Hausdorff

This approximation is a special case of the Baker-Campbell-Hausdorff lemma for additional discu.ssion and extensions to more general classes of methods. 

While each term in the resulting series has the form required by equation 12,29, the caveat is that as each commutator connects sites x and x that are a distance d apart, the terms of order n in Ai and Aj yield contributions to H x) that obey the third condition in equation 12.29 only for d = nd. In other words, we also need to require that the Baker-Campbell-Hausdorff series has a sufficient degree of convergence, t... 

Consider a set X. The topological space (X, T) is called a Hausdorff space if for any two distinct points x, y eX there exist disjoint T-open sets Tx, Ty,... 

If the topology T is chosen as the metric topology, that is, if the T-open sets are precisely those which are open in some metric d introduced into the set X, then one obtains the metric topological space (X, T). Note that the metric topological space (X, T) is a Hausdorff space and also a normal space. 

More precisely, a Hausdorff space X covered by countable many T-open sets G(1), G(2),. . . , is an K-dimensional differentiable manifold if it satisfies the following conditions ... 

Hausdorff, W. P., Caron, M. G. Lefkowitz, R. J. (1990). Turning off the signal desensitization of beta-adrenergic receptor function. EASES J. 4, 2881-9. 

In cases where the Hamiltonians (typically due to phase or amplitude switching in the rf fields) are discontinuously time-dependent, the average Hamiltonian may conveniently be set up using the semi-continuous Baker-Campbell-Hausdorff (scBCH) expansion [56] as... 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

Uwe T. Bornscheuer, Department of Biotechnology and Enzyme Catalysis, Institute of Biochemistry, Greifswald University, Felix-Hausdorff-Str. 4, 17487 Greifswald, Germany... 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

Since Ruelle s function vanishes if its argument takes the unit value P(l) =0, the Hausdorff dimension can be expanded in powers of the wavenumber as... 

In systems with two degrees of freedom such as the two-dimensional Lorentz gases, there is a single positive Lyapunov exponent X and the partial Hausdorff dimension of the set of nonescaping trajectories can be estimated by the ratio of the Kolmogorov-Sinai entropy to the Lyapunov exponent [ 1, 38]... 

Bouvier, M., Hausdorff, W. P., de Blasi, A., et al. (1988) Removal of phosphorylation sites from the beta 2-adrenergic receptor delays onset of agonist-promoted desensitization. Nature. 333, 370-373. 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

To simplify the right-hand side of (38), we exploit the Campbell-Hausdorff formula... 

Computing the exponents with the help of the Campbell-Hausdorff formula yields... 

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