Irreducible equations

Once the complete system of equations has been partitioned into the irreducible subsystems of simultaneous equations, it is desirable to decompose further these irreducible blocks of equations so that their solution can be simplified. The decomposition of the irreducible subsystems is called tearing. In the remainder of this section the subsystems of irreducible equations found by partitioning will be referred to as blocks to distinguish them from the smaller subsystems of simultaneous equations obtained within a block after the tearing is accomplished. 

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... 

Let us summarize the general procedure we would follow for an arbitrary reaction system. For the batch or PFTR we write S first-order differential equations describing each of the S species. We then eliminate as many equations as possible by finding S — R suitable stoichiometric relations among species to obtain R irreducible equations. Sometimes some of these are uncoupled from others so that we may need to solve smaller sets simultaneously. 

In Sec. 2.3 you learned how to formulate uncoupled material balances that could be solved one equation at a time for the values of the unknown variables. All the equations were quite simple, and you did not have to solve two or more equations (called irreducible equations because of the nature of their coupling) simultaneously. You should have observed in Sec. 2.3 that the information provided with respect to the problems such as flows and concentrations was of a special structure that led to the... 

A method for precedence ordering so as to partition a model into a sequence of smaller models containing sets of irreducible equations (equations that have to be solved simultaneously as illustrated in Fig. 5.6). 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a synnnetry operation on the molecular Sclnodinger equation. 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Theorem C establishes the fact that there exists essentially only one irreducible representation of the y-matrices (all others are related to it by a similarity transformation). As a corollary to this theorem, theorem B allows us to assert that 8 is uniquely defined up to a factor. For suppose there were two such 8% say Sx and S2 such that y = Stf8x 1 and y u = 82yu82 1 then by equating y M in these equations... 

The Ether is not useful to teach MT. The EM field is most effectively viewed as an irreducible entity completely defined by Maxwell s equations. (If one wants to make the interaction with point charges in N.M or QM explicit, one can add the Lorentz force or the minimal coupling.) All physical properties of th EM field and its interaction with matter follow from Maxwell s equations and the matter equations. 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

FIGURE 18. The symmetry-adapted, orthogonal linear combinations of the localized a-orbitals of norbornadiene 75 belonging to the irreducible representations / and 62 of the point group C2v- The A] and B2 combinations are the relay orbitals for through-bond interaction between jra and 7Tb which define, according to equation 56, the orbitals jt and jt... 

This equation (16) is known as the great orthogonality theorem for the irreducible representations of a group and occupies a central position in the theory of group representations. 

To improve the formatting of the equations that represent a plant, many commercial codes partition the equations into groups of irreducible sets of equations, that is, those that have to be solved simultaneously. If a plant is represented by thousands of equations, the overall time consumed in their solution via either a GRG or SQP algorithm is reduced by partitioning and rearranging the order of the equations with the result indicated in Figure 15.6. Organization of the set of equations into irreducible sets can be carried out by the use of permutation matrices or by one of... 

The subscript d.i. indicates the dynamical irreducible part.21 The superscript indicates the diagonal part. To evaluate the kernel of this integral equation in the limit of long times, we apply the usual asymptotic formula... 

In Equation 6.22, Sa at Zow is maximum with the limiting value of 1 - Sr, where ST is the irreducible aqueous-phase saturation in the porous medium under study. 

B. How the Irreducible Sets of Equations (Those to be Solved Simultaneously)... 

This section treats the partitioning of the system equations into the smallest irreducible subsystems, that is, the smallest groups of equations that must be solved simultaneously. Partitioning represents the first and easiest of the two phases of decomposition tearing (which is discussed in Section VI) is more difficult. 

It should order the equations within the irreducible subsystems so that the minimum number of variables need be iterated or specified as system inputs to obtain a complete solution of the system. 

B. How the Irreducible Sets of Equations (Those to be Solved Simultaneously) Correspond to the Maximal Loops in the Adjacency Matrix and are Invariant of the Output Set... 

The nonzero elements of a column, j, of the matrix P indicate the loops that are fed information by the equations of loop j, and the nonzero elements of any row, i, indicate the loops that feed information to loop i. The ordering procedure for the irreducible sets of equatins (maximal loops) is as follows ... 

To tackle this problem, we first need to know how a given representation F is reduced to its irreducible representations T) in other words, to determine the coefficients a, in the equation F= Sat Fi. Although this is a key problem in group theory, here we only explain how to perform this reduction without entering into formal details, which can easily be found in specialized textbooks. 

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