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Parallel Architectures
Published in Pranabananda Chakraborty, Computer Organisation and Architecture, 2020
Hypercube is a binary n-cube architecture in which an n-cube consists of N = 2n nodes that are connected in an n-dimensional cube with two nodes per dimension. Each node is represented by small circle that consists of a processor, local memory and communication circuits. Each processor Pi has bi-directional links to n other neighbouring processors; these links actually form the edges of the hypercube. (Hayes, J. P., et al.). In an n-dimensional hypercube, each node is directly connected to immediately adjacent n neighbours. The cube’s side is of length 1, so each co-ordinate is either a 0 or a 1. Moving from any arbitrary vertex to any of its adjacent vertices changes exactly one co-ordinate, so adjacent vertices differ in exactly one position. This is illustrated in Figure 10.12a. A set of 2n distinct n-bit binary addresses can be assigned to the processors in such a way that the address of Pi differs from that of each of its neighbours in exactly 1 bit. Figure 10.12b illustrates a 3-cube hypercube with 8 nodes. A 4-cube can be formed by interconnecting the corresponding nodes of two 3-cubes as shown in Figure 10.12c. The node degree of an n-cube equals n.
Supercomputing in the 1990s A Distributed Solution
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
S. Ashley Burns, Charlie F. Bender
Perhaps an example best illustrates the patterns of acceptance of groundbreaking computing systems between academia and industry. The Cosmic Cube developed at the California Institute of Technology in the early 1980s has led to other hypercube systems, of which the Intel Scientific Computers’ 80386-based iPSC is an example. A hypercube can be visualized as a “cube” scalable to any dimension, with the points of intersection of sides representing processors; the lines formed by intersections of sides represent interconnections between the processors. Around 1985, hypercubes appeared in university and national computer center settings. Especially early on, the technology has widely been seen as unsuited for real-world computing problems. After several years of availability, hypercube systems enjoy limited success for large-scale computing in industrial settings. The iPSC owes some popularity to its hypercube topology, which has proved easily applicable to specialized applications such as petroleum research and image processing (Fox, 1988). Whether the hypercube will at some later time be applied to a wider range of problems remains to be seen.
FUZZY SETS, FUZZY OPERATORS, AND FUZZY RELATIONS
Published in Kumar S. Ray, Soft Computing and Its Applications, Volume One, 2014
Kosko introduced a very useful graphical representation of fuzzy sets. Figure 35 shows an example in which the universal set consists only of the two elements x1 and x2. Each point in the interior of the unit square represents a subset of X. The convention is that the coordinates of the representation correspond to the membership values of the elements in the fuzzy set. The point (1, 1), for example, represents the universal set X, with membership function Ju.j4(x1) = 1 and I*a(x2) = 1. The point (1,0) represents the set { i} and the point (0, 1) the set { x2}. The crisp subsets of X are located at the vertices of the unit square. The geometric visualization can be extended to an n-dimensional hypercube.
Chaos on the hypercube and other places
Published in International Journal of Mathematical Education in Science and Technology, 2022
For a hypercube in N dimensional space each of the 2N vertices will be attached to N others. We assume the coupling strength between the connected vertices is and the coupling of a vertex to itself is . We next note that the diagonal elements of the adjacency matrix (16) are all zero, with all other elements either zero or one. Thus, the coupling matrix for the N dimensional hypercube can be written as: where is the n × n unit matrix. Finally, we note that for the sawtooth map the Jacobian of this system can be written as, and using (15) give the eigenvalues to be with the ith eigenvalue being of multiplicity and thus the spectrum of N+1 distinct global Lyapunov exponents is
A general approach to deriving diagnosability results of interconnection networks*
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
Eddie Cheng, Yaping Mao, Ke Qiu, Zhizhang Shen
The hypercube, , Harary et al. [2] is perhaps one of the most studied, also the simplest, interconnection networks, with commercial applications [44,45]. It is n-regular, both vertex and edge transitive, with small diameter. Several hypercube variants have also been suggested, including augmented cubes, crossed cubes, enhanced cubes, folded cubes, möbius cubes, twisted cubes, and ( generalised) exchanged cubes. Many algorithms have been designed to run on these hypercube based architectures to solve realistic issues in applications. As a recent example, Bcube, a general hypercube-based structure, was suggested in [46] as a network structure to support reconfigurable modular data centres.
Modification of the Maximin and ϕp (Phi) Criteria to Achieve Statistically Uniform Distribution of Sampling Points
Published in Technometrics, 2020
Miroslav Vořechovský, Jan Eliáš
The extension of this concept into an -dimensional unit hypercube is straightforward. Our basis functions are now -dimensional products of unidimensional polynomials of arbitrary order