Two-dimensional minimization

In the special case of optimization problems with two variables, the follovdng alternatives may be used  

The third relation of (5.8) means that for each value of the variable xi, a onedimensional minimum of the function F xi,X2) is searched for with respect to the variable X. The variable X2 is, in turn, used to rninimize the function 

A new MinimizationMonoVeryRobust class object is used to find this minimum. It finds the value of X2 that minimizes the function, which is optimal with respect to x. This function has the following seven minima in 0 X2 10  

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The proposed model [29,30] is based on the possibility of forming sections of a cubic lipid-water phase which gives a two-dimensional minimal surface with "holes" facing alternate sides of the bilayer. If these holes through the bilayer are plugged with protein molecules, we can form a bilayer that is closely related to the "planar" bilayer conformation (cf. Fig. 5.6). As this new conformation is related to the cubic phase, we will call it whereas the "normal" conformation, which corresponds to the La phase, is called L 2. ... 

Another possibility is to perform a two-dimensional search in the plane generated by the Newton and gradient directions. A two-dimensional minimization problem takes place. 

The three-dimensional synnnetry that is present in the bulk of a crystalline solid is abruptly lost at the surface. In order to minimize the surface energy, the themiodynamically stable surface atomic structures of many materials differ considerably from the structure of the bulk. These materials are still crystalline at the surface, in that one can define a two-dimensional surface unit cell parallel to the surface, but the atomic positions in the unit cell differ from those of the bulk structure. Such a change in the local structure at the surface is called a reconstruction. 

One fiirther method for obtaining surface sensitivity in diffraction relies on the presence of two-dimensional superlattices on the surface. As we shall see fiirtlrer below, these correspond to periodicities that are different from those present in the bulk material. As a result, additional diffracted beams occur (often called fractional-order beams), which are uniquely created by and therefore sensitive to this kind of surface structure. XRD, in particular, makes frequent use of this property [4]. Transmission electron diffraction (TED) also has used this property, in conjunction with ultrathin samples to minimize bulk contributions [9]. 

A systematic comparison of two sets of data requires a numerical evaluation of their likeliness. TOF-SARS and SARIS produce one- and two-dhnensional data plots, respectively. Comparison of sunulated and experimental data is accomplished by calculating a one- or two-dimensional reliability (R) factor [33], respectively, based on the R-factors developed for FEED [34]. The R-factor between tire experimental and simulated data is minimized by means of a multiparameter simplex method [33]. 

For example one forms, within a two-dimensional (2D) sub-Hilbert space, a 2x2 diabatic potential matrix, which is not single valued. This implies that the 2D transformation matrix yields an invalid diabatization and therefore the required dimension of the transformation matrix has to be at least three. The same applies to the size of the sub-Hilbert space, which also has to be at least three. In this section, we intend to discuss this type of problems. It also leads us to term the conditions for reaching the minimal relevant sub-Hilbert space as the necessary conditions for diabatization. ... 

The shape of a droplet or of the front end of a film can be determined from the surface energies and interaction forces between the interfaces. These also determine the equilibrium thickness of a liquid film that completely wets a surface. The calculation is done by minimization of the free energy of the total system. In a two-dimensional case the free energy of a cylindrical droplet can be expressed as [5] ... 

It may be safely assumed that each of these sets of receptor locations is structurally somewhat heterogeneous, and that it occupies a specific area of the sensory epithelium, with a characteristic, possibly very complex, topology. Because knowledge of this taste modality is still minimal, the topological structure of the taste modality is pure speculation. One such speculation was offered by Beets. To simplify his reasoning, Beets assumed the topology to refer to a two-dimensional area of the sensory... 

Greek indices a, p = x,y,z of the Cartesian coordinate axes is meant). Minimization of expression (2.1.1) for an arbitrary two-dimensional Bravais lattice becomes possible since the Fourier representation in q implies the reduction of the double sum over j and/ to the single sum over q. Then the ground state energy is given by... 

Dark-field electron microscopy (in which the image is formed from the scattered beam), when combined with improved techniques of sample handling and preparation and minimal radiation exposure, can lead to images of sufficiently undamaged DNA at a resolution of 10 A (116). Figure 45 shows such an image in which the two-dimensional projection of the helix is clearly visible on the undamaged part of the molecule. 

FIGURE 5.15 Different misclassification measures are used for two-dimensional data with three groups. They all depend on the choice of the split variable (here the horizontal axis) and the split point (here the point, v). The task is to find the variable and the split point which minimize a chosen error measure. 

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