Dynamical equivalence

Perhaps the simplest way to prove that a system is capable of universal computation - certainly the most straightforward way - is to show that the system in question is formally equivalent to another system that has already been proven to be a universal computer. In this section we sketch a proof of the computational universality of Conway s Life-rule by explicitly constructing dynamical equivalents of all of the computational ingredients required by a conventional digital computer. 

Nitrogen Content and Dynamic Equivalency of Selected Items of Clothing. . . . ... 

To be consistent with original work [42], the data shown in Pigs. 13-16 were obtained for a slightly modified (but dynamically equivalent) version of Model IVa, where gu = 0.152eV... 

Once the mapping (p x) is available, it is possible to find a dynamically-equivalent system by means of the coordinate transformation 2 = (x), where 2 is the independent variable in the new system. 

If two systems, such as a model and its prototype or two pipelines with different fluids, are to be dynamically equivalent so far as inertia and viscous friction are concerned, they must both have the same value of Nr. For the same fluid in both cases, the equation shows that a high velocity must be used with a model of small linear dimensions. It is also possible to compare the action of fluids of very different viscosities, provided only that L and V are so chosen as to give the same value of Nr. 

The above development can be extended to the dynamic systems. The dynamic equivalent of equations (2) and (5) are ... 

Equation (11) represents the time-discrete dynamic equivalent of the steady-state balance equations (2). The dynamic balance equations (11) present some characteristic properties of the sampled-data input and output relationship, that are not present in the corresponding steady-state equations 1) There are as many equations as the number of outputs 2) Each equation contain only one output 3) Each equation contain, except for special cases, all the inputs variables. 

The seas may also act as a receptor for depositing aerosol. Deposition velocities of particles to the sea are a function of particle size, density, and shape, as well as the state of the sea. Experimental determination of aerosol deposition velocities to the sea is almost impossible and has to rely upon data derived from wind tunnel studies and theoretical models. The results from two such models appear in Figure 4, in which particle size is expressed as aerodynamic diameter, or the diameter of an aero-dynamically equivalent sphere of unit specific gravity.If the airborne concentration in size fraction of diameter d is c then... 

A useful technique is the method of symbolic dynamics. We will encounter this topic in Section 2.4. Symbolic dynamics enables us to establish connections between dynamical systems. Often it is possible to map a given dynamical system onto an older one which has been studied before. If such a mapping is possible on the symbolic level, the two systems are dynamically equivalent. 

Symbolic dynamics is one of the most powerful tools in the theory of chaotic systems. It is a qualitative method for characterizing the dynamics of a given nonlinear system. The power of symbolic dynamics shows whenever a new system N, whose properties are not yet known, can be mapped via symbolic dynamics onto an old dynamical system O that has already been thoroughly studied, and whose properties are understood. If such a mapping exists, the two systems N and O axe dynamically equivalent. Obviously, symbolic dynamics has the potential to save a lot of work. 

This equation implies perfect dynamical equivalence of all the chain atoms this is physically true only for the ring, whereas it is a model assumption in the periodic case. However, it should be noted that, apart from the first few collective modes, the periodic chain gives a good description of the open-chain dynamics and may be safely retained when investigating local chain motions, as suggested by Akcasu, Benmouna, and Han [81] and shown by us [82, 83]. The general solution of Eqn. (3.1.5) may be cast in the form... 

As in Eqs (18.19) we impose outgoing boundary conditions on the out state, by assigning a small damping rate rj to this channel. The corresponding terms containing r/ that appear in (18.58d,e) insure that a steady state is achieved if we also impose a constant CT,n,in on the dynamics (equivalent to the driving boundary condition f i (Z) = exp(—iEiat/K)Cin(O ) imposed on Eqs (18.19)). 

The classification of logical network structures imposed by the hypercube description depends on the signs of the focal point coordinates - associated with each orthant of phase space, which leads to the hypercube representation of the allowed flows. We consider that two different networks are in the same dynamical equivalence class if their directed A-cube representations can be superimiposed under a symmetry of the A-cube. For example, in three dimensions there is only one cyclic attractor (see Fig. 3b), but this can appear in eight different orientations on the 3-cube. From a dynamical perspective, exactly the same qualitative dynamics can be found in any of these networks provided the focal points are chosen in an identical fashion. However, from a biological... 

Using the Polya enumeration theorem, it is possible to compute the number of dynamical equivalence classes as a function of N [34, 45]. A lower limit on this number can be obtained by dividing the number of different networks by the order of the symmetry group of the A-cube. Thus, the number of dynamical equivalence classes is greater than or equal to... 

Equation 18.52 becomes the dynamic equivalent of Equation 18.35 where D is the probability of... 

Hernandez, S., J. Gabriel Segovia Hemmdez, and V. Rico Ramirez, Thermo dynamically equivalent distillation schemes to the Petlyuk colutrm for ternary mixtures. 

Although this offers an equivalent fall rate for particle/tablet bed surface, it does not fuUy account for the ratio of forces throughout the pan, which would result in a dynamic difference from the original process. A generally accepted means to scale-up to the dynamic equivalent is to scale to a constant ratio of inertial to gravitational forces. This ratio is commonly referred to as the Froude Number (Fr), a dimensionless value that indicates the system dynamics. A generally accepted Froude Number expression for rotating drums is as follows ... 

According to the arguments presented in Section 4 it follows that the relaxation equations for these four variables consist of two coupled equations for n (g) and g (q) and two totally uncoupled equations for (q) and gy (q). This breakup is a consequence of our choice of coordinate axes such that the wave vector q defines the z axis, g (q) is the momentum density parallel to q, namely the longitudinal momentum, and g iq) and gy(q) are the momentum densities perpendicular to q, namely the transverse components of the momentum density. By symmetry, gz(q),gx(q), and gy(q) are independent, and furthermore gxiq) and gy q) should be dynamically equivalent. Let us therefore only treat g ,(q). 

---