Discretization of space

The artificial discretization of space, however, is awkward. In particular, it is responsible for introducing a large set of constants w A as a substitute for the diffusion coefficient D. This difficulty is even more serious if one has to take into account the free flight in space of the particles, as in the case of neutrons in a reactor, see 4. Hence it is desirable to cast (1.1) and (1.2) in a form in which the spatial coordinates occur as continuously varying parameters instead of the discrete subscripts k. 

Nandanwar, M. Kumar, S. 2008 A new discretization of space for the solution of multidimensional population balance equations. Chemical Engineering Science 63, 2198-2210. 

Figure 1 Illustration of the discretization of space by a cubic lattice centered on grid point i. The potential < > is located on the solid circles and the dielectric e is placed on the open circles. The difference equation resulting from this is...

In practice, however, all these numerical techniques use a discretization of space, real-space and pseudospectral methods additionally discretize the chain contour. Thereby, a microscopic cutoff is introduced via the numerical methodology. 

Minimal, Particle-Based, Coarse-Grained Model Discretization of Space and Molecular Contour... 

The parameter, e, plays the same role as the Ginzburg parameter, Gi, does for the mean field theory. The essential difference is that Gi = 1 is a coarsegrained parameter, that is, it is a property of the physical system. The parameter, e, that controls the accuracy of the quasi-instantaneous field approximation, however, also depends on the discretization of space, AI, and molecular contour, N. By a careful choice of these discretization parameters of the soft, coarse-grained model, one can reduce e even if Gi will not be small and fluctuation effects will be important. 

For the numerical solution of the Re5molds equation (2) a similar method is used as the one presented in [1]. The finite differences method is applied for the discretization of space (inner bearing surface) and time (shaft position). 

The reader may think of a finite dimensional subspace of the original state space. This subspace may, e.g., be associated with a suitable discretization in space. For a generalization of Thm. 3 to the infinitely dimensioned case, see [5]. 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

Following the discretization of the solution domain Q (i.e. line AB) into two-node Lagrange elements, and representation of T as T = Ni(x)Ti) in terms of shape functions A, (.v), i = 1,2 within the space of a finite element Q, the elemental Galerkin-weighted residual statement of the differential equation is written as... 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

The 8 functions limit the non-vanishing regions of / -space to discrete layer planes perpendicular to k. These layer planes are infinitely sharp, because the helix was assumed to be infinitely long. Limiting the summation to a finite length of the helix would lead to broadening of these layer planes. 

In tlie case of a discrete sample space (i.e., a sample space consisting of a finite number or countable infinitude of elements), tliese postulates require tliat tlie numbers assigned as probabilities to tlie elements of S be noimegative and have a sum equal to 1. These requirements do not result in complete specification of tlie numbers assigned as probabilities. The desired interpretation of probability must also be considered, as indicated in Section 19.2. The matliematical properties of the probability of any event are tlie same regardless of how tliis probability is interpreted. These properties are formulated in tlieorems logically deduced from tlie postulates above without tlie need for appeal to interpretation. Tliree basic tlieorems are ... 

A completely discrete phase space (i.e. discrete values of lattice-site positions, particle velocities and time, so that particles move from site-to-site and collisions taken within discrete time steps). 

Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. We call models of physics that assume Finite Nature Digital Mechanics. . ..we take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata. ... 

Fredkin s finite nature hypothesis makes no assumptions about the actual scale of space-time s discretization. It might be as large as current estimates of the... 

The 80-bp right operator. Op, can be subdivided into three discrete, evenly spaced, 17-bp cis-active DNA elements that represent the binding sites for either of two bacteriophage A, regulatory proteins. Impor-... 

The discretized momentum-space wave function corresponding to a momentum of ki% is denoted by 44. As with the discretized spatial wave function [Eq. (37)], the discretized momentum wave functions are also normalized so that 4/ p = 1 (i.e., = i/ ki) V ). 

Usually the space over which the objective function is minimized is not defined as the p-dimensional space of p continuously variable parameters. Instead it is a discrete configuration space of very high dimensionality. In general the number of elements in the configuration space is exceptionally large so that they cannot be fully explored with a reasonable computation time. 

On the other hand, the permanent EDM of an elementary particle vanishes when the discrete symmetries of space inversion (P) and time reversal (T) are both violated. This naturally makes the EDM small in fundamental particles of ordinary matter. For instance, in the standard model (SM) of elementary particle physics, the expected value of the electron EDM de is less than 10 38 e.cm [7] (which is effectively zero), where e is the charge of the electron. Some popular extensions of the SM, on the other hand, predict the value of the electron EDM in the range 10 26-10-28 e.cm. (see Ref. 8 for further details). The search for a nonzero electron EDM is therefore a search for physics beyond the SM and particularly it is a search for T violation. This is, at present, an important and active held of research because the prospects of discovering new physics seems possible. 

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