Anchored Space
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The World in Your Head
The World In Your Head: A Gestalt View of the Mechanism of Conscious Experience represents a bold assault on one of the greatest unsolved mysteries in science: the nature of consciousness and the human mind. Rather than examining the brain and nervous system to see what they tell us about the mind, this book begins with an examination of conscious experience to see what it can tell us about the brain. Through this analysis, the first and most obvious observation is that consciousness appears as a volumetric spatial void, containing colored objects and surfaces. This reveals that the representation in the brain takes the form of an explicit volumetric spatial model of external reality. Therefore, the world we see around us is not the real world itself, but merely a miniature virtual-reality replica of that world in an internal representation. In fact, the phenomena of dreams and hallucinations clearly demonstrate the capacity of the brain to construct complete virtual worlds even in the absence of sensory input. Perception is somewhat like a guided hallucination, based on sensory stimulation. This insight allows us to examine the world of visual experience not as scientists exploring the external world, but as perceptual scientists examining a rich and complex internal representation. This unique approach to investigating mental function has implications in a wide variety of related fields, including the nature of language and abstract thought, and motor control and behavior. It also has implications to the world of music, art, and dance, showing how the patterns of regularity and periodicity in space and time--apparent in those aesthetic domains--reflect the periodic basis set of the underlying harmonic resonance representation in the brain.
Spatial Information Theory
Author: Anthony G. Cohn
language: en
Publisher: Springer Science & Business Media
Release Date: 2005-09
This book constitutes the refereed proceedings of the International Conference on Spatial Information Theory, COSIT 2005, held in Elliottville, NY, USA in September 2005. The 30 revised full papers presented were carefully reviewed and selected from 82 submissions. The papers are organized in topical sections on vagueness, uncertainty, and gradation; paths and routes; ontologies and semantics; ontologies and spatial relations; spatial reasoning: cognitive maps and spatial reasoning; time, change, and dynamics; landmarks and navigation; geographic information, and spatial behaviour.
Tractability of Multivariate Problems: Standard information for functionals
Author: Erich Novak
language: en
Publisher: European Mathematical Society
Release Date: 2008
This is the second volume of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. The second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration. The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds. Tractability results, even for linear functionals, are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability. The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed. This volume is of interest to researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. There are 61 open problems listed to stimulate future research in tractability.