A priori constraints

For concreteness, let us suppose that the universe has a temporal depth of two to accommodate a Fi edkin-type reversibility i.e. the present and immediate past are used to determine the future, and from which the past can be recovered uniquely. The RUGA itself is deterministic, is applied synchronously at each site in the lattice, and is characterized by three basic dimensional units (1) digit transition, D, which represents the minimal informational change at a given site (2) the length, L, which is the shortest distance between neighboring sites and (3) an integer time, T, which, while locally similar to the time in physics, is not Lorentz invariant and is not to be confused with a macroscopic (or observed) time t. While there are no a priori constraints on any of these units - for example, they may be real or complex - because of the basic assumption of finite nature, they must all have finite representations. All other units of physics in DM are derived from D, L and T. 

The a priori penalty prior(x) oc — log Pr x allows us to account for additional constraints not carried out by the data alone (i.e. by the likelihood term). For instance, the prior can enforce agreement with some preferred (e.g. smoothness) and/or exact (e.g. non-negativity) properties of the solution. At least, the prior penalty is responsible of regularizing the inverse problem. This implies that the prior must provide information where the data alone fail to do so (in particular in regions where the noise dominates the signal or where data are missing). Not all prior constraints have such properties and the enforced a priori must be chosen with care. Taking into account additional a priori constraints has also some drawbacks it must be realized that the solution will be biased toward the prior. 

Thiebaut, E., Conan, J.-M., 1995, Strict a priori constraints for maximum likelihood blind deconvolution, JOSA.A, 12, 485 Thiebaut, E., 2002, Optimization issues in blind deconvolution algorithms, SPIE 4847, 174... 

A Monte Carlo simulation (Fig. 3) can be made as usual (that is, without constraints on the output age), in which case only about 24% of the trials will yield ratios corresponding to a finite age, and a younger limit of >821 ka (95% confidence) or >531 ka (68% conf) is indicated. If, however, the a priori assumption of a closed system with no initial °Th is made, the failed trials can be ignored (since they violate the a priori constraints), and solution of both age and age-error (630 +370/-210 ka at 95% conf., or +150/-140 ka at 68% conf) can be obtained from the Monte Carlo simulations. 

Purely mathematical approaches using the PSF of the system and/or a priori object information rarely exceeded a factor of two [17,22,23] and were prone to producing artifacts. They can be augmented through additional a priori constraints, such as the objects featuring different absorption or emission spectra [24]. In this case, the resolution problem can become almost trivial, because objects with different spectra can be separated with suitable spectral filters. However, because of the difficulty to mark all features in a sample with different labels, reducing the resolution problem to a color separation... 

Thus, Eq. (38b) minimizes the quadratic form [Eq. (30a)] with two terms ( =1,2), where the second term 2(3) represents a priori constraints on the with derivatives. The inclusion of 2( ) ill th minimization can be considered as applying limitations on the quadratic norm of m-th derivatives of y(x) that are commonly used as a measure of smoothness (e. g. see [32]). Indeed, if one assumes the diagonal covariance matrix Cg. with diagonal elements... 

The consideration of a priori constraints as an equal component in the multi-term inversion Eqs. (30-32) is a useful tool for applying multiple constraints in a retrieval algorithm. For example, a simultaneously constraining solution by both a priori estimates and smoothness assumptions can be considered as inversion of the data from three independent sources and Eq. (25) is... 

Thus, applying multiple a priori constraints is straightforward using multi-term LSM formulations, while multiple a priori constraints usually are not considered in the scope of basic formulas [Eqs. (19-20) and (23-24)]. 

Equations (50) are helpful for analyzing the effects of constraints on the solution. It follows from Eq. (50b) that by strengthening a priori constraints (by 2 3), one formally can suppress the random errors of the retrieval to... 

Sections 4.3-4.4 provide quantitative definitions of Lagrange multipliers. However, in reality the detailed information required for an explicit definition of a priori constraints may not be available. In such situations the following recipes and discussions may be useful. 

In fact, the foregoing introduces a contradiction, since the mole fraction summation cannot be satisfied at the same time. Therefore, an a priori constraint cannot be introduced between K, and K- that is, no additional equation can be introduced. And, if so, it would pertain only to a particular situation that is, some unique combination of the variables. 

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