Equations function

Vc, an exchange-correlation term, Exc(p), and an external potential, [V , which arises primarily from nuclear-electron attraction but could include extramolecular perturbations, such as electric and magnetic fields. If the electronic wave function were expressed as a determinantal wave function, as in HF theory, then a set of equations functionally equivalent to the HF equations (A.40) emerges [324]. Thus... 

Hence, the time independent Navier-Stokes equations are put in form of three coupled, nonlinear, ordinary differential equations, functions of rj only ... 

Ordinary differential equations, difference and delay equations, functional equations, boundary-value problems, and stochastic equations... 

Note that we have equated a and b to Tc and Pc only at the critical point. Therefore these functions could have other values away from the critical point. However, as we have equated functions of V,... 

Table 1 summarizes the polymers or copolymers considered here and the experimental ranges of pressure and temperature over which data are available. In Table 2 the Tait-equation functions, with parameters obtained from the fit, are given for 90 polymer or copolymer melts. 

Liu, S. (1996). Local-density approximation, hierarchy of equations, functional expansion, and adiabatic connection in current-density-functional theory. Phys. Rev. A 54, 1328-1336. 

Characterization of a process addressing the trends of substrate depletion, cell growth, and product formation involves the solution of a set of equations. Functional forms of the equations will vary, given the different representative models for growth and stages of product formation and of substrate consumption, and the solution procedure often requires a numerical approach. However, several simplifications are often made by attending to the relevance of several variables as compared to others, namely, often the maintenance coefficient is negligible. 

It can also be described as the parallel form. This is because Equation (3.33) can be represented diagrammaticaUy, as in Figure 3.10, as number of operations working in parallel. To convert the box diagram to an equation, functions in parallel are additive, those in series are multiplicative. 

The latter inequality was derived for the case of zero chemical potential gradients but, since/, p ai are insensitive to the gradients of the state variables, it holds true also in general. Inequality 18 represents a constraint for the constitutive equation function / in equation 3, insofar as it requires that / be positive when the actual pressure is lower that the equilibrium value and that/be negative in the opposite case while both/and the difference p - p are both demanded to be zero in equilibrium conditions. 

The description of the behavior of particulate materials relies on constitutive equations, functions of stress, strain, and other physical quantities describing the system. It is rather difficult to extract macroscopic observables like the stress from experiments, e.g. in a two-dimensional (2D) geometry with photo-elastic material, where stress is visualized via crossed polarizers [6, 7]. The alternative is, to perform discrete element simulations [2, 4] and to average over the microscopic quantities in the simulation, in order to obtain some averaged macroscopic quantity. The averages over scalar quantities like density, velocity and particle-spin are strai tforward, but for the stress and the deformation gradient, one finds slightly different definitions in the literature [3, 8-11]. 

Once the differential equation function is written and saved, a script (i.e., an m-file) containing the call to the integrator must be written. Here, parameter values, initial conditions, and options are specified, and the integration routine is called with the following command ... 

This is rather glib. We have work to do in providing an exact semantics for expressions involving universal. Similarly, we need to define more precisely the intended meaning of == when used to equate functions. 

In the field of mathematics, the requirements for linearity are the same whether they be applied to differential equations, functions, operators, transforms, functionals or other mathematical operations. When these linearity requirements are applied to constitutive theories they are applied best to functional equations [21] since in continuum mechanics a simple material is defined as material wherein the present state of stress can depend upon the history of the deformation gradients [18,19,20]. For the case of small displacements and rotations, the state of stress can be taken as being dependent upon the history of the strain tensor [21-26] and can be expressed functionally as... 

Careful examination of these two conditions indicates that the first rule of linearity, called scalar multiplication or homogeneity of degree one [21,27], is contained in the second rule of linearity, called additivity or Boltzmann superposition. This duplication can simply be shown for all scalars that are rational numbers [1]. Therefore only one mathematical requirement for linearity exists if a reasonable form of continuity requirement is enforced, and that is Boltzmann superposition. It can be shown also that scalar multiplication in no way implies superposition. In fact scalar multiplication is simply a homogeneity condition of degree one in the constitutive law, and many non-linear differential equations, functions, and functionals are homogeneous but not linear. 

Linear equation function f5(x) return 5 x print( f5 newton f5,0))... 

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