Husserl on Geometry and Spatial Representation

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl's ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl.

Geometry and physics of knots

KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number-the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility(5).

Methods for Clearance Influence Analysis in Planar and Spatial Mechanisms

This doctoral dissertation presents a new method to asses the influence of clearancein the kinematic pairs on the configuration of planar and spatial mechanisms. The subject has been widely investigated in both past and present scientific literature, and is approached in different ways: a static/kinetostatic way, which looks for the clearance take-up due to the external loads on the mechanism; a probabilistic way, which expresses clearance-due displacements using probability density functions; a dynamic way, which evaluates dynamic effects like the actual forces in the pairs caused by impacts, or the consequent vibrations. This dissertation presents a new method to approach the problem of clearance. The problem is studied from a purely kinematic perspective. With reference to a given mechanism configuration, the pose (position and orientation) error of the mechanism link of interest is expressed as a vector function of the degrees of freedom introduced in each pair by clearance: the presence of clearance in a kinematic pair, in facts, causes the actual pair to have more degrees of freedom than the theoretical clearance-free one. The clearance-due degrees of freedom are bounded by the pair geometry. A proper modelling of clearance-affected pairs allows expressing such bounding through analytical functions. It is then possible to study the problem as a maximization problem, where a continuous function (the pose error of the link of interest) subject to some constraints (the analytical functions bounding clearance- due degrees of freedom) has to be maximize. Revolute, prismatic, cylindrical, and spherical clearance-affected pairs have been analytically modelled; with reference to mechanisms involving such pairs, the solution to the maximization problem has been obtained in a closed form.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

An electrical heart model incorporating real geometry and motion

Abstract—This paper describes an electrical model of the ventricles incorporating real geometry and motion. Cardiac geometry and motion is obtained from segmentations of multipleslice MRI time sequences. A static heart model developed previously is deformed to match the observed geometry using a novel shape registration algorithm. The resulting electrocardiograms and body surface potential maps are compared to a static simulation in the resting heart. These results demonstrate that introducing motion into the cardiac model modifies the ECG during the T wave at peak contraction of the ventricles.

Thinking in three dimensions: Exploring students' geometric thinking and spatial ability with the Geometer's Sketchpad

Current reform initiatives recommend that geometry instruction include the study of three-dimensional geometric objects and provide students with opportunities to use spatial skills in problem-solving tasks. Geometer's Sketchpad (GSP) is a dynamic and interactive computer program that enables the user to investigate and explore geometric concepts and manipulate geometric structures. Research using GSP as an instructional tool has focused primarily on teaching and learning two-dimensional geometry. This study explored the effect of a GSP based instructional environment on students' geometric thinking and three-dimensional spatial ability as they used GSP to learn three-dimensional geometry. For 10 weeks, 18 tenth-grade students from an urban school district used GSP to construct and analyze dynamic, two-dimensional representations of three-dimensional objects in a classroom environment that encouraged exploration, discussion, conjecture, and verification. The data were collected primarily from participant observations and clinical interviews and analyzed using qualitative methods of analysis. In addition, pretest and posttest measures of three-dimensional spatial ability and van Hiele level of geometric thinking were obtained. Spatial ability measures were analyzed using standard t-test analysis. ^ The data from this study indicate that GSP is a viable tool to teach students about three-dimensional geometric objects. A comparison of students' pretest and posttest van Hiele levels showed an improvement in geometric thinking, especially for students on lower levels of the van Hiele theory. Evidence at the p < .05 level indicated that students' spatial ability improved significantly. Specifically, the GSP dynamic, visual environment supported students' visualization and reasoning processes as students attempted to solve challenging tasks about three-dimensional geometric objects. The GSP instructional activities also provided students with an experiential base and an intuitive understanding about three-dimensional objects from which more formal work in geometry could be pursued. This study demonstrates that by designing appropriate GSP based instructional environments, it is possible to help students improve their spatial skills, develop more coherent and accurate intuitions about three-dimensional geometric objects, and progress through the levels of geometric thinking proposed by van Hiele. ^

Effect of soil structure on temporal and spatial dynamics of bacteria

Soil is a complex heterogeneous system comprising of highly variable and dynamic micro-habitats that have significant impacts on the growth and activity of resident microbiota. A question addressed in this research is how soil structure affects the temporal dynamics and spatial distribution of bacteria. Using repacked microcosms, the effect of bulk-density, aggregate sizes and water content on growth and distribution of introduced Pseudomonas fluorescens and Bacillus subtilis bacteria was determined. Soil bulk-density and aggregate sizes were altered to manipulate the characteristics of the pore volume where bacteria reside and through which distribution of solutes and nutrients is controlled. X-ray CT was used to characterise the pore geometry of repacked soil microcosms. Soil porosity, connectivity and soil-pore interface area declined with increasing bulk-density. In samples that differ in pore geometry, its effect on growth and extent of spread of introduced bacteria was investigated. The growth rate of bacteria reduced with increasing bulk-density, consistent with a significant difference in pore geometry. To measure the ability of bacteria to spread thorough soil, placement experiments were developed. Bacteria were capable of spreading several cm’s through soil. The extent of spread of bacteria was faster and further in soil with larger and better connected pore volumes. To study the spatial distribution in detail, a methodology was developed where a combination of X-ray microtopography, to characterize the soil structure, and fluorescence microscopy, to visualize and quantify bacteria in soil sections was used. The influence of pore characteristics on distribution of bacteria was analysed at macro- and microscales. Soil porosity, connectivity and soil-pore interface influenced bacterial distribution only at the macroscale. The method developed was applied to investigate the effect of soil pore characteristics on the extent of spread of bacteria introduced locally towards a C source in soil. Soil-pore interface influenced spread of bacteria and colonization, therefore higher bacterial densities were found in soil with higher pore volumes. Therefore the results in this showed that pore geometry affects the growth and spread of bacteria in soil. The method developed showed showed how thin sectioning technique can be combined with 3D X-ray CT to visualize bacterial colonization of a 3D pore volume. This novel combination of methods is a significant step towards a full mechanistic understanding of microbial dynamics in structured soils.

Temporal Auditory Processing And Phonological Awareness In Children With Benign Epilepsy With Centrotemporal Spikes.

The aim of this research was to analyze temporal auditory processing and phonological awareness in school-age children with benign childhood epilepsy with centrotemporal spikes (BECTS). Patient group (GI) consisted of 13 children diagnosed with BECTS. Control group (GII) consisted of 17 healthy children. After neurological and peripheral audiological assessment, children underwent a behavioral auditory evaluation and phonological awareness assessment. The procedures applied were: Gaps-in-Noise test (GIN), Duration Pattern test, and Phonological Awareness test (PCF). Results were compared between the groups and a correlation analysis was performed between temporal tasks and phonological awareness performance. GII performed significantly better than the children with BECTS (GI) in both GIN and Duration Pattern test (P < 0.001). GI performed significantly worse in all of the 4 categories of phonological awareness assessed: syllabic (P = 0.001), phonemic (P = 0.006), rhyme (P = 0.015) and alliteration (P = 0.010). Statistical analysis showed a significant positive correlation between the phonological awareness assessment and Duration Pattern test (P < 0.001). From the analysis of the results, it was concluded that children with BECTS may have difficulties in temporal resolution, temporal ordering, and phonological awareness skills. A correlation was observed between auditory temporal processing and phonological awareness in the suited sample.

Rhinometric evaluation of nasal cavity geometry and its relation to the upper arch transverse distance

The objective of this study was to evaluate children's respiratory patterns in the mixed dentition, by means of acoustic rhinometry, and its relation to the upper arch width development. Fifty patients were examined, 25 females and 25 males with mean age of eight years and seven months. All of them were submitted to acoustic rhinometry and upper and lower arch impressions to obtain plaster models. The upper arch analysis was accomplished by measuring the interdental transverse distance of the upper teeth, deciduous canines (measurement 1), deciduous first molars (measurement 2), deciduous second molars (measurement 3) and the first molars (measurement 4). The results showed that an increased left nasal cavity area in females means an increased interdental distance of the deciduous first molars and deciduous second molars and an increased interdental distance of the deciduous canines, deciduous first and second molars in males. It was concluded that there is a correlation between the nasal cavity area and the upper arch transverse distance in the anterior and mid maxillary regions for both genders.

High-level and phonological awareness abilities of children following management for supratentorial tumour: Part II

Background. Limited information is available regarding the impact of childhood tumour on the cerebral hemispheres and supratentorial cranial fossa. However, a recent study found that children managed for a tumour located in this region may demonstrate reduced general language abilities. However, the indirect or direct impact of a tumour in this region on higher-level language abilities in childhood is at present largely unknown. Materials and methods. The present study examined the higher language and phonological awareness abilities of five children treated for supratentorial tumour ranging in age from seven to fourteen years in age. Assessments included measures of receptive and expressive semantic abilities, inferencing, figurative language, and problem solving, as well as a comprehensive pre-literacy test. Results. As a group, reductions were evident in problem solving, and in the ability to receive and decode content of high-level language when compared to a group of age- and gender-matched peers. At an individual level, only two of five children managed for supratentorial tumour demonstrated language deficits. These two cases were noted to be the same children previously identified as also having general language deficits. More widespread findings were noted in phonological awareness, with four of the five children previously managed for supratentorial tumour demonstrating weaknesses in one or more areas. Conclusions. Findings demonstrated that weaknesses in general language ability in children managed for supratentorial tumour may indicate higher-level language difficulties. Language abilities beyond general measures of language should be monitored, as well as long-term consideration of phonological awareness abilities in this population.

The octahedral manipulator: Geometry and mobility

In most Of the practical six-actuator in-parallel manipulators, the octahedral form is either taken as it stands or is approximated. Yet considerable theoretical attention is paid in the literature to more general forms. Here we touch on the general form, and describe some aspects of its behavior that vitiate strongly against its adoption as a pattern for a realistic manipulate,: We reach the conclusion that the structure for in-parallel manipulators must be triangulated as fully as possible, so leading to the octahedral form. In describing some of the geometrical properties of the general octahedron, we show how they apply to manipulators. We examine in detail the special configurations at which the 6 x 6 matrix of leg lines is singular presenting results from the point of view of geometry in preference to analysis. In extending and enlarging on some known properties, a few behavioral surprises materialize. In studying special configurations, we start with the most general situation, and every other case derives from this. Our coverage is more comprehensive than any that we have found. We bring to light material that is, we think, of significant use to a designer.

Integrating attribute and space characteristics in choropleth display and spatial data mining

This paper develops an interactive approach for exploratory spatial data analysis. Measures of attribute similarity and spatial proximity are combined in a clustering model to support the identification of patterns in spatial information. Relationships between the developed clustering approach, spatial data mining and choropleth display are discussed. Analysis of property crime rates in Brisbane, Australia is presented. A surprising finding in this research is that there are substantial inconsistencies in standard choropleth display options found in two widely used commercial geographical information systems, both in terms of definition and performance. The comparative results demonstrate the usefulness and appeal of the developed approach in a geographical information system environment for exploratory spatial data analysis.

Population demography and spatial distribution of the mantis shrimp Squilla biformis (Stomatopoda, Squillidae) from Pacific Costa Rica

The mantis shrimp Squilla biformis is the most conspicuous and abundant stomatopod captured during benthic trawling operations off the Pacific coast of Costa Rica. Due to its abundance, this species is considered a potential fisheries resource for the region. Nevertheless, its life history is practically unknown. The present study describes the population demography, spatial distribution and behaviour of S. biformis from Pacific Costa Rica. The population was principally composed of individuals between 20 and 32 mm carapace length (CL), forming 2 age groups. Individuals of 35 to 45 mm CL and > 45 mm CL were poorly represented. We assume that larger individuals are more frequent at greater depths (probably on the continental slope), thus out of the reach of the fishing vessels used in our study. Males outnumbered females, as observed in other stomatopods. Visual evidence of their behaviour demonstrates that the adults in this species possess a benthic and pelagic life style. Largest numbers of individuals (50% of the total) were found between 240 to 260 m, the same bathymetric range that was historically occupied by commercial shrimps. This shift may be related to intense fishing activities. We observed a synchronized moulting of females and males during less luminous (third and fourth) lunar phases. The evolutionary development of a group moulting system could confer advantages to S. biformis in comparison to other stomatopods whose moulting process is individual and asynchronous.

Temporal and spatial distribution of anurans in the Pampa Region (Santa Maria, RS).

The present study aimed to determine the richness, occurrence constancy, reproductive modes. standard of abundance distribution, season of vocalization and to test correlation among climatic variables and activity of vocalization of anurans in a region of the Pampa Biome, Santa Maria, Rio Grande do Sul State. During the period of Novernber/2001 to October/2002 monthly collections were carried out utilizing the `survey at breeding site` method and examination of specimens kept in the Colecao Herpetologica do Setor de Zoologia da Universidade Federal de Santa Maria (ZUFSM). Tire Occurrence of 25 species of anurans was recorded. The anurofauna recorded represents 30% of the species known to Occur in Rio Grande do Sul, and comprises species generally associated with grasslands in this state and neighboring countries. Four reproductive modes were recorded: mode 1 (14 species: 58.3%) mode 11 and 30 (9 species` 37.5%) and mode 24 (1 species; 4.2%). The low diversification of reproductive modes is likely related to the homogeneity of the grassland habitat. Most species were constant or accessory in the Study area and the species abundance distribution patterns fit in the Broken Stick and Log-normal models. characterized by homogeneity of species abundance distribution. Most species showed great plasticity in habitat. but few were plastic in vocalization sites use. There was a weak positive correlation between species richness and precipitation. There was also a weak positive correlation between the abundance of species calling activity and maximum average temperatures. These correlations indicated that, in the study area. the abundance of calling males is more affected by the temperature, and species richness is more affected by precipitation, despite the fact that significantly higher species richness occurs during the hottest period of the year. These results showed that the climatological variables examined were not enough to explain the seasonal occurrence of species, thus the influence of other environmental variables merit to be tested in future studies.

Implications of prevalent and incident diabetes mellitus on left ventricular geometry and function in the ageing heart: The MONICA/KORA Augsburg cohort study

Background and Aim: It is unclear to what extent diabetes modulates the ageing-related adaptations of cardiac geometry and function. Methods and Results: We examined 1005 adults, aged 25-74 years, from a population-based survey at baseline in 1994/5 and at follow-up in 2004/5. We compared persistently non-diabetic individuals (ND; no diabetes at baseline and at follow-up, n = 833) with incident (ID; non-diabetic at baseline and diabetic at follow-up, n = 36) and with prevalent diabetics (PD; diabetes at baseline and follow-up examination, n = 21). Left ventricular (LV) geometry and function were evaluated by echocardiography. Statistical analyses were performed with multivariate linear regression models. Over ten years the PD group displayed a significantly stronger relative increase of LV mass (+9.34% vs. +23.7%) that was mediated by a more pronounced increase of LV end-diastolic diameter (+0% vs. +6.95%) compared to the ND group. In parallel, LA diameter increased (+4.50% vs. +12.7%), whereas ejection fraction decreased (+3.02% vs. -4.92%) more significantly in the PD group. Moreover, at the follow-up examination the PD and ID groups showed a significantly worse diastolic function, indicated by a higher E/EM ratio compared with the ND group (11.6 and 11.8 vs. 9.79, respectively). Conclusions: Long-standing diabetes was associated with an acceleration of age-related changes of left ventricular geometry accumulating in an eccentric remodelling of the left ventricle. Likewise, echocardiographic measures of systolic and diastolic ventricular function deteriorated more rapidly in individuals with diabetes. (C) 2009 Elsevier B.V. All rights reserved.