Spatial description

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

The unrotated spatial stress rate relation may also be related to its counterpart in the current spatial description. Using (5.1792), (5.1892), (A.36), and... 

The modern theory of the behavior Of matter, called quantum mechanics, was developed by several workers in the years 1925-1927. For our purposes the most important result of the quantum mechanical theory is that the motion of an electron is described by the quantum numbers and orbitals. Quantum numbers are integers that identify the stationary states of an atom the word orbital means a spatial description of the motion of an electron corresponding to a particular stationary state. 

Example of an ASL spatial description. An English translation of the ASL could be I enter the room. There is a table to the left, a TV on the far side, and a chair to the right. ... 

Levinson, S. C. (1997). Cognitive consequences of spatial description in Guugu Yimithir. Linguistic Anthropology, 7, 98-131. 

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

Locally conducted investigations are the base of all spatial descriptions. Depending on the subject under study, the quality of regional and global observations, in principle, improves in accordance with the quantity of single studies and the type of their spatial distribution. However, scientific pro-... 

Chapter 3 adds also the description of spatial distribution (gradients). Only single fluid is considered for the sake of simplicity and preparation of the basics for the subsequent treatment of mixtures. Mathematics necessary for the spatial description is introduced in Sect. 3.1. Section 3.2 in the same chapter stresses the importance of the referential frame (coordinate system) and its change in the mathematical description. Sections 3.3—3.6 shows the development of final material model (of a fluid) within our thermodynamic framework, consistent with general laws (balances) as well as with thermodynamic principles (the First and Second Laws and the principles of rational thermodynamics). The results of this development are simplified in Sect. 3.7 to the model of (single) fluid with linear transport properties. Sections 3.6 and 3.7 also show that the local equilibrium hypothesis is proved for fluid models. The linear fluid model is used in Sect. 3.8 to demonstrate how the stability of equilibrium is analysed in our approach. 

The family of dynamic elastic-contractile model proteins that form the basis for the assertions of the central message of this volume do not lend themselves to the precise spatial descriptions of proteins that form crystals. Nonetheless, important structural description is possible for the poly(GVGVP) family. Indeed, the experimental and computational elucidation of... 

See equations (3.12) - (3. ).4). Note that we are here using a material description, whereas in Section 3 we used a spatial description of the strain field. 

For the spatial description it is relevant to know whether the system can be considered to be Inmped or not. Lnmped means that all variables are independent of the location. An example is a thermometer, which will be discussed in the sequel. A good approximation is that the mercuiy has the same temperature eveiywhere, independent of the height and the cross sectional location. 

Here k is the wave vector of the incident atom at infinity, and U(r) is the potential of the optical gradient force defined by eqn (6.1). By solving the Schrodinger equation (7.17) by some method or other, we obtain a spatial description of the atomic beam, including in its focal region (Klimov and Letokhov 2003). 

Although image acquisition systems improve their spatial resolution, there are instrumental limits related to this parameter. Superresolution strategies are mathematical procedures devoted to obtain spatial descriptions of the sample surface that go beyond the limits of the instrumental spatial resolution [135,136]. 

There is another way of describing the motion of a body consisting of particles, which does not require knowledge of the paths of individual particles. In this description, called the spatial description, the particle velocity v(f) at time t is considered as a dependent variable ... 

The distinction between material and spatial descriptions is clear in that in the former x(r) is the dependent variable and X and t are the independent variables, whereas in the latter w(r) is the dependent variable and x and t the independent variables. Frequently, the material coordinates are called Lagrangian, and the spatial coordinates Eulerian. ... 

The spatial description of this motion may be obtained by first substituting t = t into Eq. (2.8), yielding... 

Obtain the material and spatial descriptions of velocity and acceleration for the following motions ... 

Taken together, the accumulated evidence is that the WCC complex plays the role of the primary photoreceptor of the Neurospora circadian clock but is inextricably linked to the function of the clock. In light, the WCC upregulates FRQ, WC-1, and WD transcription, while FRQ feeds back to block the transcriptional activity of the WCC, thereby downregulating its own transcription. However, FRQ also promotes WC-1 synthesis, indirectly contributing to increased WCC levels, because WC-1 is the limiting partner of the pair. WD also acts to limit WCC complex activity. The interacting feedback loops of these complex interactions form the core of the clock. What remains to be added are the full temporal and spatial descriptions that sum to a 24 h cycle. 

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