The hypercube

One example of such a new approach is the hypercube architecture, in which many processors are linked together as a team to solve a single problem. A well-integrated team of cheap processing units can potentially outperform the most sophisticated single-processor machine. In addition, since each processor can have its own dedicated memory, the total system can have both more memory than current supercomputers and more memory in use at any given moment. 

Bit vectors live in an -dimensional, discrete hypercubic space, where each vertex of the hypercube corresponds to a set. Figure 2 provides an example of sets with three elements. Distances between two bit vectors, vA and vB, measured in this space correspond to Hamming distances, which are based on the city-block Zj metric... 

You stare into Sally s eyes with pupils slightly dilated in the dim room light. Take a look at the hypercube drawing. Can you see the sixteen corners The number of corners (or vertices) doubles each time we increase the dimension of the object. The hypercube has thirty-two edges. To get the volumes of each object, all you have to do is multiply the length of the sides. For example, the volume of a cube is l, where / is the length of a side. The hypervolume of a hypercube is l. The hyperhypervolume of a 5-D cube is P, and so on. 

You then bring out a paper model of an unfolded hypercube. Sally, we can cut a hypercube and flatten it to the third dimension in the same way we flattened a cube by unfolding it into the second dimension. In the case of the hypercube, the faces are really cubes (Fig. 4.7). 

You point at a poster on the wall. The hypercube has often been used in art. My favorite is the unfolded hypercube from Salvador Dali s 1954 painting Corpus Hypercubus (Fig. 4.8). By making the cross an unfolded tesseract, Dali represents the orthodox Christian belief that Christ s death was a metahistorical event, taking place in a region outside of our space... 

Figure 4.7 One way to unfold a hypercube. Just as with the cube in Figure 4.6, the bottom cubical face must join with the top face when folding the cubes to reform the hypercube. This folding must be done in the fourth dimension. (The forwardmost cubical face is shaded to help clarify the drawing.)...

I would like to delve further into the fourth dimension by discussing hyperspheres in greater detail. Let s start by considering some exciting experiments you can conduct using a pencil and paper or calculator. My favorite 4-D object is not the hypercube but rather its close cousin, the hypersphere. Just as a circle of radius r can be define by the equation = r, and a sphere can be defined by... 

Heinlein, R. (1958) —And he built a crooked house, in Fantasia Mathematical C. Fadiman, ed. New York Simon and Schuster. (Original story published in 1940.) The misadventures of a California architect who built his house to resemble the projection in 3-D space of a 4-D hypercube. When the hypercube house folds, it looks like an ordinary cube from the outside because it rests in our space on its cubical face—just as a folded paper cube, sitting on a plane, would look to Flatlanders like a square. Eventually the hypercube house falls out of space altogether. 

Chemical image data sets are visualized as a three-dimensional cube spanning one wavelength and two spatial dimensions called a hypercube (Figure 7.2). Each element within the cube contains the intensity-response information measured at that spatial and spectral index. The hypercube can be treated as a series of spatially resolved spectra (called... 

Kono s designs [46] are attempts to reduce the number of design points in continuous D-optimality designs (constructed on a hypercube), by replacing all points in centers of two-dimensional planes with one point in the hypercube center. The number of design points by Kono s designs is defined by this expression ... 

If the optimal solution is in the limit of the hypercube then do not reduce the size of the hypercube, simply move it to do that this last point be the centre of the hypercube. The limits of the hypercube can go further than the bounds of the variables, although the sampling is always performed inside those bounds. Therefore a contraction step is only performed if the last point is in the boundary of the hypercube, but not if it is only in the limit of the domain of a variable. 

It will be usefuljto rewrite Eq. (73 ) into the more familiar form. With the equality C = ( 1/f) for the hypercubic lattices together with the definition [Pg.172]

Jimenez-Montano, M. A., de la Mora-Basflnez, C. R., and Poschel. T., The hypercube structure of the genetic code explains conservative and non-conservative amino acid substitutions in vivo and in vitro. BioSystems 39,117-125 (1996). 

To appreciate this latter point, consider the trajectory of a single molecule in space. Let us discretize this trajectory such that we represent the trajectory of the molecule by a succession of regularly spaced points. Thus, these points may be viewed as the p nodes of a cubic lattice. Clearly, in each spatial dimension, 2 otit of the total number of the p nodes in that direction lie on the surface of the cube. Extending these considerations to N instead of just a single molecule, it is immediately clear that we need to replace the original cube by an Af-dimensionaJ hypercube such that in eacli dimension the fraction p — 2/p represents the ratio of nodes not on the surface of the hypercube relative to the total number of nodes. To estimate this fraction... 

Moreover, to represent the continuous trajectory of each molecule by a succession of regularly spaced nodes, the lattice constant of the hypercube should be sufficiently small that is, p should be large enough so that x = 2/p C 1. In this case, we can approximate the logarithm in Ekj. (5.3) through ln(l — x) = -X -h O (j ) —.T such that... 

In the remainder of this review we focus on the mathematical analysis of the piecewise linear equations largely based on earlier studies [34-48]. In Section IV we show that the logical structure can be mapped onto a hypercube in N dimensions, where each vertex of the hypercube represents the state of each of the N variables, and the dynamics and logical structure in the network are represented as directed edges on the hypercube. 

We then use the hypercube representation to carry out a nonlinear dynamical analysis of these networks. The key insight is that quantitative aspects of flows in phase space can be computed from linear fractional maps that represent the flows between boundaries on the hypercube. Analysis is possible because the composition of two linear fractional maps is a hnear fractional map. This analysis is useful for analyzing steady states, limit cycles, and chaotic dynamics in these networks. 

The hypercube representation is also useful for examining problems in discrete mathematics suggested by this work. In Section VI we count the number of distinct networks under the symmetry of the hypercube, and we also show how to classify chaotic dynamics. 

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... 

The hypercube structure arises as a consequence of the binary nature of the state space assumed in Eq. (4). It is possible to modify the number of thresholds and still have a tractable mathematical problem, though many of the simplifications associated with the hypercube would disappear [23, 60, 61]. In principle, many of the same notions apply, although the area is not yet very developed. Finally, software that is capable of incorporating multiple thresholds into the piecewise linear equations with multiple time constants and thresholds has recently been developed by deJong et al. [62]. 

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