Recurrence theorem

Fig. 9. A ring diagram and the recurrence rule, (a) A ring diagram, (b) The recurrence theorem for ring diagrams.

This second point is quite an interesting one, for there is a theorem known as the Poincare recurrence theorem which states that an isolated system (like our molecule left to itself) will in the course of time return to any of its previous states (e.g. the initial state), no matter how improbable that state may be. This recurrence can be observed with very small molecules but not with polyatomic molecules, because in the latter there are far too many levels of the final state the recurrence time is then far longer than any practicable observation time. 

Polva s Recurrence Theorem dictates one of the most important differences between 7.B and AB. It states that every random-walk trajectory in one or two dimensions passes through every point in space but that this is not true in three dimensions 17 J.SO. The Recurrence Theorem is the subject of a famous (among mathematicians) joke. 

ZB diffusion is one-dimensional, die Recurrence Theorem guarantees lhat every Z and If constitute a ZB. since they always have a contact iraiectory. As a conscciucncc ol die three-dimensional nature of effective AB diffusion, the same theorem guar-antces that some AB will gain non-contact ua fc lories. 

Radiation boundary condition 197 Radical coupling/disproportionaiion 248 Radical mechanism and early evidence 7-8 Radical probes 11-13 Reactivity of Grignard reagents 171 Reactivity scries 8-11 Recurrence theorem 199 Reduction potentials ol organic halides 36 Redueiive dimeri/ation ol carbonyl compounds 21 Rieke-ntayncsium 289 Grignard reactions 257... 

The question stated above was formulated in two ways, each using an exact result from classical mechanics. One way, associated with the physicist Loschmidt, is fairly obvious. If classical mechanics provides a correct description of the gas, then associated with any physical motion of a gas, there is a time-reversed motion, which is also a solution of Newton s equations. Therefore if decreases in one of these motions, there ought to be a physical motion of the gas where H increases. This is contrary to the /f-theorem. The other objection is based on the recurrence theorem of Poincare [15], and is associated with the mathematician Zermelo. Poincare s theorem states that in a bounded mechanical system with finite energy, any initial state of the gas will eventually recur as a state of the gas, to within any preassigned accuracy. Thus, if H decreases during part of the motion, it must eventually increase so as to approach, arbitrarily closely, its initial value. 

P. Bocchieri, A. Loigner, The quantum recurrence theorem, Phys. Rev. 107 (1957) 337. 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

Theorem. For any three consecutive orthonormal polynomials defined by Eq. (7.4), we have the recurrence formula... 

Theorem C.6 establishes that the dynamical system generated by the vector field (C.l) on K", restricted to the limit set L, is topologically equivalent to the dynamical system generated by a Lipschitz vector field on ZZ restricted to Q L). As a consequence, these two dynamical systems share common dynamical properties. Since Z, is a compact invariant set, so too is Q L) a compact invariant set. It is also known that limit sets are chain-recurrent (see [C] for the definition) and so Q(L) has this property as well. Therefore, the dynamics on L are that of a compact, invariant, chain-recurrent set in one less dimension. 

According to Theorem C.6, the limit set can be deformed to a compact invariant set A, without rest points, of a planar vector field. By the Poin-care-Bendixson theorem, A must contain at least one periodic orbit and possibly entire orbits which have as their alpha and omega limits sets distinct periodic orbits belonging to A. Using the fact that A is chain-recurrent, Hirsch [Hil] shows that these latter orbits cannot exist. Since A is connected it must consist entirely of periodic orbits that is, it must be an annulus foliated by closed orbits. Monotonicity is used to show... 

For instance, as for the recurrence phenomenon discovered in the FPU nonlinear lattices [25], whose explicit Hamiltonian will be presented in Section IV, there may exist a gap between the results of numerical simulations and the mathematical theorem, but a rigorous result certainly plays a significant role and theoretical arguments based on it can be deduced [26],... 

This theorem is of fundamental importance in fractional dynamics because use of it coupled with continued fraction methods [8] allows recurrence relations associated with normal diffusion to be generalized to fractional dynamics in an obvious fashion.) Regarding the ac response, if we assume that the above result may be analytically continued into the domain of the imaginaries or if we equivalently note that if D denotes the operator d/dt and G(oo) denotes an arbitrary function of oo, then [42]... 

Here, as above, y is the Sack inertial parameter. Noting the initial condition, Eq. (238), all the cn j (0) in Eq. (256) will vanish with the exception n = 0. On using the integration theorem of Laplace transformation as generalized to fractional calculus, we have from Eq. (256) the three-term recurrence relation [cf. Eq. (240)] for the only case of interest q — 1 (since the linear dielectric response is all that is considered) ... 

In dielectric relaxation l = 1 so that by taking the Laplace transform of Eqs. (273)-(275) over the time variables and noting the generalized integral theorem for Laplace transforms, we then have a system of algebraic recurrence relations for the Laplace transform of cln m(t) (m = 0, 1) governing the dielectric response, namely,... 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

Theorem 3.1 Any set of orthogonal polynomials Paif) has a recurrence formula relating any three consecutive polynomials in the following sequence ... 

Theorem 1 (Leopold, 1982). Let gArhvao be a sequence of degree N polynomials which satisfy the recurrence relation... 

Bloch s theorem is a corollary of the lattice periodicity applied to the potential energy. The eigenfunctions of the Hamiltonian is given in Equation 6.D.1 with lJ(r I R) U(r) with R of the Bravais lattice, which can be selected as planar waves that are recurrent to the Bravais lattice. 

Robinson, Harary and Balaban [54], by plying Pdlya s theorem and Otter s dissimilarity characteristic equation [85], presented for the first time recurrence formulas for counting the achiral isomers of alkyl radicals and alkanes. If the constitutional isomers of alkanes are denoted by Vn (quartic trees), and if their number, including stereoisomers, is denoted hy fn (steric trees), the latter number may be decomposed into achiral (tn) chiral (cn) ismners. The numbers fcH n = 1-14 are presented in Table 3 ... 

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

The omega notation is used to provide a lower bound, while the theta notation is used when the obtained bound is both a lower and an upper bound. The little oh notation is a very precise notation that does not find much use in the asymptotic analysis of algorithms. With these additional notations available, the solution to the recurrence for insertion and merge sort are, respectively, 0(n ) and 0(n logn). The definitions of O, 2, 0, and o are easily extended to include functions of more than one variable. For example, f(n,m) = 0(g(n, m)) if there exist positive constants c, uq and mo such that /(n, m) < cg(n, m) for all n> no and all m > mo. As in the case of the big oh notation, there are several functions g(n) for which /(n) = Q(g(n)). The g(n) is only a lower bound on f(n). The 0 notation is more precise that both the big oh and omega notations. The following theorem obtains a very useful result about the order of f(n) when f(n) is a polynomial in n. 

Theorem 2. The reliability fimction of the ageing series-consecutive m out of k F system is given by the following recurrent formula... 

Interlude 3.2 Poincare Recurrence Times We have seen that Boltzmann s entropy theorem leads not only to an expression for the equilibrium distribution function, but also to a specific direction of change with time or irreversibility for a system of particles or molecules. The entropy theorem states that the entropy of a closed system can never decrease so, whatever entropy state the system is in, it will always change to a higher entropy state. At that time, Boltzmann s entropy theorem was viewed to be contradictory to a well-known theorem in dynamics due to Poincare. This theorem states that... 

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