Reducing the number of independent variables

By reducing the number of independent variables, e g. space or time, a partial differential equation can, in the best case, be reduced to an ordinary differential equation or even an algebraic equation, and the computational effort is reduced substantially. The most obvious approach is to use the symmetry in geometry. Spherical and cylindrical symmetry is often easy to find. This will reduce a 3D problem to a ID or 2D problem with a significant decrease in computational time. 

Geometrical symmetry requires that the boundary conditions as well as the equations are symmetrical. Problems including gravity can only be symmetrical in a cylinder with gravity in the axial direction. Reduction to a ID or 2D problem may also change the physical appearance, e g. bubbles do not exist in axisymmetric 2D except on the symmetry axis they become toroid in 3D elsewhere. A toroid bubble moving in a radial direction must alter the diameter to maintain the volume, and the forces around the bubble will be unphysical. 

Symmetry boundary conditions should be used with care. The outflow across a symmetry boundary is balanced by an identical inflow, which means that the net transport will be zero. Transport along the boundary is allowed, but no transport across the boundary will occur. It is not sufficient for a conclusive result that the differential equations and the boundary conditions are symmetrical the symmetrical solution may be only one of many possible solutions. 

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The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

One way to reduce the number of independent variables in the FRET-adjusted spectral equation is to use samples with a fixed donor-to-acceptor ratio. Under these conditions, the values of d and a are no longer independent, but rather the concentration of d is now a function of a and vice-versa. This approach is typical for the situation of FRET-based biosensor constructs. These sensors normally are designed to have a donor fluorophore attached to an acceptor by a domain whose structure is altered either as a result of a biological activity (such as proteolysis or phosphorylation), or by its interaction with a specific ligand with which it has high affinity. In general, FRET based biosensors have a stoichiometry of one... 

The symmetry of the structure is imposed on the field (j)(r) by building the field inside a unit cell of smaller polyhedron, replicating it by reflections, translations, and rotations [21-23]. This procedure reduces the number of independent variables one order of magnitude for G structure and two orders of magnitude for D structure. 

A goal in deriving the governing equations is to reduce the number of independent variables by eliminating the molalities nij of the secondary species. To this end, we can rearrange the equation above to give the value of ntj,... 

Providing an additional piece of information about the size of each phase predicts that a total of Ni + N, or Nc, values is needed to constrain the system s state and extent. This total matches the number of variables we must supply in order to solve the governing equations. Hence, although we can make no claim that we have cast the governing equations in simplest form, we can say that we have reduced the number of independent variables to the minimum allowed by thermodynamics. 

The behavior of a reactive wave depen ds on the flow of its reacting and product-gases. The conservation laws lead to systems of partial differential equations of the first order which are quasilinear, ie, equations in which partial derivatives appear linearly. In practical cases special symmetry of boundary and initial conditions is often invoked to reduce the number of independent variables. 

Although a fixed value, 0.306, is not necessarily adequate and the best factor had better to be determined by the regression analysis using Eq. 22 or its counterparts, we used E values defined by Eq. 20 for the steric effect in the following examples. The E value works very well for a number of examples. This might be simply fortuitous but rather favorable in reducing the number of independent variables for the analysis. 

NondLmensionalizaiion reduces the number of independent variables in one-dimensional transient conduction problems from 8 to 3, offering great convetiience in the presentation of results. 

Fixed combinations of selected descriptors accounting for molecular properties of interest. The simplest combined descriptors are the differences and average values of basis descriptors such as -> connectivity indices or - path numbers, and the ratios of different descriptors defined with the aim of normalization to obtain, for example, size-independent indices. Moreover, optimal linear combinations of highly correlated descriptors are combined descriptors calculated so as to reduce the number of independent variables (e.g. - principal properties). 

In order to reduce the number of independent variables, we use the definition of the moments in Eq. (2.2) to find the moment-transport equation corresponding to Eq. (2.50) ... 

The total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

The recognition of such interrelationships between properties has two benefits. Firstly, it can significantly reduce the number of independent variables which have to be considered, and thus the amount of effort required, for the design and development of a new polymer or a new fabrication process. Secondly, it can provide a more realistic set of expectations concerning the maximum attainable performance and the tradeoffs which often have to be made between mutually conflicting "ideal" performance requirements. 

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

The whole set of these equations may then be combined with that for the rate limiting process the latter equation is simplified by reducing the number of independent variables which are related to each other by the balanced equations. 

Buckingham is best known for his early work on thermodynamics and for his later study of dimensional theory. Attracted to problems that could not be solved by pure calculation but requiring experimentation as well, he demonstrated more clearly than anybody before him how the planning and interpretation of experiments can be facilitated by the method of dimensions, later referred to as dimensional analysis. He pointed out the advantages of dimensionless variables and how to generalize empirical equations. His frequently cited ir-theorem serves to reduce the number of independent variables and shows how to experiment on geometrically similar models so as to satisfy the most general requirements of physieal as well as dynamic similarity. 

From this equation we can see the number of independent intensive variables in any homogeneous phase. There are c terms containing p, i.e., c independent compositional intensive variables, plus two other intensive terms, T and P, for a total of c -l- 2 intensive variables. In a single homogeneous phase, these c -I- 2 variables are linked by one equation (4.68), so only c -f- 2 -1 of them are independent. If there are p phases, there are still only c-l-2 intensive variables, because they all have the same value in every phase (at equilibrium), but now there is one equation (4.68) for each phase. Each additional equation reduces the number of independent variables by one, so there are now c + 2-p independent intensive variables. These independent intensive variables are called degrees of freedom, /, so... 

The method of weighted residuals is a way of reducing the number of independent variables or the problem domain dimension. The basic idea of the method is to approximate the solution of the problem over a domain by a functional form called a trial function. The trial function s form is specified but it has adjustable constants. The trial function is chosen so as to give a good solution to the original differential equation. An excellent treatment of the method is given in the book by Finlayson (1972). As an example of how the method works, let us consider the heat conduction equation... 

There usually are, however, various equiUbria and other conditions that reduce the number of independent variables. For instance, each phase may have the same temperature and the same pressure equilibrium may exist with respect to chemical reaction and transfer between phases (Sec. 2.4.4) and the system may be closed. (While these various conditions do not have to be present, the relations among T, p,V, and amounts described by an equation of state of a phase are always present.) On the other hand, additional independent variables are required if we consider properties such as the surface area of a liquid to be relevant. ... 

Each of these relations is an independent restriction that reduces the number of independent variables by one. When we substitute expressions for dU, dF , and d from these relations into Eq. 8.1.1, make the further substitution diS = dS — d > and collect... 

Each independent relation resulting from equilibrium imposes a restriction on the system and reduces the number of independent variables by one. A two-phase system with thermal equilibrium has the single relation = r . For a three-phase system, there are two such relations that are independent, for instance and (The additional relation is not independent since we may deduce it from the other two.) In... 

If a component a does not exist in one of the phases b , then the corresponding mole fraction x = 0, reducing the number of independent variables by one. However, this also decreases the number of constraining equations (7.2.2) by one. Hence there is no overall change in the number of degrees of freedom. 

Contrary to the /cc-based system for which frustration is manifested, a simple pair approximation has been proved to provide fairly reasonable results for a Acc-based system. And we formulate the entropy of B2 ordered phase within the pair approximation and the internal energy of the system is limited to the nearest neighbor pair interaction. Also, by considering the symmetry of 1 1 stoichiometry, we reduce the number of independent variables of free energy and, then, the free energy of B2 phase is symbolically written as... 

Symmetry can be used to limit the volume over which the problem has to be solved. If the boundary conditions are the same on all sides, a square can be reduced to 1/8 of its surface area, as shown in Figure 5.1, and a cube can be reduced to 1/8 of its volume by modeling one comer, or even to 1/16 or 1/32 by finding more mirror planes. The boundary conditions must be die same on all external surfaces. The use of mirror planes does not reduce the number of independent variables, but it results in a smaller volume and fewer mesh points to solve. 

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