:
1677:
1671:
1667:
1663:
1658:
1653:
1649:
1643:
1639:
1635:
1630:
1626:
1605:
1600:
1596:
1592:
1589:
1586:
1581:
1577:
1573:
1568:
1564:
1543:
1531:
1529:
1526:
1508:
1505:
1502:
1496:
1493:
1489:
1483:
1479:
1475:
1472:
1469:
1464:
1461:
1458:
1454:
1448:
1444:
1440:
1435:
1432:
1429:
1425:
1421:
1416:
1413:
1410:
1406:
1400:
1396:
1392:
1389:
1386:
1381:
1378:
1375:
1371:
1365:
1361:
1357:
1352:
1348:
1339:
1332:
1319:
1316:
1308:
1305:
1302:
1296:
1293:
1289:
1283:
1279:
1275:
1272:
1269:
1264:
1261:
1258:
1254:
1248:
1244:
1240:
1235:
1232:
1229:
1225:
1216:
1213:
1200:
1197:
1192:
1189:
1186:
1182:
1176:
1172:
1168:
1165:
1162:
1157:
1154:
1151:
1147:
1141:
1137:
1133:
1128:
1124:
1115:
1112:
1108:
1104:
1094:
1079:
1076:
1073:
1069:
1063:
1059:
1055:
1052:
1049:
1044:
1041:
1038:
1034:
1028:
1024:
1020:
1015:
1011:
1002:
999:
985:
979:
975:
971:
966:
963:
960:
956:
951:
945:
941:
937:
934:
931:
927:
921:
917:
913:
908:
905:
902:
898:
893:
887:
883:
879:
875:
869:
865:
861:
856:
852:
847:
838:
835:
832:
814:
810:
806:
803:
800:
795:
791:
787:
784:
780:
775:
770:
766:
757:
753:
738:
733:
729:
724:
711:
708:
703:
700:
697:
693:
687:
683:
679:
676:
673:
668:
665:
662:
658:
652:
648:
644:
639:
635:
626:
622:
613:
611:
608:
593:
589:
585:
582:
579:
574:
571:
568:
564:
558:
554:
550:
545:
542:
539:
535:
529:
525:
521:
516:
512:
508:
502:
496:
488:
486:
485:
479:
475:
470:
464:
459:
455:
439:
430:
417:
414:
411:
406:
402:
396:
392:
388:
385:
382:
377:
374:
371:
368:
365:
361:
355:
351:
347:
342:
339:
336:
332:
323:
320:
307:
304:
301:
296:
293:
290:
286:
280:
276:
272:
269:
266:
261:
258:
255:
251:
245:
241:
237:
232:
228:
219:
211:
207:
200:
196:
192:
184:
182:
180:
176:
172:
168:
164:
160:
156:
155:exchange rate
152:
148:
144:
139:
137:
136:
131:
130:
117:
116:
107:
106:
100:
96:
89:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
39:combinatorics
36:
30:
19:
9514:
9510:
9500:
9492:
9487:
9479:
9474:
9464:
9454:
9433:
9398:
9390:
9346:
9342:
9338:determinants
9331:
9324:
9313:
9309:
9302:
9295:
9278:
9185:
9182:
9167:
9164:z-transforms
9157:
9052:
8965:
8962:
8922:
8919:
8861:
8856:
8855:
8852:
8720:
8717:
8540:
8538:is given by
8537:
8420:
8415:
8414:
8238:
8235:
8232:
8160:
8060:
8053:
7992:
7756:
7625:
7524:
7362:
7304:
7134:
7059:
6958:
6955:
6936:
6926:
6919:
6915:
6908:
6902:
6898:
6892:
6889:
6825:
6822:
6710:
6703:
6699:
6696:
6413:
6406:
6279:
6267:
6262:
6256:
6252:
6247:
6242:
6234:
6229:
6223:
6219:
6214:
6209:
6202:
6198:
6193:
6188:
6177:
6092:
6079:
6072:
6069:
6058:
6051:
6046:
6029:
6025:
6014:
5998:
5994:
5983:
5972:
5890:
5887:
5813:
5807:
5803:
5795:
5791:
5784:
5780:
5777:
5763:
5755:
5741:
5733:
5722:
5205:
5202:
5148:
5129:
5025:
5018:
5014:
5012:, the roots
5004:
4997:
4992:
4988:
4979:
4972:
4968:
4964:
4961:
4949:
4945:
4918:
4914:
4911:
4821:
4818:
4726:
4723:
4703:
4684:
4673:
4669:
4662:
4659:
4532:
4529:
4437:
4434:
4283:
4257:
4200:
3967:
3910:
3797:
3794:
3577:
3323:
3320:
3175:
3172:
3102:
3001:
2904:
2841:
2796:
2726:
2721:
2641:
2541:
2505:
2475:
2358:
2282:
2205:
2166:) is simply
1963:
1535:
1527:
1340:
1333:
1217:
1214:
1116:
1113:
1106:
1102:
1095:
1003:
1000:
839:
836:
833:
758:
751:
736:
731:
727:
725:
627:
620:
617:
609:
489:
482:
480:
473:
468:
462:
457:
453:
431:
324:
321:
220:
209:
205:
198:
190:
188:
140:
135:steady state
133:
127:
113:
103:
101:
94:
87:
58:
54:
50:
32:
9511:J. Int. Seq
9334:Issai Schur
2839:), we have
2722:eigenvalues
458:homogeneous
185:Definitions
163:econometric
159:time series
79:time period
37:(including
35:mathematics
9529:Categories
9382:References
6057:sin
6050:cos
4424:See also:
195:parameters
177:(VAR) and
63:polynomial
9517:: 16.2.2.
9251:λ
9234:⋯
9217:λ
9179:Stability
8791:−
8728:−
8662:−
8635:−
8611:−
8557:−
8475:−
8450:−
8395:−
8365:−
8341:−
8329:→
8310:∗
8262:→
8134:−
8084:∞
8069:∑
8039:λ
8014:λ
7829:−
7713:−
7674:∑
7485:−
7482:⋯
7479:−
7456:−
7440:−
7428:⋅
7399:−
7345:−
7270:−
7220:⋯
7095:≥
7088:∑
7032:−
7008:⋯
6997:−
6863:−
6781:⋮
6671:−
6659:−
6621:⋮
6603:−
6548:−
6518:⋯
6507:−
6472:−
6382:−
6358:⋯
6347:−
6318:−
6150:λ
6124:λ
6043:Cyclicity
5956:ψ
5947:θ
5941:
5917:δ
5904:γ
5866:θ
5863:
5857:δ
5854:−
5848:θ
5845:
5839:γ
5690:θ
5687:
5678:−
5672:θ
5669:
5631:θ
5628:
5613:θ
5610:
5546:θ
5543:
5534:−
5531:θ
5528:
5486:θ
5483:
5471:θ
5468:
5397:β
5392:−
5384:α
5335:β
5322:α
5262:λ
5228:λ
5178:β
5165:α
5103:β
5098:±
5090:α
5071:β
5068:±
5065:α
5047:λ
5034:λ
4887:λ
4870:⋯
4853:λ
4800:−
4776:⋯
4765:−
4633:λ
4625:−
4605:λ
4596:−
4582:⋯
4571:−
4564:λ
4541:λ
4511:−
4487:⋯
4476:−
4401:−
4317:−
4237:λ
4210:λ
4171:−
4153:
4140:θ
4070:−
4041:−
3992:−
3883:θ
3877:
3859:θ
3853:
3821:−
3775:−
3763:β
3750:α
3737:λ
3733:⋅
3724:λ
3695:α
3680:λ
3667:λ
3646:δ
3542:δ
3536:
3510:δ
3504:
3451:∓
3423:β
3411:θ
3405:
3391:α
3379:θ
3373:
3357:β
3344:α
3301:δ
3298:−
3292:θ
3286:
3250:θ
3244:
3226:θ
3220:
3148:λ
3127:λ
3082:β
3079:±
3076:α
3064:λ
3051:λ
2885:λ
2866:λ
2772:λ
2751:λ
2698:λ
2671:λ
2608:−
2599:−
2522:−
2457:−
2435:−
2337:−
2315:−
2177:−
1996:−
1800:−
1695:⋯
1590:…
1497:−
1476:−
1473:⋯
1470:−
1462:−
1441:−
1433:−
1414:−
1393:−
1390:⋯
1387:−
1379:−
1358:−
1297:−
1273:⋯
1262:−
1233:−
1190:−
1166:⋯
1155:−
1077:−
1053:⋯
1042:−
980:∗
972:−
964:−
935:⋯
922:∗
914:−
906:−
870:∗
862:−
807:−
804:⋯
801:−
788:−
771:∗
749:equal to
734:—a value
701:−
677:⋯
666:−
586:−
583:⋯
580:−
572:−
565:λ
551:−
543:−
536:λ
522:−
513:λ
503:λ
386:⋯
375:−
294:−
270:⋯
259:−
143:economics
9430:(1970).
9355:See also
8890:′
8876:″
8857:Example:
8416:Example:
4939:for the
105:solution
71:sequence
67:variable
6914:, ...,
6036:
6023:√
6011:
6005:
5992:√
5980:
5768:
5752:
5746:
5730:
5144:modulus
5142:is the
5134:is the
4716:pairs.
4668:, ...,
2202:Order 2
1960:Order 1
1100:equals
204:, ...,
99:, etc.
9442:
9407:
9166:. The
8031:where
7999:ansatz
7305:where
6896:is an
5973:where
5723:where
5130:where
4689:, but
4260:stable
4150:arccos
3911:where
3321:where
2401:ansatz
1096:where
153:, the
149:, the
129:stable
75:degree
45:, and
6890:Here
3578:Here
2129:with
2053:with
454:order
118:) of
49:), a
9440:ISBN
9405:ISBN
6055:and
5789:and
5138:and
4986:and
4706:real
4395:<
4374:and
4359:<
4323:<
4311:<
4284:both
4228:and
3938:and
3638:and
3598:and
3022:and
2972:and
2925:and
2488:>
467:and
214:and
102:The
9123:is
8718:or
6708:as
5938:cos
5860:sin
5842:cos
5684:sin
5666:cos
5625:sin
5607:cos
5540:sin
5525:cos
5480:sin
5465:cos
4687:≤ 4
4682:if
3874:sin
3850:cos
3533:sin
3501:cos
3402:sin
3370:cos
3283:cos
3241:sin
3217:cos
2720:or
2476:all
623:≠ 0
618:If
476:≠ 0
471:if
465:= 0
460:if
97:+ 1
90:− 1
57:or
33:In
9531::
9515:19
9513:.
9509:.
9419:^
9150:.
9039:0.
8839:0.
7985:.
6270:−1
6266:2,
6261:=
6259:−2
6251:=
6246:3,
6240:,
6237:−1
6233:1,
6228:=
6226:−1
6218:=
6213:2,
6207:,
6197:=
6192:1,
6078:=
6062:.
6059:θt
6052:θt
6039:.
6028:+
5997:+
5808:δi
5806:±
5798:+1
5775:.
5007:+1
4995:+1
4701:.
4416:.
4255:.
2689:,
2503:.
2198:.
1338:)
1105:−
478:.
218::
189:A
138:.
41:,
9469:.
9448:.
9413:.
9349:*
9347:y
9327:x
9320:x
9314:i
9310:c
9305:x
9298:x
9290:M
9285:t
9281:t
9265:,
9260:t
9255:n
9245:n
9241:c
9237:+
9231:+
9226:t
9221:1
9211:1
9207:c
9203:=
9198:t
9194:x
9168:z
9136:n
9132:a
9111:0
9089:x
9086:a
9082:e
9061:n
9036:=
9033:]
9030:n
9027:[
9024:f
9021:c
9018:+
9015:]
9012:1
9009:+
9006:n
9003:[
9000:f
8997:b
8994:+
8991:]
8988:2
8985:+
8982:n
8979:[
8976:f
8973:a
8949:.
8944:x
8941:a
8937:e
8933:=
8930:y
8906:0
8903:=
8900:y
8897:c
8894:+
8887:y
8883:b
8880:+
8873:y
8869:a
8836:=
8833:]
8830:n
8827:[
8824:f
8821:2
8818:+
8815:]
8812:1
8809:+
8806:n
8803:[
8800:f
8797:)
8794:4
8788:n
8785:(
8782:n
8779:+
8776:]
8773:2
8770:+
8767:n
8764:[
8761:f
8758:n
8755:2
8752:+
8749:]
8746:3
8743:+
8740:n
8737:[
8734:f
8731:4
8704:0
8701:=
8698:]
8695:n
8692:[
8689:f
8686:2
8683:+
8680:]
8677:2
8674:+
8671:n
8668:[
8665:f
8659:]
8656:1
8653:+
8650:n
8647:[
8644:f
8641:n
8638:3
8632:]
8629:3
8626:+
8623:n
8620:[
8617:f
8614:4
8608:]
8605:2
8602:+
8599:n
8596:[
8593:f
8590:n
8587:3
8584:+
8581:]
8578:1
8575:+
8572:n
8569:[
8566:f
8563:)
8560:1
8554:n
8551:(
8548:n
8524:0
8521:=
8518:y
8515:2
8512:+
8507:]
8504:2
8501:[
8497:y
8493:)
8490:1
8487:+
8484:x
8481:3
8478:(
8470:]
8467:3
8464:[
8460:y
8456:)
8453:4
8447:x
8444:3
8441:+
8436:2
8432:x
8428:(
8401:]
8398:m
8392:k
8389:+
8386:n
8383:[
8380:f
8377:)
8374:1
8371:+
8368:m
8362:n
8359:(
8356:.
8353:.
8350:.
8347:)
8344:1
8338:n
8335:(
8332:n
8324:]
8321:k
8318:[
8314:y
8305:m
8301:x
8280:]
8277:k
8274:+
8271:n
8268:[
8265:f
8257:]
8254:k
8251:[
8247:y
8218:a
8198:)
8195:x
8192:(
8189:f
8169:n
8145:n
8141:)
8137:a
8131:x
8128:(
8122:!
8119:n
8114:)
8111:a
8108:(
8103:)
8100:n
8097:(
8093:f
8079:0
8076:=
8073:n
8017:x
8010:e
7973:)
7970:n
7967:(
7962:i
7958:k
7937:)
7934:x
7931:(
7926:i
7922:f
7901:m
7879:n
7874:i
7870:r
7847:m
7843:)
7837:i
7833:r
7826:x
7823:(
7803:)
7800:n
7797:(
7794:z
7774:)
7771:x
7768:(
7765:p
7736:i
7732:m
7727:)
7721:i
7717:r
7710:x
7707:(
7702:)
7699:x
7696:(
7691:i
7687:f
7678:i
7670:=
7664:)
7661:x
7658:(
7655:g
7650:)
7647:x
7644:(
7641:f
7612:.
7606:)
7603:x
7600:(
7597:g
7592:)
7589:x
7586:(
7583:f
7577:=
7574:)
7571:x
7568:(
7565:Y
7542:)
7539:x
7536:(
7533:Y
7511:.
7503:n
7499:x
7493:n
7489:a
7474:2
7470:x
7464:2
7460:a
7453:x
7448:1
7444:a
7437:1
7433:1
7424:)
7420:)
7417:x
7414:(
7411:p
7408:+
7402:x
7396:1
7392:b
7386:(
7382:=
7379:)
7376:x
7373:(
7370:Y
7348:1
7342:n
7322:)
7319:x
7316:(
7313:p
7291:)
7288:x
7285:(
7282:p
7279:+
7273:x
7267:1
7263:b
7258:+
7255:)
7252:x
7249:(
7246:Y
7241:n
7237:x
7231:n
7227:a
7223:+
7217:+
7214:)
7211:x
7208:(
7205:Y
7200:2
7196:x
7190:2
7186:a
7182:+
7179:)
7176:x
7173:(
7170:Y
7167:x
7162:1
7158:a
7154:=
7151:)
7148:x
7145:(
7142:Y
7118:t
7114:x
7108:t
7104:y
7098:0
7092:t
7084:=
7081:)
7078:x
7075:(
7072:Y
7046:,
7043:b
7040:+
7035:n
7029:t
7025:y
7019:n
7015:a
7011:+
7005:+
7000:1
6994:t
6990:y
6984:1
6980:a
6976:=
6971:t
6967:y
6945:i
6943:y
6932:b
6927:b
6920:n
6916:a
6912:1
6909:a
6903:n
6899:n
6893:A
6875:b
6871:+
6866:1
6860:t
6855:w
6849:A
6845:=
6840:t
6835:w
6807:]
6799:i
6796:,
6793:n
6789:w
6772:i
6769:,
6766:2
6762:w
6752:i
6749:,
6746:1
6742:w
6735:[
6730:=
6725:i
6720:w
6704:i
6700:w
6679:.
6674:1
6668:t
6665:,
6662:1
6656:n
6652:w
6648:=
6639:t
6636:,
6633:n
6629:w
6606:1
6600:t
6597:,
6594:1
6590:w
6586:=
6577:t
6574:,
6571:2
6567:w
6559:b
6556:+
6551:1
6545:t
6542:,
6539:n
6535:w
6529:n
6525:a
6521:+
6515:+
6510:1
6504:t
6501:,
6498:2
6494:w
6488:2
6484:a
6480:+
6475:1
6469:t
6466:,
6463:1
6459:w
6453:1
6449:a
6445:=
6436:t
6433:,
6430:1
6426:w
6409:n
6393:b
6390:+
6385:n
6379:t
6375:y
6369:n
6365:a
6361:+
6355:+
6350:2
6344:t
6340:y
6334:2
6330:a
6326:+
6321:1
6315:t
6311:y
6305:1
6301:a
6297:=
6292:t
6288:y
6275:n
6268:t
6263:w
6257:t
6253:y
6248:t
6243:w
6235:t
6230:w
6224:t
6220:y
6215:t
6210:w
6203:t
6199:y
6194:t
6189:w
6180:n
6159:.
6154:t
6146:t
6141:2
6137:c
6133:+
6128:t
6118:1
6114:c
6110:=
6105:t
6101:x
6088:λ
6083:2
6080:λ
6076:1
6073:λ
6030:δ
6026:γ
6019:/
6015:δ
5999:δ
5995:γ
5988:/
5984:γ
5975:ψ
5959:)
5953:+
5950:t
5944:(
5933:t
5929:M
5921:2
5913:+
5908:2
5898:2
5873:)
5869:t
5851:t
5835:(
5829:t
5825:M
5821:2
5804:γ
5796:j
5792:c
5785:j
5781:c
5764:M
5760:/
5756:β
5742:M
5738:/
5734:α
5725:θ
5703:)
5697:)
5693:t
5681:i
5675:t
5662:(
5656:1
5653:+
5650:j
5646:c
5642:+
5638:)
5634:t
5622:i
5619:+
5616:t
5603:(
5597:j
5593:c
5587:(
5580:t
5576:M
5572:=
5561:)
5555:t
5550:)
5537:i
5521:(
5514:1
5511:+
5508:j
5504:c
5500:+
5495:t
5490:)
5477:i
5474:+
5461:(
5454:j
5450:c
5445:(
5439:t
5435:M
5431:=
5420:)
5414:t
5409:)
5405:i
5400:M
5387:M
5378:(
5371:1
5368:+
5365:j
5361:c
5357:+
5352:t
5347:)
5343:i
5338:M
5330:+
5325:M
5316:(
5309:j
5305:c
5300:(
5294:t
5290:M
5286:=
5277:t
5272:1
5269:+
5266:j
5256:1
5253:+
5250:j
5246:c
5242:+
5237:t
5232:j
5222:j
5218:c
5189:.
5182:2
5174:+
5169:2
5159:=
5156:M
5140:M
5132:i
5115:)
5111:i
5106:M
5093:M
5084:(
5080:M
5077:=
5074:i
5062:=
5057:1
5054:+
5051:j
5043:,
5038:j
5019:j
5015:λ
5005:j
4998:λ
4993:j
4989:c
4980:j
4973:λ
4969:j
4965:c
4950:i
4946:c
4941:n
4933:n
4929:n
4925:t
4919:i
4915:c
4896:t
4891:n
4881:n
4877:c
4873:+
4867:+
4862:t
4857:1
4847:1
4843:c
4839:=
4834:t
4830:x
4803:n
4797:t
4793:x
4787:n
4783:a
4779:+
4773:+
4768:1
4762:t
4758:x
4752:1
4748:a
4744:=
4739:t
4735:x
4685:n
4674:n
4670:λ
4666:1
4663:λ
4644:n
4640:a
4636:+
4628:1
4622:n
4618:a
4614:+
4609:2
4599:2
4593:n
4589:a
4585:+
4579:+
4574:1
4568:n
4558:1
4554:a
4550:=
4545:n
4514:n
4508:t
4504:x
4498:n
4494:a
4490:+
4484:+
4479:1
4473:t
4469:x
4463:1
4459:a
4455:=
4450:t
4446:x
4404:B
4398:1
4391:|
4387:A
4383:|
4362:1
4355:|
4351:B
4347:|
4326:2
4320:B
4314:1
4307:|
4303:A
4299:|
4270:a
4241:2
4214:1
4182:)
4174:B
4166:2
4162:A
4157:(
4147:=
4128:B
4125:4
4122:+
4117:2
4113:A
4107:B
4102:B
4097:1
4093:a
4089:2
4086:+
4081:2
4077:a
4073:A
4065:1
4061:a
4055:2
4051:A
4044:i
4038:=
4031:F
4022:B
4016:2
4012:a
4008:+
4003:1
3999:a
3995:A
3986:=
3979:E
3951:2
3947:a
3924:1
3920:a
3897:,
3893:)
3889:)
3886:n
3880:(
3871:F
3868:+
3865:)
3862:n
3856:(
3847:E
3843:(
3836:2
3833:n
3828:)
3824:B
3818:(
3815:=
3810:n
3806:a
3781:,
3778:B
3772:=
3767:2
3759:+
3754:2
3746:=
3741:2
3728:1
3704:,
3701:A
3698:=
3692:2
3689:=
3684:2
3676:+
3671:1
3626:G
3606:F
3586:E
3557:G
3554:F
3548:=
3545:)
3539:(
3525:G
3522:E
3516:=
3513:)
3507:(
3492:2
3488:F
3484:+
3479:2
3475:E
3469:=
3466:G
3457:i
3454:F
3448:E
3445:=
3442:D
3439:,
3436:C
3426:M
3417:=
3414:)
3408:(
3394:M
3385:=
3382:)
3376:(
3361:2
3353:+
3348:2
3338:=
3335:M
3307:,
3304:)
3295:n
3289:(
3278:n
3274:M
3270:G
3267:2
3264:=
3260:)
3256:)
3253:n
3247:(
3238:F
3235:+
3232:)
3229:n
3223:(
3214:E
3210:(
3204:n
3200:M
3196:2
3193:=
3188:n
3184:a
3157:n
3152:2
3144:D
3141:+
3136:n
3131:1
3123:C
3120:=
3115:n
3111:a
3088:.
3085:i
3073:=
3068:2
3060:,
3055:1
3030:D
3010:C
2985:1
2981:a
2958:0
2954:a
2933:D
2913:C
2889:n
2881:n
2878:D
2875:+
2870:n
2862:C
2859:=
2854:n
2850:a
2827:0
2824:=
2821:B
2818:4
2815:+
2810:2
2806:A
2781:n
2776:2
2768:D
2765:+
2760:n
2755:1
2747:C
2744:=
2739:n
2735:a
2702:2
2675:1
2650:r
2624:,
2621:0
2618:=
2611:B
2605:r
2602:A
2594:2
2590:r
2582:,
2579:B
2576:+
2573:r
2570:A
2567:=
2558:2
2554:r
2525:2
2519:n
2515:r
2491:1
2485:n
2460:2
2454:n
2450:r
2446:B
2443:+
2438:1
2432:n
2428:r
2424:A
2421:=
2416:n
2412:r
2385:n
2381:r
2377:=
2372:n
2368:a
2345:.
2340:2
2334:n
2330:a
2326:B
2323:+
2318:1
2312:n
2308:a
2304:A
2301:=
2296:n
2292:a
2276:(
2260:r
2257:=
2254:t
2232:n
2228:r
2224:=
2219:n
2215:a
2186:0
2183:=
2180:r
2174:t
2150:k
2147:=
2142:0
2138:a
2115:n
2111:r
2107:k
2104:=
2099:n
2095:a
2074:1
2071:=
2066:0
2062:a
2039:n
2035:r
2031:=
2026:n
2022:a
1999:1
1993:n
1989:a
1985:r
1982:=
1977:n
1973:a
1945:.
1940:n
1936:r
1930:2
1926:n
1920:3
1916:k
1912:+
1907:n
1903:r
1899:n
1894:2
1890:k
1886:+
1881:n
1877:r
1871:1
1867:k
1863:=
1858:n
1854:a
1833:r
1811:3
1807:)
1803:r
1797:x
1794:(
1774:n
1751:i
1747:k
1726:,
1721:n
1716:d
1712:r
1706:d
1702:k
1698:+
1692:+
1687:n
1682:2
1678:r
1672:2
1668:k
1664:+
1659:n
1654:1
1650:r
1644:1
1640:k
1636:=
1631:n
1627:a
1606:,
1601:d
1597:r
1593:,
1587:,
1582:2
1578:r
1574:,
1569:1
1565:r
1544:d
1512:)
1509:1
1506:+
1503:n
1500:(
1494:t
1490:y
1484:n
1480:a
1465:2
1459:t
1455:y
1449:1
1445:a
1436:1
1430:t
1426:y
1422:=
1417:n
1411:t
1407:y
1401:n
1397:a
1382:1
1376:t
1372:y
1366:1
1362:a
1353:t
1349:y
1336:b
1320:b
1317:+
1312:)
1309:1
1306:+
1303:n
1300:(
1294:t
1290:y
1284:n
1280:a
1276:+
1270:+
1265:2
1259:t
1255:y
1249:1
1245:a
1241:=
1236:1
1230:t
1226:y
1201:b
1198:+
1193:n
1187:t
1183:y
1177:n
1173:a
1169:+
1163:+
1158:1
1152:t
1148:y
1142:1
1138:a
1134:=
1129:t
1125:y
1109:*
1107:y
1103:y
1098:x
1080:n
1074:t
1070:x
1064:n
1060:a
1056:+
1050:+
1045:1
1039:t
1035:x
1029:1
1025:a
1021:=
1016:t
1012:x
986:)
976:y
967:n
961:t
957:y
952:(
946:n
942:a
938:+
932:+
928:)
918:y
909:1
903:t
899:y
894:(
888:1
884:a
880:=
876:)
866:y
857:t
853:y
848:(
815:n
811:a
796:1
792:a
785:1
781:b
776:=
767:y
754:*
752:y
747:y
743:n
739:*
737:y
712:b
709:+
704:n
698:t
694:y
688:n
684:a
680:+
674:+
669:1
663:t
659:y
653:1
649:a
645:=
640:t
636:y
621:b
594:n
590:a
575:2
569:n
559:2
555:a
546:1
540:n
530:1
526:a
517:n
509:=
506:)
500:(
497:p
474:b
463:b
440:n
418:.
415:b
412:+
407:t
403:y
397:n
393:a
389:+
383:+
378:1
372:n
369:+
366:t
362:y
356:1
352:a
348:=
343:n
340:+
337:t
333:y
308:,
305:b
302:+
297:n
291:t
287:y
281:n
277:a
273:+
267:+
262:1
256:t
252:y
246:1
242:a
238:=
233:t
229:y
216:b
210:n
206:a
202:1
199:a
124:n
120:n
110:t
95:t
88:t
83:t
20:)
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