Frobenius

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Setting up the Frobenius-Perron operator with respect to this subset. 

Then, the discretized Frobenius-Perron operator v = PdU can be written componentwise as... 

Almost Invariant Sets Recall that the relevant almost invariant sets correspond to eigenvalues X 1 with A < 1 of the associated Frobenius-Perron operator. 

Fig. 7. Eigenmeasure V2 of the Frobenius-Perron operator to the second largest eigenvalue A2 = 0.9963 for the test system (15) with 7 = 3. iV2 was computed via our new subdivision algorithm (cf. Section 4).

Professor Schur also made me aware of a consideration by Frobenius which is closely related to the argument given in Sec. 19. 

G. Frobenius and J. Sohur, Sitzber. preuss. AJcad. Wits., Physik.-math. Kl., 49, 180 (1806). 

With alkoxide and phenoxide ions diazo ethers are formed. The latter played a considerable part in the classic controversy about diazo isomerism between Hantzsch and Bamberger at the turn of the 19th century (see Sec. 1.1). Von Pechmann and Frobenius (1894 a) showed that the most convenient synthesis for preparative pur-... 

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

There are two procedures available for solving this differential equation. The older procedure is the Frobenius or series solution method. The solution of equation (4.17) by this method is presented in Appendix G. In this chapter we use the more modem ladder operator procedure. Both methods give exactly the same results. 

The eigenvalues and eigenfimctions of the orbital angular momentum operator may also be obtained by solving the differential equation I ip = Xh ip using the Frobenius or series solution method. The application of this method is presented in Appendix G and, of course, gives the same results... 

Equation (6.24) may be solved by the Frobenius or series solution method as presented in Appendix G. However, in this chapter we employ the newer procedure using ladder operators. 

The Frobenius or series solution method for solving equation (G.l) assumes that the solution may be expressed as a power series in x... 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonie oseillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

It is often possible to find a solution of homogeneous differential equations in the form of a power series. According to Frobenius, the power series should have the general form... 

This equation, which determines the powers of the Frobenius series is called the indicial equation and it has solutions m = 0orm = l. The coefficient of the next power must also vanish ... 

In this way, all the coefficients Ar of the Frobenius series can be determined step by step. The recursion formula generates two independent series for odd and even values of r. For... 

The Frobenius series can be shown not to converge forjx > 1. An important special case occurs when c = q = 1(1 + 1), l = 0,1,2,..., any non-negative integer. For each of these special values, one of the two chains of coefficients terminates at A because... 

In particular the even chain terminates when / is even, while the odd chain terminates when l is odd. The corresponding Frobenius solution then simplifies to a polynomial. It is called the Legendre polynomial of degree /, or Pi(x). The modified form of equation (16) becomes... 

This equation can be solved by the Frobenius method, assuming a series solution... 

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