It representation

We consider the problem of splitting for 2-D tensor field by using its representation by means of the tensor u... 

It is possible to represent molecules with feature trees at various levels of resolution. The maximum simplification of a molecule is its representation as a feature tree with a single node. On the other hand, each acyclic atom forms a node at the highest level. Due to the hierarchical nature of feature trees, all levels of resolution can be derived from the highest level. A subtree is replaced by a single node which represents the union of the atom sets of the nodes belonging to this subtree. 

The theme of this book has a strong emphasis on the structure of chemical compounds, its representation, and its correlation with properties. However, there are compounds whose structure is either unknown or ill-defincd. This is true, for example, for many polymers, particularly those that have been prepared from several components, or for many technical materials, such as glues, washing powder, etc. 

If the dynamics produce a third-order differential equation, its representation would be... 

Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

Modularity. Since we would like to use the sufficient theory in a variety of contexts and problems, we need a theory that was easy to extend and modify depending on the context. In our state-space formulation the sufficient theory is couched in terms of constraints on variables. This theory gives us the opportunity to modularize its representation, partitioning the information necessary to prove the looseness of one type of constraint from that required to prove the looseness of a different constraint type. The ability to achieve modularity is a function not only of the theory but also of the representation, which should have sufficient granularity to support the natural partitioning of the components of the theory. 

A book about the mystery of fire and the groups which use flame in their rituals or esoteric practices. He deals with five groupings of initiates who use flame or its representation in rituals or otherwise, addressing Kabbalism (Qabbalism), Hermeticism and Alchemy, Rosecrucianism, Freemasonry, and the Egyptians"... 

The Operation which changes the signs of all three components of is called e inversion. Its representation is obviously a negative unit matrix. Then,... 

Let us look at the standard Hamiltonian (13). Its representation restricted to the ground state and the first excited state of the fast mode may be written according to the wave operator procedure [62] by aid of the four equations... 

Implementations always include some element of specification (but not vice versa), so the two kinds of document are never entirely separate. First, an implementation need not be complete down to the finest detail it might interconnect a few smaller component specifications, leaving their implementations to be chosen separately. Second, the concepts the participants talk about when they interact can also be specified rather than implemented using types, we can define what information must be sent without deciding its representation. 

Legacy adapter This thin layer shields business objects from idiosyncrasies of the legacy system and its representations. 

The three parameters in the Morse function D, B, re are positive and are usually chosen to fit the bond dissociation energy, the harmonic vibrational frequency and the equilibrium bond length. At r = re, the Morse function V = 0. As r — D, V approaches D. For r re, V is large and positive, corresponding to short range repulsion. Although the Morse function has been used extensively, its representation of the potential away from re is not satisfactory. Several modifications have been proposed in Morse function. 

Figure 5-8. The spectrum vector y , measured at three wavelengths and its representation in a 3-dimensional space.

Figure 5-61. Relationship between a new sample spectrum y and its representation u in the eigenvector space.

The rest of this chapter is organized as follows In the next section 1 describe the basics of the theoretical description of SD, including the approximations that are often used to simplify its representation. Sec. Ill will deal with the analysis of the molecular origin of SD under the conditions when the LRA is valid. In Sec. IV, the breakdown of the LRA is discussed and its causes analyzed. The chapter will be concluded in Sec. V. 

Tang et al. [20] have examined the population dynamics in a three-level system, and its representation in a surrogate two-level system, to test the scheme outlined above. In the model system considered state 3 is weakly coupled with states 1 and 2, so that population transfer between states 1 and 2 should dominate the dynamics, with only a small contribution from population transfer to and from state 3. The coupling of state 3 with states 1 and 2 was taken to be one-tenth of the coupling between states 1 and 2, that is, M)3 = M23 = Mn/10 = -1/10. Using the formalism sketched above, the exact system dynamics is governed by the coupled equations of motion for the three states,... 

Denote by DE(G) the set of directed edges of Conn(G). Denote by V(G) the vector space with canonical basis feOdeOEto- For every cycle c of Conn(G), choose an orientation of it and denote by f(c) its representation in V(G). Denote... 

If the specific tensorial structure (14.58) of a two-electron operator is known, then we can obtain its representation in terms of the product of operators (14.30) acting in the space of states of one shell. In fact, if we substitute into the two-electron matrix element which enters into (13.23), the operator (14.58) in the form... 

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

Fig. 2.7. The bonding potential V(r) and its representation by a harmonic potential energy function and an additional cubic term.

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

It should be remarked that the subgroup E, F4 and its representations form the internal isometric group JF for the choice [0, tr] of the domain of t. The fixed... 

Figure 1.19 Schematic drawing of the cation-rr interaction showing the contact between the two. The quadrupole moment of benzene, along with its representation as two opposing dipoles, is also shown.

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