Some Comments on Non-Euclidean Mental Maps

R G Golledge; L J Hubert

doi:10.1068/a140107

Some Comments on Non-Euclidean Mental Maps

Environment and Planning A Economy and Space ◽

10.1068/a140107 ◽

1982 ◽

Vol 14 (1) ◽

pp. 107-118 ◽

Cited By ~ 27

Author(s):

R G Golledge ◽

L J Hubert

Keyword(s):

Spatial Representation ◽

Mental Maps ◽

Curved Spaces ◽

Euclidean Metric ◽

General Properties ◽

Cognitive Information ◽

Euclidean Metrics

The Euclidean metric is perhaps the most commonly used and most convenient one for representing mapped phenomena. In this paper we examine the suitability of representing cognitive phenomena via the Euclidean metric. Some general properties of spaces are examined with particular emphasis on the properties of isotropy, incompleteness, and curvature, and a more detailed discussion is undertaken of the suitability of using curved spaces (particularly Reimannian spaces) for the representation of cognitive information. A final discussion is presented on the problems of handling manifolds with folds, warps, and tears; and speculations are made concerning the appropriateness of non-Euclidean metrics for the spatial representation of mental maps.

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On What Manifold do Diffusion Tensors Live?

10.54294/7kibc5 ◽

2008 ◽

Author(s):

Ofer Pasternak ◽

Ragini Verma ◽

Nir Sochen ◽

Peter J. Basser

Keyword(s):

Diffusion Tensor ◽

Affine Invariant ◽

Clinical Tool ◽

Basic Notion ◽

Euclidean Metric ◽

Statistical Framework ◽

Euclidean Metrics ◽

Diffusion Tensors ◽

Tensor Data ◽

Confidence Estimates

Diﬀusion tensor imaging has become an important research and clinical tool, owing to its unique ability to infer microstructural properties of living tissue. Increased use has led to a demand for statistical tools to analyze diﬀusion tensor data and perform, for example, conﬁdence estimates, ROI analysis, and group comparisons. A ﬁrst step towards developing a statistical framework is establishing the basic notion of distance between tensors. We investigate the properties of two previously proposed metrics that deﬁne a Riemannian manifold: the aﬃne-invariant and Euclidean metrics. We ﬁnd that the Euclidean metric is more appropriate for intra-voxel comparisons, and suggest that a context-dependent metric may be required for inter-voxel comparisons.

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New approaches to obtaining scientific innovation in morphological studies of bladder transitional epithelium

Galician Medical Journal ◽

10.21802/gmj.2017.2.11 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

O Popadynets ◽

O Yurakh ◽

N Tokaruk ◽

T Kotyk ◽

I Pukach ◽

...

Keyword(s):

Cluster Analysis ◽

Transitional Epithelium ◽

Cellular Composition ◽

Structural System ◽

Euclidean Metric ◽

Morphological Studies ◽

Scientific Innovation ◽

Euclidean Metrics ◽

Hematoxylin And Eosin ◽

Multiple Clusters

Objective: To demonstrate the capabilities of cluster analysis in receiving scientific innovation results in morphological studies of cells of the bladder urothelium.Materials and methods.10 Wistar rats were used. Histological sections were stained with hematoxylin and eosin; electron microscope studies were conducted; morphometry was performed in ImageJ and statistics – in studio-R using nonparametric methods and multivariate statistics.Results. A brief description of the main stages of cluster analysis shows way to determine the most important features of uroteliocytes and to reveal their heterogeneity, algorithms of Euclidean metrics and methods of clustering were described, the features of the application of the analysis in morphological studies were presented, an example of using these methods in searching for new results was presented, the models of morphological substantiation of clustering results were showed. Conclusion: 1) cluster analysis provides a scientific novelty in studies of transitional epithelium of the bladder; 2) it is used in case of heterogeneity of cellular composition of urothelium that is detected with a help of coefficient of variation; 3) the most significant features of uroteliocytes are their cell area and their nuclei area; 4) new results on the number of clusters were obtained by method of Ward, and new data on their indicators – by k-means; 5) Euclidean metric is the best to use; 6) to assess the adequacy of the results pairwise comparisons between multiple clusters were carried out according to their indicators; 7) results are presented in dimentional projection and they characterize cellular composition of the urothelium as structural system and detect systemic effects.

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Identifying groups of airborne particles by ordination

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100172012 ◽

1994 ◽

Vol 52 ◽

pp. 856-857

Author(s):

M. Shlepr ◽

C. M. Vicroy

Keyword(s):

Elemental Analysis ◽

Energy Dispersive Spectroscopy ◽

Manufacturing Process ◽

Spatial Representation ◽

Airborne Particles ◽

Common Source ◽

Data Set ◽

Microelectronics Industry ◽

Induced Changes ◽

Source Materials

The microelectronics industry is heavily tasked with minimizing contaminates at all steps of the manufacturing process. Particles are generated by physical and/or chemical fragmentation from a mothersource. The tools and macrovolumes of chemicals used for processing, the environment surrounding the process, and the circuits themselves are all potential particle sources. A first step in eliminating these contaminants is to identify their source. Elemental analysis of the particles often proves useful toward this goal, and energy dispersive spectroscopy (EDS) is a commonly used technique. However, the large variety of source materials and process induced changes in the particles often make it difficult to discern if the particles are from a common source.Ordination is commonly used in ecology to understand community relationships. This technique usespair-wise measures of similarity. Separation of the data set is based on discrimination functions. Theend product is a spatial representation of the data with the distance between points equaling the degree of dissimilarity.

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