Positional constraints

Lanteri, H., Roche, M., and Aime, C., 2002, Penalized maximum likelihood image restoration with positivity constraints multiplicative algorithms. Inverse Problems 18, 1397... 

Until recently, the theory did not allow the configuration of the positive state to be described, due to entry/exit DNAs interpenetration upon application of the positive constraint to the loop. A recent development [63] takes the DNA impenetrability into account and deals with the resulting DNA self-contacts, which were allowed to slide freely, following the needs of the energy minimization process. 

The A -representability conditions on the 2-RDM can be systematically strengthened by adding some of the 3-positivity constraints to the 2-positivity conditions. For three molecules in valence double-zeta basis sets Table II shows... 

Although restorations by this method are superior to those obtained by inverse filtering alone, it should be no surprise that they do not show the dramatic improvements obtainable through the positivity constraint. When, in fact, the integration limits of Eq. (44) are taken well outside the true extent of the object, very little improvement is noted. The applied finite-extent bound must literally butt against the true object for greatest effect. How, then, can the more powerful and useful constraint of positivity be incorporated ... 

For the present work, we chose the constrained method described by Jansson (1968) and Jansson et al (1968, 1970). See also Section V.A of Chapter 4 and supporting material in Chapter III. This method has also been applied to ESCA spectra by McLachlan et al (1974). In our adaptation (Jansson and Davies, 1974) the procedure was identical to that used in the original application to infrared spectra except that the data were presmoothed three times instead of once, and the variable relaxation factor was modified to accommodate the lack of an upper bound. Referring to Eqs. (15) and (16) of Section V.A.2 of Chapter 4, we set k = 2o(k)K0 for 6(k) < j and k = Kq exp[3 — for o(k) > This function is seen to apply the positivity constraint in a manner similar to that previously employed but eliminates the upper bound in favor of an exponential falloff. We also experimented with k = k0 for o(k) > j, and found it to be equally effective. As in the infrared application, only 10 iterations were needed. 

Note from the form of Eq. (59a) that hm can only be positive, regardless of parameters and Xn. Hence, a positivity constraint is a natural consequence of the overall approach. For other cases of prior knowledge, such as the case of emission spectra, boundedness would result. By comparison, all other restoration methods that enforce positivity or boundedness do it in an ad hoc manner. 

The same nonlinear fitting routine could determine the best values for w sind 17 which might not necessarily have physical meaning but could be used to compute r and We used a regularized inversion of equation (15) that incorporates a positivity constraint on G(r,K) in order to obtain more detailed information on the characteristic linewidth profile. A discussion of this approach may be pursued in reference (10). 

For optimal fitting, rotation and translation of the objects are usually necessary. If an object Ay can be exactly superimposed on object A by translation and rotation (in general, by motions allowed by the actual orientation and position constraints), then we regard Ay as a version of object A. 

Table 2.1 Examples of special position constraints on coordinates, anisotropic displacement parameters and site occupancy factors...

Special position Constraints on coordinates Constraints on Ijy values Constraints on occupancies... 

The refinement of such disorders is relatively easy the second site of each atom can be calculated directly from the positions of the atoms of the first component by means of the symmetry operator of the special position. Therefore, it is not necessary to have two parts in the. ins file. Instead of part 1, PART 2, and PART 0, the disordered atoms are flanked with part -1 and part 0. The negative part number suppresses the generation of special position constraints, and bonds to symmetry-related atoms are excluded from the connectivity table. Moreover, the use of the second free variable is not indicated in such a case, as the ratio between the components is determined by the multiphcity of the special position. 

Molecules that are located very close to special positions, so that the symmetry would lead to chemically unreasonable arrangements, are treated the same way. In such a case the spec instruction, which generates all appropriate special position constraints for the specified atoms, may be helpful too. 

Another possible explanation for Q(2) could be a 95 5 or so disorder of Zr(2) and its ligands. Such a disorder, however, should also result in a higher U q value for Zr(2), which is not observed. Actually, the opposite is the case C/eq(Zr(l)) = 0.030, 7eq(Zr(2)) = 0.024. This difference can be explained with the special position constraints (see Section 2.5.2) that restrict the shape of the displacement ellipsoid of Zr(2) to fulfil the fourfold symmetry. This, in turn, artificially lowers the calculated value of C/eq for Zr(2). 

Constraints (3.17) and (3.18) define the last position at which each chemical s is scheduled. Constraints (3.19)-(3.20) assure that each chemical is at least scheduled once in the fundamental cycle. No more than one chemical is scheduled at each position. Constraints (3.21) indicate that all chemicals are scheduled in the (previous) fundamental cycle. 

Using an input interface, such as a mouse or joystick the camera position, the field of view of the camera, and the focal point are interactively changed. The software may provide collision detection for the camera and anatomic surfaces, thus applying a force model to keep the camera inside the sinonasal structures (Jolesz et al. 1997). These approaches, however, require user training and the introduction of positional constraints via software to hinder the perforation of the imaged structure by the virtual endoscope (Carvalho 2003). 

The advantage of the custom code approach is that a wide range of other checks can be readily added and help improve process integrity. For example validity check can be made on aU flows, temperatures and control valve positions. Constraints such as maximum flow imbalance, maximum skin temperature etc. can be included. Logic can be included to handle out of the ordinary situations, such as instrument failure, and the application degraded gracefiilly rather than an all-or-nothing approach. 

We assume that we are given a nominal position pn, which satisfies the position constraint... 

In the last paragraph we distinguished two different types of velocity variables p and V, The velocity constraint was based on p, whereas the Drazin ODE was formulated in terms of v. This scheme will be repeated in this paragraph, when constructing the Drazin ODE of the position constrained problem. There, we will introduce two types of position variables p and q. The position constraint will be formulated in terms of p, whereas the Drazin ODE is based on g-position variables. Let us start again from the second order formulation of the problem ... 

Let us consider Eqs. (5.1.3a), (5.1.3b) and (5.1.5b). This set of equations was obtained by twice differentiating the position constraint with respect to time and some algebraic manipulations of the equations. A further time differentiation of the algebraic equation leads to an ODE in all variables p, v and A ... 

In the index-2 and index-1 formulations the position constraint is no longer incorporated. This constraint is violated when using one of these formulations. The result is given in Fig. 5.2, where the development of the residual of the position constraint is visualized for the pendulum example. 

Figure 5.2 Residual of the position constraint, when integrating the pendulum with the implicit Euler method and h = 0.005.

Index-3 (Position constraint) problems when solving the linear system in corrector iteration step size changes unstable in v and A... 

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