Contracted matrix

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

Solution of the original matrix by a standard program would give incorrect statistical parameters if the restrictive equations were entered directly into the matrix. Solution of the contracted matrix in Table III by a standard program gives exactly the same results as are obtained from the original matrix by the special program of Free and co-workers. However, the substituent constants for A, E, and / must be calculated by use of the restrictive equations when the contracted matrix is used. 

Trapped fat cells and free fat in raw matrix. After cooking, contracted matrix trapped fat but some areas of coalesced fat visible... 

Table 3.1 Assignment of contracted matrix indices to their corresponding tensor index pairs...

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

For the transverse buckling mode in Figure 3-55, the matrix material expands or contracts in the y-direction. However, the matrix strain in the y-direction (transverse to the fibers) is presumed to be independent of y, i.e., simply twice the two adjacent fiber displacements, v, divided by the original distance between the fibers ... 

The stress-strain relations in this book are typically expressed in matrix form by use of contracted notation. Both the stresses and strains as well as the stress-strain relations must be transformed. First, the stresses transform for a rotation about the z-axis as in Figure A-1 according to... 

NAS. 1975. Matrix of Electrical and Fire Hazard Properties and Classification of Chemicals. National Academy of Sciences Report to US Coast Guard, Contract No. DOT-CG-41680-A(1975). 

In order to study lattice relaxation effect by ASR we assume a simple model. As a first step we consider the terminal point approximation. Here the distortion of the lattice taken into account is the stretching or the contraction and angular distortion of the bond connecting two sites in a lattice and the effect of neighbouring site is neglected. As a result of such distortion the structure matrix takes the form ... 

Therefore, before a final wall structure can be selected, it is necessary to conduct a combined strain analysis in both the longitudinal and hoop directions. This analysis will consider thermal contraction strains, the internal pressure, and the pipe s ability to bridge soft spots in the trench s bedding. In order to do this we must know more about the inherent properties of the material we are dealing with that is a structure made up of successive layers of continuous filament-wound fiberglass strands embedded within a plastic matrix. We must know the modulus of the material in the longitudinal direction and the... 

The contractions that are encountered in applying Wick s theorem in the computation of the -matrix are given by... 

Rare Earth Stabilized Zirconia Matrix Bricks in The Pilot Test Unit (PTU) at AEDC , Report No AEDC-TR-72-161, Contract F40600-73-C-0004 (Nov 1972) 49) A.D. Kirshenbaum et al,... 

First, the function e(t) computed from e(x) (Fig. 33), is divided into a number of time intervals which are sufficiently short to justify the approximation of a constant average strain-rate within each period. Only the region of space where the strain rate is significantly different from zero, i.e. from — 4r 5 x +r0 in the case of abrupt contraction flow (Fig. 33), will contribute to the degradation and needs to be considered in the calculations. The system of Eq. (87) is then solved locally using the previously mentioned matrix technique [153]. 

Alle the deformation zones contain a finite and equal number of extended chains in their most highly stretched strands. This surprising conformity of the deformation zones may well be the consequence of the imposed plane-strain fracture condition which impedes lateral contraction of the material. However, no quantitative explanation has been presented as yet. A plausible explanation would be to assume that due to the hindered lateral contraction additional tensile stresses are transferred to the most extended strand with each additional chain pulled out of the matrix [112]. 

This in turn leads to the initial one-electron density matrix by contraction over electron variables,(48)... 

Figure 12-14. The creatine phosphate shuttle of heart and skeletal muscle. The shuttle allows rapid transport of high-energy phosphate from the mitochondrial matrix into the cytosol. CKg, creatine kinase concerned with large requirements for ATP, eg, muscular contraction CIC, creatine kinase for maintaining equilibrium between creatine and creatine phosphate and ATP/ADP CKg, creatine kinase coupling glycolysis to creatine phosphate synthesis CK, , mitochondrial creatine kinase mediating creatine phosphate production from ATP formed in oxidative phosphorylation P, pore protein in outer mitochondrial membrane.

Michaeli (1960) opposed these views. He concluded that whatever the exact mechanism was, the binding of divalent cations caused contraction and coiling of the polyelectrolyte as was the case with adds. He disagreed with the concept of ionic crosslinking. The phenomenon of precipitation could be explained simply in terms of reduced solubility. From this he concluded that precipitation took place in an already coiled molecule and the matrix consisted of spherical macromolecules containing embedded cations. 

The RDM s are therefore much simpler objects than the A-electron Wave Function (WF) which depends on the variables of N electrons. Unfortunately, the search for the 7V-representability conditions has not been completed and this has hindered the direct use of the RDM s in Quantum Chemistry. In 1963 A. J. Coleman [4] defined the A -representability conditions as the limitations of an RDM due to the fact that it is derived by contraction from a matrix represented in the N-electron space. In other words, an antisymmetric A-electron WF must exist from which this RDM could have been derived by integrating with respect to a set of electron variables. 

Two different approaehes to this problem will be described in this work. They are based in quite different philosophies, but both are aimed at determining the RDM without a previous knowledge of the WF. Another common feature of these two approaches is that they both employ the discrete Matrix representation of the Contraction Mapping (MCM) [17,18]. Applying this MCM is the alternative, in discrete form, to integrating with respect to a set of electron variables and it is a much simpler tool to use. 

In many cases a simpler form of this mapping may be used. Thus, the RDM by itself, when it is not involved in matrix operations it can be contracted by using... 

The interest of contracting the matrix form of the Schrodinger equation by employing the MCM, is that the resulting equation is easy to handle since only matrix operations are involved in it. Thus, when the MCM is employed up to the two electron space, the geminal representation of the CSchE has the form [35] ... 

Most contract research organizations and pharmaceutical companies are organized in a matrix management structure. This structure is called a matrix because there are project teams that span various functional departments. It may help to visualize the relationship like this ... 

Changes in the natures of individual phases of or phase separation within a formulation are reasons to discontinue use of a product. Phase separation may result from emulsion breakage, clearly an acute instability. More often it appears more subtly as bleeding—the formation of visible droplets of an emulsion s internal phase in the continuum of the semisolid. This problem is the result of slow rearrangement and contraction of internal structure. Eventually, here and there, globules of what is often clear liquid internal phase are squeezed out of the matrix. Warm storage temperatures can induce or accelerate structural crenulation such as this thus,... 

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