Geometry as Transfer

It is generally accepted that intelligent action involves considerable use of transfer. For example, Carbonell [1] has argued that learning proceeds by analogical reasoning; Rosch [12] has argued that categorization proceeds by seeing objects in terms of prototypes; and Leyton [9] has argued that the human perceptual system is organized as a hierarchy of transfer. The role of geometry is also seen as fundamental to the representations produced by the cognitive system. For example, Gallistel [2] has elaborated the powerful role of geometry in animal learning and navigation; Lakoff [3] has emphasized the role of geometry in semantics; and Leyton [9] has proposed an extensive role for geometry in causal explanation. We bring together the two above factors, transfer and geometry, in the book, Leyton [10], by developing a generative theory of shape in which transfer is a fundamental organizing principle. In this approach, transfer is basic to the very meaning of geometry. The purpose of the present paper is to give an introduction to this transfer-based theory of geometry

Generalized iterated wreath products of cyclic groups and rooted trees correspondence

Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving literature's fastest FFT upper bound estimate.Comment: 15 pages, to appear in Advances in the Mathematical Science

Topography of Generically Folded Spacecapes: Towards a Cognitive Metatheory in Architectural Design

If we visualize philosophy as a science of the sciences (in the tradition of Hans Heinz Holz), then theories within the philosophical framework are always meta-theories of the respective field of interest in question with which philosophy is actually dealing. Hence, a philosophy of architecture in particular, shows up as a meta-theory of theories of architecture with respect to all the possible conditions that are necessary to actually constitute its characteristic field of interest (including the social modes of communication). As we can easily see, it is also a cognitive meta-theory in the first place (i.e. one of the human condition). If then, traditionally, philosophy can also be visualized as being primarily constituted by four components that turn to its topic, namely by means of historical analysis, scientific approach, aesthetical and ethical conceptualizations, respectively, we can recognize that the second and the third overlap somehow with the architectural activities. This is certainly true since the establishment of architecture as a field of activity of its own, separated from building and engineering, as was achieved in the epoch of the Italian Renaissance by Brunelleschi, Alberti, and others.Peer Reviewe

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060