Algebraic solutions operators

This may be accomplished by writing stationary relationships (necessary n-add relationships) involving conditional and unconditional probabilities, followed by solving such relationships simultaneously for the conditional probabilities. While this can be done algebraically, it is simpler to do this by conducting operations on a matrix constructed from the conditional probabilities. The procedure described by Price (5) can provide an algebraic solution if desired, but it can be the basis of a program that provides numerical results. The matrix multiplication method (1), provides numerical results only, but it seems to be the preferred approach when the polymerization system is complex. 

In order to solve a set of simultaneous equations, there must be the same number of equations as there are independent variables. Quite an analogous thing occurs with the simultaneous equations in ordinary algebra. The methods used for the solution of these equations are analogous to those employed for similar equations in algebra. The operations here involved are chiefly processes of elimination and substitution, supplemented by differentiation or integration at various stages of the computation. The use of the symbol of operation D often shortens the work. 

In the language of linear algebra, these operations rotate A and B, with T referred to as a transformation matrix or a rotation matrix. Since any T will leave n unaltered, each of the elements of T must be specified to define a unique solution. Normalization of each column determines F elements, leaving F F - 1) parameters undefined. Extra information is necessary to determine these parameters. This is known as the rotation problem. Methods for addressing the rotation problem are discussed in Section IV. 

One can see that the integral transform indeed facilitates the resolution of ODE boundary value problems and also partial differential equations comprised of Sturm-Liouville operators (e.g., Eq. 11.45). The simplicity of such operational methods lead to algebraic solutions and also give a clearer view on how the solution is represented in Hilbert space. Moreover, students may find that the Sturm-Liouville integral transform is a faster and fail-safe way of getting the solution. Thus, Eq. 11.52 represents the solution to an almost infinite variety of ordinary differential equations, as we see in more detail in the homework section. 

In the following examples a, b, and c represent known quantities, and x is the unknown. The object in each case is to solve the equation for x. The steps of the algebraic solution are shown, as well as the operation performed on both sides of the equation. Each example is accompanied by a practice problem that is solved by the same method. You should be able to solve the problem, even if you have not yet reached the point in the book where such a problem is likely to appear. Answers to these practice problems may be found at the end of Appendix I. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

Section 5.3. There, the operator T> s d/dx was used in the solution of ordinary differential equations. In Chapter 5 the vector operator del , represented by the symbol V, was introduced. It was shown that its algebraic form is dependent on the choice of curvilinear coordinates. 

In order to apply the concepts of modern control theory to this problem it is necessary to linearize Equations 1-9 about some steady state. This steady state is found by setting the time derivatives to zero and solving the resulting system of non-linear algebraic equations, given a set of inputs Q, I., and Min In the vicinity of the chosen steady state, the solution thus obtained is unique. No attempts have been made to determine possible state multiplicities at other operating conditions. Table II lists inputs, state variables, and outputs at steady state. This particular steady state was actually observed by fialsetia (8). 

The preceding classic set of algebraic equations form a well-defined sparse structure that has been analyzed extensively. Innumerable techniques of solution have been proposed for problems with 0 degrees of freedom, that is, the column operating or design variables are completely specified. 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

The connection between the algebra of U(2)9 and the solutions of the Schrodinger equation with a Morse potential can be explicitly demonstrated in a variety of ways. One of these is that of realizing the creation and annihilation operators as differential operators acting on two coordinates x and x",... 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

Another ingredient one needs in the application of algebraic methods to problems in physics and chemistry is the eigenvalues of Casimir operators in the representations of Section A.8. The known solution is given in Table A.5. 

We will first give a discussion of some results of general spin-operator algebra not much is needed. This is followed by a derivation of the requirements spatial functions must satisfy. These are required even of the exact solution of the ESE. We then discuss how the orbital approximation influences the wave functions. A short qualitative discussion of the effects of dynamics upon the functions is also given. 

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