Five-dimensional Space-time

Normally, solids are crystalline, i.e. they have a three-dimensional periodic order with three-dimensional translational symmetry. However, this is not always so. Aperiodic crystals do have a long-distance order, but no three-dimensional translational symmetry. In a formal (mathematical) way, they can be treated with lattices having translational symmetry in four- or five-dimensional space , the so-called superspace their symmetry corresponds to a four- or five-dimensional superspace group. The additional dimensions are not dimensions in real space, but have to be taken in a similar way to the fourth dimension in space-time. In space-time the position of an object is specified by its spatial coordinates x, y, z the coordinate of the fourth dimension is the time at which the object is located at the site x, y, z. 

Here the index s runs over the relevant saddle points, those that are visited by an appropriate deformation of the real integration contour, which is the real five-dimensional (t,t, k) space, to complex values, and Sp(t,t, k) s denotes the five-dimensional matrix of the second derivatives of the action (4.5) with respect to t,t and fc, evaluated at the saddle points. The time dependence of the form factors (4.6) and (4.7) is considered as slow, unless stated otherwise (see Sect. 4.5 and [27]). 

The theory of Kaluza and Klein [89, 90] is based on an observation that of two macroscopic forces of Nature only gravitation can be ascribed to geometric features of four-dimensional space-time. In order to incorporate another interaction the logical development would be to consider an additional dimension and to examine if extra degrees of freedom provided by 15 covariant components of the five-dimensional symmetric tensor needed to specify the line element... 

An elegant but simple model of a five-dimensional universe has been proposed by Thierrin [224]. It is of particular interest as a convincing demonstration of how a curved four-dimensional manifold can be embedded in a Euclidean five-dimensional space-time in which the perceived anomalies such as coordinate contraction simply disappear. The novel proposal is that the constant speed of light that defines special relativity has a counterpart for all types of particle/wave entities, such that the constant speed for each type, in an appropriate inertial system, are given by the relationship... 

Of all the models considered in the previous chapter only the original de Sitter model meets the relativistic requirements. It is the only one to assume a space-time metric rather than a privileged time coordinate. It transcends special relativity in assuming a space-time embedded in five-dimensional... 

This model can hardly be more unUke a universe that expands in three-dimensional Euclidean space. The special-relativistic requirement of foru -dimensional space, with the cru vature of general relativity superimposed, seems to demand that space-time has a minimum of five dimensions, which is equivalent to four-dimensional projective space, described by five homogeneous coordinates. Locally perceived three-dimensional space therefore is an illusion and extrapolation of local structure, beyond the Galactic borders, a gross distortion. 

In all of these statements the emphasis is on unified fields and not on cosmology. As a matter of fact, the equivalence of the projective unified model to five-dimensional spaces in general, and to that of Einstein and Mayer in particular, was first demonstrated by Veblen himself (Monograph Chapter Vlll). The crucial observation is that this equivalence mapping is done in the tangent space, without implying the equivalence of the Einstein-Mayer five-dimensional construct with four-dimensional projective space-time. The five-dimensional spaces are not projective, but affine spaces. 

We can also use equation (1) to map the five-dimensional associated spaces in space-time-point x to the tangent space belonging to x. This mapping... 

In an other vein, DCMs may be considered as five-dimensional materials with three dimensions of space, one dimension of time/dynamics, and one dimension of constitution, representing the... 

The full resolution of a turbulent mixing problem would require full field measurements of three instantaneous velocity components over time [u(x,y,z,t), v(x,y,z,t), w(x,y,z,t)], plus full field concentration(s) for each component [c(x,y,z,t)]. This five dimensional space is not easily attainable with current methods, and the postprocessing requirements of this quantity of data suggest that some averaging will be required, hi Section 2-3.4.1, we consider the various common experimental methods and what dimensions of this problem they measure. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

Considep two-dimensional transient heat transfer in an L-shaped solid body that is initially at a uniform temperalure of 90°C and whose cross section is given in Fig. 5-51. The thermal conductivity and diffusivity of the body are k = 15 W/m C and a - 3.2 x 10 rriVs, respectively, and heat is generated in Ihe body at a rate of e = 2 x 10 W/m. The left sutface of the body is insulated, and the bottom surface is maintained at a uniform temperalure of 90°C at all times. A1 time f = 0, the entire top surface is subjected to convection to ambient air at = 25°C with a convection coefficient of h = 80 W/m C, and the right surface is subjected to heat flux at a uniform rate of r/p -5000 W/m. The nodal network of the problem consists of 15 equally spaced nodes vrith Ax = Ay = 1.2 cm, as shown in the figure, Five of the nodes are at the bottom surface, and thus their temperatures are known. Using the explicit method, determine the temperature at the top corner (node 3) of the body after 1,3, 5, 10, and 60 min. 

For the experiment array, I prefer an orthogonal central-composite design (2), (3), which consists of three main parts, as shown in Table I. The first is a conventional 16-experiment fractional factorial design for five variables at two levels. The second comprises three identical experiments at the average, or center-point, conditions for the first 16 experiments. The final part comprises two out-lier experiments for each variable. These augment the basic two level design to provide an estimate of curvature for the response to each variable. The overall effect of the design is to saturate effectively the multi-dimensional variable space. It is more effective than the conventional "one-variable-at-a-time" approa.ch. 

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