Symmetric extension

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

The simplest GUT model [10], namely the SU(5) unification and its super-symmetric extension, is almost certainly ruled out from many reasons lack of proton decay at the predicted lifetime level, inability to produce neutrino masses, and so on. On the other hand, the next target, SO(IO) models, and their SUSY extensions in particular, are very promising. It has just the needed component of the right handed Majorana lepton for realization of the seesaw mechanism. The models also accommodates B - L violation which is needed for baryogenesis, as is explained later. 

The major advantage of periodic extension is that no additional coefficients are required (if the number of data points is a power of 2) and it preserves the orthogonality of the DWT. However, if singular end-effects owing to nonperiodicity is a major concern, then consideration should be given to using other extension techniques, such as symmetric extension. 

A symmetric extension involves the symmetric reflection of a function about the boundaries of the interval [0, K] in which the finite-length function is defined. This is shown in Fig. 13. 

Symmetric extension has the advantage, compared with periodic extension, that the function at the interval boundaries is continuous. However, the first... 

The periodised DWT developed in the previous section can be used for the data subject to symmetric or anti-symmetric extension. However, the data will have to be extended to 2 data points, where... 

Fig. 15 WPT results for a linear ramp signal IN = 128, Dauhechies wavelet. N/ = 6) with boundary handling using. symmetric extension. First six levels shown only.

Unfortunately, the presence of additional coefficients can introduce dependencies in the coefficients close to the boundaries as well as prevent perfect reconstruction upon back-transformation. Furthermore, the symmetric extension property should be applied at each level of the decomposition. Perfect reconstruction can only be achieved through the use of symmetric wavelets such as bi-orthogonal wavelets [2]. 

As in the case of symmetric extension, the data will need to be padded (at both ends of the sequence) to 2- data points prior to applying periodic extension. Fig. 17 shows the resulting WPT for a ramp signal when zeropadding boundary extension has been applied. ... 

The tension T across a normal cross-section of a fiber undergoing an axially symmetric extension equals the integral over the cross-section of the S component of the Cauchy stress tensor ... 

The crystal orbitals were calculated by the DFT method in the plane-wave basis set with the CASTEP code [377] in the GGA density functional. A set of special points k in the Brillouin zone for all the crystals was generated by the snperceU method (see Chap. 3) with a 5 x 5 x 5 diagonal symmetric extension, which corresponds to 125 points. In all cases, the pseudopotentials were represented by the normconserving optimized atomic pseudopotentials [621], which were also used to calculate the atomic potentials of free atoms. In the framework of both techniques (A, B), the population analysis was performed in the minimal atomic basis set i.e. the basis set involved only occupied or partially occupied atomic orbitals of free atoms. It is well known that the inclusion of diffuse vacant atomic orbitals in the basis set can substantially change the results of the population analysis. For example, if the Mg 2p vacant atomic orbitals are included in the basis set, the charge calculated by technique A for the Mg... 

Histologically, a diffuse, symmetrical, extensive demyelination is seen. There may be almost complete lack of myelin sheaths in the central nervous system (Jervis 1960) with the exception, in most cases, of some U-fibers and of myelin sheaths within the central grey nuclei and the optic radiation. Demyelination is most marked in the internal capsule and pyramidal tracts spinal roots and peripheral nerves are involved to a lesser degree (Jacobi 1947 Bertrand et al. 1954). There may be partial loss, or in the center of involved regions, complete loss of axis cylinders. In the periphery of demyelinized areas they may be swollen and terminally distended (Hollander and Pilz 1964). The axis cylinders of U-fibers are, in general, well preserved. Small inflammatory lymphocytic infiltrates may be seen perivascularly. 

An expression similar to the above was obtained by Insarova [168] for the extensional viscosity of uniformly distributed rigid rods subjected to axially symmetric extension as follows ... 

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