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Bewildered, the form-maker stands alone
Published in Theodora Vardouli, Olga Touloumi, Computer Architectures, 2019
The problem posed by the BRI, or at least the way that Alexander interpreted it, had to do with developing a proper organization (structure) on which the abundant knowledge and data about building that were becoming available would be able to hang from. This was a concern that had preoccupied a good portion of his doctoral work. To the arbitrary matching between the organization of building information and the needs of a specific design situation, Alexander counter-proposed “to set up temporary isomorphisms between the library’s organization and the cognitive organization of the process” (Alexander 1962: 120). “Isomorphism” etymologically translated as equality of form, was a mathematical term indicating a one-to-one mapping between the elements of two different systems. Achieving Alexander’s goal necessitated “some logical or mathematical relation between the two classification systems,” the source of which would be “the topological structure of the problem” (Alexander 1962: 120).
Topological Analysis of Local Structure in Atomic Systems
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Emanuel A. Lazar, David J. Srolovitz
The language and tools of graph theory can be used to record complete topological information of a Voronoi cell by looking at it as a planar graph. Briefly, a graph is a set of points called vertices, and a set of connections between those vertices are called edges. A planar graph is one whose vertices and edges can be drawn in the plane without any edges crossing. Two graphs are isomorphic if there is a correspondence between their vertices so that two vertices are connected by an edge in one graph if and only if corresponding vertices are connected by an edge in the other graph [1059]. Mathematical theorems from the early twentieth century [965, 1142] guarantee that every Voronoi cell can be uniquely represented as a planar graph, thus allowing us to make precise statements about Voronoi cells using the language of graph theory. Figure 15.8 illustrates planar graphs corresponding to the three Voronoi cells of Figure 15.7.
General Ideas of Natural Science, Signal Theory, and Information Theory
Published in Andrey Popoff, Fundamentals of Signal Processing in Metric Spaces with Lattice Properties, 2017
Symmetry is almost impossible to define exactly. In every single phenomenon, symmetry inevitably takes the form corresponding to it. Examples of symmetry include the metrics of poetry and music, coloring in painting, word arrangements in literature, constellations in astronomy. Symmetry reveals itself within limitations of physical processes passing. These constraints are described by laws of preservation of energy, mass, momentum, electrical charge, etc. The creation of relativistic quantum theory evoked the discovery of a new type of symmetry. This is symmetry of nature’s laws with respect to simultaneous transformations of charge conjugation, parity transformation, and time reversal designated CPT-symmetry. Symmetry underlies elementary particle classification in chemistry and allowed Dmitri Mendeleev to devise his periodic table of elements. Gregor Mendel applied the idea of symmetry to characterize hereditary factors. In abstract algebra and topology, symmetry appears in isomorphic and homomorphic mappings respectively.
Organizational patterns of RRI: how organizational properties relate to RRI implementation
Published in Journal of Responsible Innovation, 2021
Thomas Kjeldager Ryan, Niels Mejlgaard, Lise Degn
Organizational change has within this theoretical approach been conceptualized as processes of adaption to unstable environments through isomorphic processes. Isomorphism describes the process wherein organizations within the same field adopts similar structures or processes, as a result of coercive, mimetic or normative pressures (DiMaggio and Powell 1983). Coercive isomorphism takes place when organizations change in response to e.g. governmental requirements or mandates, but also emerges in more subtle ways through: formal and informal pressures exerted on organizations by other organizations upon which they are dependent and by cultural expectations in the society within which organizations function. Such pressures may be felt as force, as persuasion, or as invitations to join in collusion. (DiMaggio and Powell 1983)
Algorithmic graph theory for post-processing molecular dynamics trajectories
Published in Molecular Physics, 2023
Sana Bougueroua, Ylène Aboulfath, Dominique Barth, Marie-Pierre Gaigeot
Graph isomorphism as defined in ref. [45] allows to represent each conformer with a fingerprint graph and allows comparisons between graphs. In our applications, isomorphism consists in comparing the distribution of edges between two 2D-MolGraphs: if the two graphs being compared have the same set of bonds/interactions connected to the same set of atoms (in terms of chemical types/colours for the graphs), these graphs are then isomorphic, i.e. the two graphs are identical, see refs. [45,47–49].
A general system for heuristic minimization of convex functions over non-convex sets
Published in Optimization Methods and Software, 2018
S. Diamond, R. Takapoui, S. Boyd
Two (undirected) graphs are isomorphic if we can permute the vertices of one so it is the same as the other (i.e. the same pairs of vertices are connected by edges). If we describe them by their adjacency matrices A and B, isomorphism is equivalent to the existence of a permutation matrix such that , or equivalently ZA=BZ.