Theorems, Poincare

Fortunately, the earth rotates with a relatively small angular velocity, when the force of attraction plays the dominant role. It is interesting to raise the following 

Here S is a surface surrounding the earth and the normal n directed outward. 

To preserve the earth, the component of the gravitational field along the normal has to be negative and this means that the surface integral satisfies an inequality 

Here V is the volume of the earth and the integral represents its total mass. Introducing the average density, 8 , of the earth, Equation (2.101) becomes 

This relationship was derived by Poincare and defines the range of frequencies, where the earth or any planet is not broken. The remarkable feature of this inequality is the fact that it is independent of the dimensions of the planet, and only the density defines the maximal permissible frequency. Introducing the period T, we represent Equation (2.102) as 

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Bendixon, negative criterion of, 333 Bendixon-Poincare Theorem, 333 Berezetski, V. B., 723... 

Poincare theorem. Given n random events A ,.. ., An, the probability of their union is given by... 

This theorem is known as the Poincare theorem (1854-1912) [35] [43] [52], stating that dn = Udpidqi = UdPidQi. 

Buslaev, V.I. Buslaeva, S.F. 2005. Poincare theorem for difference equations. Mathematical Notes 78 (5 6) 877-882. 

The Poincare theorem states that the sum of all charges k of the field n, defined at a closed surface, is equal to the Euler characteristic of the surface... 

For a sphere, E = 2-, thus the two point defects at the poles of the nematic droplets in Figure 5.3 illustrate the Poincare theorem it does not matter if the interior structure is twisted or not. 

The transformation from one pair of canonically conjugate coordinates q and momenta p to another set of coordinates Q = Q(p, q, t) and momenta P = P(p, q, t) is called a canonical transformation or point transformation. In this transformation it is required that the new coordinates (P, Q) again satisfy the Hamiltonian equations with a new Hamiltonian I/ (P, Q, t) [49, 60, 81]. This theorem is known as the Poincare theorem (1854-1912) [49, 60, 81], stating that dfi =... 

On the other hand, in view of the assumed analyticity of the function, we can make use of the theorem of Poincare, namely that the solution x(t,pup2,fi) can be represented by a series arranged according to the ascending powers of the parameters, that is... 

Theorem 6.1. The generating function of the Poincare polynomials of the Hilbert scheme parameterizing n-points in X, is given by... 

MSN. 112. T. Petrosky and II. Prigogine, Poincare s theorem and unitary transformations for classical and quantum systems, Physica, 147A, 439 60 (1987). 

The first of these equations is an equation of the cyclic theorem, which therefore emerges from the symmetry of the Poincare group in free space. Similarly, Eq. (772c) gives ... 

Consideration of the symmetry of the Poincare group also shows that the cyclic theorem is independent of Lorentz boosts in any direction, and also reveals the physical meaning of the E(2) little group of Wigner. This group is unphysical for a photon without mass, but is physical for a photon with mass. This proves that Poincare symmetry leads to a photon with identically nonzero mass. The proof is as follows. Consider in the particle interpretation the PL vector... 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i < 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. 

Because we have only a single stationary state, we can use the Poincare-Bendixson theorem to recognize that sustained oscillatory responses will be found at least over the whole range of K-p parameter space corresponding to instability. (Although we must also check that the concentration a and... 

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. 

As argued, infinitesimal field generators appear as a by-product of this novel quantization scheme, so that B° is rigorously nonzero from the symmetry of the Poincare group and the B cyclic theorem is an invariant of the classical field. The basics of infinitesimal field generators on the classical level are to be found in the theory of relativistic spin angular momentum [42,46] and relies on the Pauli-Lubanski pseudo-4-vector ... 

To complete the derivation, we multiply both sides of Eq. (828) by the phase factor el s>e lis> to obtain the B cyclic theorem. The latter is therefore equivalent to a commutator relation of the Poincare group between infinitesimal magnetic held generators. Similarly... 

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